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Many people don't understand Taranovsky's ordinal notation (TON). Some of them understand normal ordinal collapsing functions (OCFs), and want to know comparisons between other notations and OCFs. However, OCFs beyond \(\Pi_3\)-reflection suddenly become complicated, such as the ones for \(\Pi_4\)-reflection, 1-stable, \(\beta\)-stable for any constant \(\beta\), and further, yet TON remains simple. Here I show some details of TON with my understanding.

Definition of TON

To understand the definition of this kind of notation is not easy, although it might look simpler than normal OCF. We need 5 steps to understand it.

Terms

Terms are the things the system work on. In the n-th system of TON, terms are constructed using these following 2 rules.

  1. \(0\) and \(\Omega_n\) are terms
  2. If \(a\) and \(b\) are terms, then \(C(a,b)\) is a term

Comparisons

Terms can be compared and connected with ">", "<" or "=".

To compare terms \(a\) and \(b\), first write them in postfix form, i.e. delete all the "(", ")" and "," and then reverse the string. In postfix form, "terms" are constructed only with 3 symbols - "0", "\(\Omega_n\)" and "C".

Then the ordering of "terms" are done in lexicographical order. i.e.

  1. \(a=b\) if \(a\) is the same string as \(b\)
  2. Empty string is less than any other string
  3. If both \(a\) and \(b\) are not empty, then let \(a_1\) (\(b_1\) respectively) be the first symbol of \(a\) (\(b\) respectively), and \(a^-\) (\(b^-\) respectively) be the string without the first symbol.
  4. If \(a_1< b_1\) (\(a_1>b_1\) respectively), then \(a< b\) (\(a>b\) respectively)
  5. If \(a_1=b_1\), then the order of \(a\) and \(b\) is the same as the order of \(a^-\) and \(b^-\)

A binary relation

Now it's the most important part of TON - the binary relation called "is m-built from below from", denoted \(B_m\) here, where m is a non-negative integer. For terms \(a\) and \(b\),

  • \(B_0(a,b)\) if \(a< b\).
  • \(B_{m+1}(a,b)\) if for every proper subterm \(c\) of \(a\), "\(c<a\)" or "\(c\) is a subterm of \(d\), where \(d\) is a subterm of \(a\) and \(B_m(d,b)\)".

To avoid "is a subterm of", this binary relation can also be described using a ternary relation \(T_m\). For terms \(a,\ b,\ c\), \(T_m(a,b,c)\) if at least one of the 5 is true.

  1. \(a< b\)
  2. \(a=\Omega_n\) and \(m\ge1\)
  3. \(a=\Omega_n<c\)
  4. \(a=C(d,e)\), where \(m\ge1\), \(T_{m-1}(d,b,a)\) and \(T_{m-1}(e,b,a)\)
  5. \(a=C(d,e)<c\), where \(T_m(d,b,c)\) and \(T_m(e,b,c)\)

Then \(B_m(a,b)\) iff \(T_m(a,b,0)\).

Standard terms

Some of the terms are standard, but some are not. In the n-th system of TON,

  • \(0\) and \(\Omega_n\) are standard.
  • \(C(a,b)\) is standard if all the following 3 are true.
    1. Both \(a\) and \(b\) are standard
    2. If \(b=C(c,d)\), then \(a\le c\)
    3. \(B_n(a,C(a,b))\)

Ordinals

In TON, one standard term means one ordinal, and different standard terms mean different ordinals. The ordering of ordinals is defined to be exactly the ordering of standard terms.

So the least standard term, 0, corresponds to the least ordinal, 0. The standard term \(a\) larger than \(b_1,\ b_2,\cdots\) corresponds to an ordinal larger than what \(b_1,\ b_2,\cdots\) correspond to.

But there is a problem: is the ordering of standard terms well-founded? It's unknown yet, and Taranovsky is working on it.

Comparisons of TON

TON has some basic properties:

  • \(C(a,b)>b\)
  • \(C(a,b)=b+\omega^a\) iff \(C(a,b)\ge a\)
  • \(C(a,b)\) is monotonic in both \(a\) and \(b\), and continuous in \(a\).

And, using \(\Omega_n=C(\Omega_{n+1},0)\) we can combine all the n-th system into one notation.

Comparing terms, checking \(T_n\), \(B_n\) and standard are computable, so all ordinals from TON are computable. But why there are some corresponence above \(\omega^\text{CK}_1\)? Because Taranovsky set gaps below some ordinals, e.g. \(\Omega_1=\omega^{CK}_1\) instead of BHO, which is the case in 1st system if there is no "gap".

To determine "how large the gaps is" is hard. For set theoretical propose, the gaps should fit large ordinal axioms in KP set theory; for googological propose, we want a diagonalizer large enough for collapsing in any possible further extensions.

Up to \(\Pi_3\)-reflection

The least example is \(\Omega_1\). It's larger than not only \(C(\Omega_1^{\Omega_1^{\Omega_1^\cdots}},0)\), but also \(C(C(\Omega_2^{\Omega_2^{\Omega_2^\cdots}},0),0)\) in 2nd system, \(C(C(C(\Omega_3^{\Omega_3^{\Omega_3^\cdots}},0),0),0)\) in 3rd system, and more systems and possible extensions beyond the limit of \(\underbrace{C(C(\cdots C(}_n\Omega_n2,0)\cdots,0),0)\). The notation may extend up to (but not including) \(\omega^\text{CK}_1\), so we can set \(\Omega_1=\omega^\text{CK}_1\).

From \(\Omega_1=\omega^\text{CK}_1\) we also have \(C(0,\Omega_1)=\omega^\text{CK}_1+1\), \(C(\Omega_1,\Omega_1)=\omega^\text{CK}_12\), \(C(C(\Omega_2,C(\Omega_2,0)),\Omega_1)=\varepsilon_{\omega^\text{CK}_1+1}\), \(C(C(\Omega_2+1,0),\Omega_1)=\theta(\Omega_\omega,\omega^\text{CK}_1+1)\), and so on. These computable application over \(\omega^\text{CK}_1\) would be very strong, but still computable. To make \(C(\Omega_2,C(\Omega_2,0))\) large enough for collapsing, we set \(C(\Omega_2,C(\Omega_2,0))=\omega^\text{CK}_2\) - the 2nd admissible ordinal.

And so on. Making \(C(\Omega_2,\beta)\) from \(\beta\) results a next admissible. Thus \(C(\Omega_2+1,0)=\omega^\text{CK}_\omega\), and \(C(\Omega_2+C(\Omega_22,0),0)\) is the first fixed point of \(\alpha\mapsto\omega^\text{CK}_\alpha\). Higher level fixed points can be \(C(\Omega_2+C(\Omega_22,0)+1,0)\), \(C(\Omega_2+C(\Omega_22,0)2,0)\), \(C(\Omega_2+C(\Omega_22,0)^{C(\Omega_22,0)},0)\), \(C(\Omega_2+C(C(\Omega_2,C(\Omega_22,0)),C(\Omega_22,0)),0)=C(\Omega_2+\varepsilon_{C(\Omega_22,0)+1},0)\), \(C(\Omega_2+C(C(\Omega_22+1,0),C(\Omega_22,0)),0)\), \(C(\Omega_2+C(C(\Omega_2^{\Omega_2^\cdots},0),C(\Omega_22,0)),0)\), \(C(\Omega_2+C(C(C(\Omega_3^{\Omega_3^\cdots},0),0),C(\Omega_22,0)),0)\), etc.. All these terms have a supremum \(C(\Omega_2+C(\Omega_2,C(\Omega_22,0)),0)\). It should corresponds to an ordinal large enough to express all these levels of admissibles, limits or fixed points. Set theories provide a candidate for this - the first recursively inaccessible ordinal, which is admissible and limit of admissibles. Then the gap below \(C(\Omega_2+C(\Omega_2,C(\Omega_22,0)),0)\) makes \(C(\Omega_2+C(\Omega_2,C(\Omega_22,0)),0)\) the first recursively inaccessible.

Let \(d=C(\Omega_2,C(\Omega_22,0))\), then

  • \(C(\Omega_2+d,0)\) is first recursively inaccessible
  • \(C(\Omega_2,C(\Omega_2+d,0))\) is first admissible after first recursively inaccessible
  • \(C(\Omega_2,\alpha)\) is next admissible ordinal after \(\alpha\)
  • \(C(\Omega_2+d,C(\Omega_2+d,0))\) is 2nd recursively inaccessible
  • \(C(\Omega_2+d+1,0)\) is first limit of recursively inaccessibles
  • \(C(\Omega_2+d,C(\Omega_2+d+1,0))\) is \(\omega\)-th recursively inaccessible
  • \(C(\Omega_2+d+C(\Omega_22,0),0)\) is first recursively inaccessible-fixed-point
  • \(C(\Omega_2+d2,0)\) is first recursively 2-inaccessible (a.k.a. level-2 recursively inaccessible)
  • \(C(\Omega_2+d\omega,0)\) is limit of recursively n-inaccessible
  • \(C(\Omega_2+d,C(\Omega_2+d\omega,0))\) is recursively inaccessible limit of recursively n-inaccessible
  • \(C(\Omega_2+d\omega+d,0)\) is first recursively \(\omega\)-inaccessible
  • \(C(\Omega_2+d C(\Omega_22,0),0)\) is the fixed point of \(\alpha\mapsto\)recursively \(\alpha\)-inaccessible
  • \(C(\Omega_2+d C(\Omega_22,0)+d,0)\) is first recursively (1,0)-inaccessible (a.k.a. level-(1,0) recursively inaccessible)
  • \(C(\Omega_2+d^2,0)\) is first recursively Mahlo
  • \(C(\Omega_2+d^2,C(\Omega_2+d^2,0))\) is 2nd recursively Mahlo
  • \(C(\Omega_2+d^2+1,0)\) is first limit of recursively Mahlos
  • \(C(\Omega_2+d^2,C(\Omega_2+d^2+1,0))\) is \(\omega\)-th recursively Mahlo
  • \(C(\Omega_2+d^2+d,0)\) is first recursively inaccessible limit of recursively Mahlos
  • \(C(\Omega_2+d^22,0)\) is first recursively Mahlo limit of recursively Mahlos
  • \(C(\Omega_2+d^3,0)\) is first recursively 2-Mahlo (a.k.a. level-2 recursively Mahlo)
  • \(C(\Omega_2+d^\omega,0)\) is limit of recursively n-Mahlo
  • \(C(\Omega_2+d^\omega+d^2,0)\) is recursively Mahlo limit of recursively n-Mahlo
  • \(C(\Omega_2+d^{\omega+1},0)\) is first recursively \(\omega\)-Mahlo (limit points of recursively n-Mahlo are stationary in this ordinal)
  • \(C(\Omega_2+d^{C(\Omega_22,0)+1},0)\) is first recursively (1,0)-Mahlo
  • \(C(\Omega_2+d^d,0)\) is first \(\Pi_3\)-reflecting
  • \(C(\Omega_2+d^d,C(\Omega_2+d^d,0))\) is 2nd \(\Pi_3\)-reflecting
  • \(C(\Omega_2+d^d+d,0)\) is first recursively inaccessible limit of \(\Pi_3\)-reflectings
  • \(C(\Omega_2+d^d+d^2,0)\) is first recursively Mahlo limit of \(\Pi_3\)-reflectings
  • \(C(\Omega_2+d^d2,0)\) is first \(\Pi_3\)-reflecting limit of \(\Pi_3\)-reflectings

Up to \(a^{++}\)-stable

Let \(d=C(\Omega_2,C(\Omega_22,0))\), then

  • \(C(\Omega_2+d^{d+1},0)\) is first \(\Pi_2\)-reflecting onto \(\Pi_3\)-reflectings (i.e. \(\Pi_3\)-reflectings are stationary in this ordinal)
  • \(C(\Omega_2+d^{d+2},0)\) is first \(\Pi_2\)-reflecting onto \(\Pi_2\)-reflectings onto \(\Pi_3\)-reflectings
  • \(C(\Omega_2+d^{d+C(\Omega_22,0)},0)\) is fixed point of \(\alpha\mapsto\)level-\(\alpha\) \(\Pi_2\)-reflecting onto \(\Pi_3\)-reflectings
  • \(C(\Omega_2+d^{d+C(\Omega_22,0)}2,0)\) is fixed point of "\(\alpha\mapsto\)level-\(\alpha\) \(\Pi_2\)-reflecting onto \(\Pi_3\)-reflectings, which is limit of fixed points of \(\alpha\mapsto\)level-\(\alpha\) \(\Pi_2\)-reflectings onto \(\Pi_3\)-reflectings"
  • \(C(\Omega_2+d^{d+C(\Omega_22,0)+1},0)\) is first level-(1,0) \(\Pi_2\)-reflecting onto \(\Pi_3\)-reflectings
  • \(C(\Omega_2+d^{d2},0)\) is first \(\Pi_3\)-reflecting that is \(\Pi_2\)-reflecting onto \(\Pi_3\)-reflectings
  • \(C(\Omega_2+d^{d2+1},0)\) is first \(\Pi_2\)-reflecting onto "\(\Pi_3\)-reflectings that are \(\Pi_2\)-reflecting onto \(\Pi_3\)-reflectings"
  • \(C(\Omega_2+d^{d3},0)\) is first \(\Pi_3\)-reflecting that is \(\Pi_2\)-reflecting onto "\(\Pi_3\)-reflectings that are \(\Pi_2\)-reflecting onto \(\Pi_3\)-reflectings"
  • \(C(\Omega_2+d^{d^2},0)\) is first \(\Pi_3\)-reflecting onto \(\Pi_3\)-reflectings
  • \(C(\Omega_2+d^{d^{C(\Omega_22,0)}},0)\) is fixed point of \(\alpha\mapsto\)level-\(\alpha\) \(\Pi_3\)-reflecting
  • \(C(\Omega_2+d^{d^{C(\Omega_22,0)}}+d^d,0)\) is \(\Pi_3\)-reflecting fixed point of \(\alpha\mapsto\)level-\(\alpha\) \(\Pi_3\)-reflecting
  • \(C(\Omega_2+d^{d^{C(\Omega_22,0)}+1},0)\) is \(\Pi_2\)-reflecting onto fixed points of \(\alpha\mapsto\)level-\(\alpha\) \(\Pi_3\)-reflecting
  • \(C(\Omega_2+d^{d^{C(\Omega_22,0)}+d},0)\) is \(\Pi_3\)-reflecting that is \(\Pi_2\)-reflecting onto fixed points of \(\alpha\mapsto\)level-\(\alpha\) \(\Pi_3\)-reflecting
  • \(C(\Omega_2+d^{d^{C(\Omega_22,0)}+d^2},0)\) is level-2 \(\Pi_3\)-reflecting that is \(\Pi_2\)-reflecting onto fixed points of \(\alpha\mapsto\)level-\(\alpha\) \(\Pi_3\)-reflecting
  • \(C(\Omega_2+d^{d^{C(\Omega_22,0)}2},0)\) is fixed point of \(\alpha\mapsto\)level-\(\alpha\) \(\Pi_3\)-reflecting onto fixed points of \(\alpha\mapsto\)level-\(\alpha\) \(\Pi_3\)-reflecting
  • \(C(\Omega_2+d^{d^{C(\Omega_22,0)+1}},0)\) is first level-(1,0) \(\Pi_3\)-reflecting
  • \(C(\Omega_2+d^{d^d},0)\) is first \(\Pi_4\)-reflecting
  • \(C(\Omega_2+d^{d^d}2,0)\) is first \(\Pi_4\)-reflecting limit of \(\Pi_4\)-reflectings
  • \(C(\Omega_2+d^{d^d+1},0)\) is first \(\Pi_2\)-reflecting onto \(\Pi_4\)-reflectings
  • \(C(\Omega_2+d^{d^d+d},0)\) is first \(\Pi_3\)-reflecting that is \(\Pi_2\)-reflecting onto \(\Pi_4\)-reflectings
  • \(C(\Omega_2+d^{d^d2},0)\) is first \(\Pi_4\)-reflecting that is \(\Pi_2\)-reflecting onto \(\Pi_4\)-reflectings
  • \(C(\Omega_2+d^{d^{d+1}},0)\) is first \(\Pi_3\)-reflecting onto \(\Pi_4\)-reflectings
  • \(C(\Omega_2+d^{d^{d2}},0)\) is first \(\Pi_4\)-reflecting that is \(\Pi_3\)-reflecting onto \(\Pi_4\)-reflectings
  • \(C(\Omega_2+d^{d^{d^2}},0)\) is first \(\Pi_4\)-reflecting onto \(\Pi_4\)-reflectings
  • \(C(\Omega_2+d^{d^{d^d}},0)\) is first \(\Pi_5\)-reflecting
  • \(C(\Omega_2+\varepsilon_{d+1},0)\) is limit of \(\Pi_n\)-reflecting
  • \(C(\Omega_2+\varepsilon_{d+1}+d^d,0)\) is \(\Pi_3\)-reflecting limit of \(\Pi_n\)-reflecting
  • \(C(\Omega_2+\varepsilon_{d+1}2,0)\) is limit of "\(\Pi_m\)-reflecting limit of \(\Pi_n\)-reflecting"
  • \(C(\Omega_2+\varepsilon_{d+1}d,0)\) is \(\Pi_2\)-reflecting onto limits of \(\Pi_n\)-reflecting
  • \(C(\Omega_2+\varepsilon_{d+1}d^d,0)\) is \(\Pi_3\)-reflecting that is \(\Pi_2\)-reflecting onto limits of \(\Pi_n\)-reflecting
  • \(C(\Omega_2+\varepsilon_{d+1}^2,0)\) is limit of "\(\Pi_m\)-reflecting that is \(\Pi_2\)-reflecting onto limits of \(\Pi_n\)-reflecting"
  • \(C(\Omega_2+\varepsilon_{d+1}^d,0)\) is \(\Pi_3\)-reflecting onto limits of \(\Pi_n\)-reflecting
  • \(C(\Omega_2+\varepsilon_{d+1}^{\varepsilon_{d+1}},0)\) is limit of "\(\Pi_m\)-reflecting that is \(\Pi_3\)-reflecting onto limits of \(\Pi_n\)-reflecting"
  • \(C(\Omega_2+\varepsilon_{d+1}^{\varepsilon_{d+1}^d},0)\) is \(\Pi_4\)-reflecting onto limits of \(\Pi_n\)-reflecting
  • \(C(\Omega_2+\varepsilon_{d+2},0)\) is limit of "\(\Pi_m\)-reflecting onto limits of \(\Pi_n\)-reflecting" (i.e. level-2 limit of \(\Pi_n\)-reflecting)
  • \(C(\Omega_2+\varepsilon_{d+C(\Omega_22,0)},0)\) is level-(1,0) limit of \(\Pi_n\)-reflecting
  • \(C(\Omega_2+\varepsilon_{d2},0)\) is first 1-stable

\(\alpha\) is \(\beta\)-stable if \(L_\alpha\prec_{\Sigma_1}L_{\alpha+\beta}\). This property can even exceed "\(\alpha\) is \(\alpha\)-stable", such as "\(\alpha\) is \(\varepsilon_{\alpha+1}\)-stable", "\(\alpha\) is \(\alpha^+\)-stable" (\(\alpha^+\) is the next admissible ordinal after \(\alpha\)), "\(\alpha\) is \(\beta\)-stable where \(\beta\) is next \(\Pi_3\)-reflecting after \(\alpha\)", "doubly 1-stable" (i.e. \(L_\alpha\prec_{\Sigma_1}L_\beta\prec_{\Sigma_1}L_{\beta+1}\)), "triply 1-stable" (i.e. \(L_\alpha\prec_{\Sigma_1}L_\beta\prec_{\Sigma_1}L_\gamma\prec_{\Sigma_1}L_{\gamma+1}\)), and "n-ply-stable" whose existence (for all n) with KP yields \(\Pi^1_2\text{-CA}_0\).

A combine concept of reflecting and stable is "\(\beta\)-\(\Pi_n\)-reflecting" ordinals. \(\alpha\) is \(\beta\)-\(\Pi_n\)-reflecting onto set \(A\) if for every \(\Pi_n\) formula \(\phi(u,\vec v)\), \(\forall\vec p\in L_\alpha(L_{\alpha+\beta}\models\phi(\alpha,\vec p)\rightarrow\exists\gamma\in\alpha\cap A\exists\delta\in\alpha\cap A(\vec p\in L_{\gamma}\land L_{\delta}\models\phi(\gamma,\vec p)\land\gamma\le\delta))\). In the following, "\(\beta\)-stable onto \(A\)" will be interpreted as "\(\beta\)-\(\Pi_0\)-reflecting onto \(A\)".

  • \(C(\Omega_2+\varepsilon_{d2+1},0)\) is limit of \(\Pi_n\)-reflecting onto 1-stable
  • \(C(\Omega_2+\varepsilon_{d2+2},0)\) is level-2 limit of \(\Pi_n\)-reflecting onto 1-stable
  • \(C(\Omega_2+\varepsilon_{d3},0)\) is first level-2 1-stable
  • \(C(\Omega_2+\varepsilon_{d^2},0)\) is first 1-\(\Pi_1\)-reflecting
  • \(C(\Omega_2+\varepsilon_{d^d},0)\) is first 1-\(\Pi_2\)-reflecting
  • \(C(\Omega_2+\varepsilon_{\varepsilon_{d2}},0)\) is first 2-stable
  • \(C(\Omega_2+\zeta_{d+1},0)\) is limit of n-stable
  • \(C(\Omega_2+\zeta_{d+2},0)\) is level-2 limit of n-stable
  • \(C(\Omega_2+\zeta_{d2},0)\) is first \(\omega\)-stable
  • \(C(\Omega_2+\zeta_{\varepsilon_{d2}},0)\) is first \(\omega+1\)-stable
  • \(C(\Omega_2+\zeta_{\zeta_{d2}},0)\) is first \(\omega2\)-stable
  • \(C(\Omega_2+\varphi(3,d2),0)\) is first \(\omega^2\)-stable
  • \(C(\Omega_2+\varphi(3,\varepsilon_{d2}),0)\) is first \(\omega^2+1\)-stable
  • \(C(\Omega_2+\varphi(4,d2),0)\) is first \(\omega^3\)-stable
  • \(C(\Omega_2+\varphi(\omega,d2),0)\) is first \(\omega^\omega\)-stable
  • \(C(\Omega_2+\varphi(C(\Omega_22,0),d+1),0)\) is fixed point of \(\alpha\mapsto\alpha\)-stable
  • \(C(\Omega_2+\varphi(\omega,\varphi(C(\Omega_22,0),d+1)+d),0)\) is \(\omega^\omega\)-stable onto fixed points of \(\alpha\mapsto\alpha\)-stable
  • \(C(\Omega_2+\varphi(C(\Omega_22,0),d+2),0)\) is level-2 fixed point of \(\alpha\mapsto\alpha\)-stable

Now we use the notation of lambda abstraction for stable levels. Ordinal \(\alpha\) is \(f(\alpha)\)-stable can be written as an \(f\)-stable ordinal, where \(f\) is an abstraction or ordinal function.

  • \(C(\Omega_2+\varphi(C(\Omega_22,0),d2),0)\) is first \(\lambda a.a\)-stable
  • \(C(\Omega_2+\varphi(C(\Omega_22,0),d2+1),0)\) is fixed point of "\(\alpha\mapsto\alpha\)-stable onto \(\lambda a.a\)-stables"
  • \(C(\Omega_2+\varphi(C(\Omega_22,0),d3),0)\) is first level-2 \(\lambda a.a\)-stable
  • \(C(\Omega_2+\varphi(C(\Omega_22,0),\varepsilon_{d2}),0)\) is first \(\lambda a.a+1\)-stable
  • \(C(\Omega_2+\varphi(C(\Omega_22,0),\varphi(C(\Omega_22,0),d+1)),0)\) is fixed point of \(\alpha\mapsto\lambda a.a+\alpha\)-stable
  • \(C(\Omega_2+\varphi(C(\Omega_22,0),\varphi(C(\Omega_22,0),d2)),0)\) is first \(\lambda a.a2\)-stable
  • \(C(\Omega_2+\varphi(C(\Omega_22,0)+1,d2),0)\) is first \(\lambda a.a\omega\)-stable
  • \(C(\Omega_2+\varphi(C(\Omega_22,0)2,d2),0)\) is first \(\lambda a.a^2\)-stable
  • \(C(\Omega_2+\varphi(C(\Omega_22,0)^2,d2),0)\) is first \(\lambda a.a^a\)-stable
  • \(C(\Omega_2+\varphi(\varepsilon_{C(\Omega_22,0)+1},d2),0)\) is first \(\lambda a.\varepsilon_{a+1}\)-stable

Beyond that, we can have \(C(\Omega_2+\varphi(C(C(\Omega_2^{\Omega_2^{\Omega_2^\cdots}},0),C(\Omega_22,0)),d2),0)\), \(C(\Omega_2+\varphi(C(C(C(\Omega_3^{\Omega_3^{\Omega_3^\cdots}},0),0),C(\Omega_22,0)),d2),0)\), and more to come. The "term supremum" of them should be -

  • \(C(\Omega_2+\varphi(d,1),0)\) is first \(\lambda a.a^+\)-stable, i.e. first \(\lambda a.C(\Omega_2,a)\)-stable
  • \(C(\Omega_2+\varphi(C(\Omega_22,0),\varphi(d,1)+1),0)\) is fixed point of \(\alpha\mapsto\alpha\)-stable onto \(\lambda a.a^+\)-stables
  • \(C(\Omega_2+\varphi(C(\Omega_22,0),\varphi(d,1)+d),0)\) is first \(\lambda a.a\)-stable onto \(\lambda a.a^+\)-stables
  • \(C(\Omega_2+\varphi(d,2),0)\) is first level-2 \(\lambda a.a^+\)-stable
  • \(C(\Omega_2+\varphi(d,\varepsilon_{d2}),0)\) is first \(\lambda a.a^++1\)-stable
  • \(C(\Omega_2+\varphi(d,\varphi(C(\Omega_22,0),d2)),0)\) is first \(\lambda a.a^++a\)-stable
  • \(C(\Omega_2+\varphi(d,\varphi(d,1)),0)\) is first \(\lambda a.a^+2\)-stable
  • \(C(\Omega_2+\varphi(d+1,0),0)\) is limit of \(\lambda a.a^+n\)-stable
  • \(C(\Omega_2+\varphi(d+1,d),0)\) is first \(\lambda a.a^+\omega\)-stable
  • \(C(\Omega_2+\varphi(d+C(\Omega_22,0),0),0)\) is fixed point of \(\alpha\mapsto\lambda a.a^+\alpha\)-stable
  • \(C(\Omega_2+\varphi(d+C(\Omega_22,0),d),0)\) is first \(\lambda a.a^+a\)-stable
  • \(C(\Omega_2+\varphi(d2,0),0)\) is first \(\lambda a.(a^+)^2\)-stable
  • \(C(\Omega_2+\varphi(d^2,0),0)\) is first \(\lambda a.(a^+)^{a^+}\)-stable
  • \(C(\Omega_2+\varphi(\varepsilon_{d+1},d),0)\) is first \(\lambda a.\varepsilon_{a^++1}\)-stable
  • \(C(\Omega_2+\varphi(\varphi(d,1),d),0)\) is first \(\lambda a.\varphi(a^+,1)\)-stable
  • \(C(\Omega_2+\varphi(\Gamma_{d+1},d),0)\) is first \(\lambda a.\Gamma_{a^++1}\)-stable

Beyond that, we can have \(C(\Omega_2+C(C(\Omega_2^{\Omega_2^{\Omega_2^\cdots}},0),d),0)\), \(C(\Omega_2+C(C(C(\Omega_3^{\Omega_3^{\Omega_3^\cdots}},0),0),d),0)\), etc. So \(C(\Omega_2+C(\Omega_2,C(\Omega_2,C(\Omega_22,0))),0)\) is first \(\lambda a.a^{++}\)-stable.

Up to inaccessibly-stable

Let \(d_0=C(\Omega_22,0)\) and \(d_{n+1}=C(\Omega_2,d_n)\).

  • \(C(\Omega_2+d_2,0)\) is first \(\lambda a.a^{++}\)-stable, i.e. first \(\lambda a.C(\Omega_2,C(\Omega_2,a))\)-stable
  • \(C(\Omega_2+d_2+d_1,0)\) is first admissible limit of \(\lambda a.a^{++}\)-stables
  • \(C(\Omega_2+d_2+\varphi(\Gamma_{d_1+1},d_1),0)\) is first \(\lambda a.\Gamma_{a^++1}\)-stable limit of \(\lambda a.a^{++}\)-stables
  • \(C(\Omega_2+d_22,0)\) is first \(\lambda a.a^{++}\)-stable limit of \(\lambda a.a^{++}\)-stables
  • \(C(\Omega_2+d_2d_1,0)\) is first \(\Pi_2\)-reflecting onto \(\lambda a.a^{++}\)-stables
  • \(C(\Omega_2+d_2^2,0)\) is first \(\lambda a.a^{++}\)-stable that is \(\Pi_2\)-reflecting onto \(\lambda a.a^{++}\)-stables
  • \(C(\Omega_2+d_2^{d_2},0)\) is first \(\lambda a.a^{++}\)-stable that is \(\Pi_3\)-reflecting onto \(\lambda a.a^{++}\)-stables
  • \(C(\Omega_2+d_2^{d_2^{d_2}},0)\) is first \(\lambda a.a^{++}\)-stable that is \(\Pi_4\)-reflecting onto \(\lambda a.a^{++}\)-stables
  • \(C(\Omega_2+\varepsilon_{d_2+d_1},0)\) is first 1-stable onto \(\lambda a.a^{++}\)-stables
  • \(C(\Omega_2+\varepsilon_{d_22},0)\) is first \(\lambda a.a^{++}\)-stable that is 1-stable onto \(\lambda a.a^{++}\)-stables
  • \(C(\Omega_2+\varphi(d_0,d_2+d_1),0)\) is first \(\lambda a.a\)-stable onto \(\lambda a.a^{++}\)-stables
  • \(C(\Omega_2+\varphi(d_0,d_22),0)\) is first \(\lambda a.a^{++}\)-stable that is \(\lambda a.a\)-stable onto \(\lambda a.a^{++}\)-stables
  • \(C(\Omega_2+\varphi(d_1,d_2+d_1),0)\) is first \(\lambda a.a^+\)-stable onto \(\lambda a.a^{++}\)-stables
  • \(C(\Omega_2+\varphi(d_1,d_22),0)\) is first \(\lambda a.a^{++}\)-stable that is \(\lambda a.a^+\)-stable onto \(\lambda a.a^{++}\)-stables
  • \(C(\Omega_2+\varphi(d_2,1),0)\) is first \(\lambda a.a^{++}\)-stable onto \(\lambda a.a^{++}\)-stables
  • \(C(\Omega_2+\varphi(d_2,\varepsilon_{d_12}),0)\) is first \(\lambda a.a^{++}+1\)-stable
  • \(C(\Omega_2+\varphi(d_2,d_2),0)\) is first \(\lambda a.a^{++}2\)-stable
  • \(C(\Omega_2+\varphi(d_2+1,0),0)\) is limit of \(\lambda a.a^{++}n\)-stable
  • \(C(\Omega_2+\varphi(d_2+1,d_1),0)\) is first \(\lambda a.a^{++}\omega\)-stable
  • \(C(\Omega_2+\varphi(d_22,0),0)\) is first \(\lambda a.(a^{++})^2\)-stable
  • \(C(\Omega_2+\varphi(d_2^2,0),0)\) is first \(\lambda a.(a^{++})^{a^{++}}\)-stable
  • \(C(\Omega_2+\varphi(\varepsilon_{d_2+1},d_1),0)\) is first \(\lambda a.\varepsilon_{a^{++}+1}\)-stable
  • \(C(\Omega_2+\Gamma_{d_2+1},0)\) is first \(\lambda a.\Gamma_{a^{++}+1}\)-stable
  • \(C(\Omega_2+d_3,0)\) is first \(\lambda a.a^{+++}\)-stable (i.e. "next 3rd admissible"-stable)
  • \(C(\Omega_2+d_4,0)\) is first \(\lambda a.a^{++++}\)-stable (i.e. "next 4th admissible"-stable)
  • \(C(\Omega_2+C(\Omega_2+1,d_0),0)\) is limit of "next n-th admissible"-stable
  • \(C(\Omega_2+\varphi(C(\Omega_2+1,d_0),d_1),0)\) is first "next \(\omega\) admissibles"-stable, i.e. first \(\lambda a.C(\Omega_2+1,a)\)-stable
  • \(C(\Omega_2+C(\Omega_2,C(\Omega_2+1,d_0)),0)\) is first "next \(\omega\)-th admissible"-stable, or "next \(\omega+1\) admissibles"-stable, i.e. first \(\lambda a.C(\Omega_2,C(\Omega_2+1,a))\)-stable
  • \(C(\Omega_2+\varphi(C(\Omega_2+2,d_0),d_1),0)\) is first "next \(\omega^2\) admissibles"-stable
  • \(C(\Omega_2+\varphi(C(\Omega_2+\varepsilon_0,d_0),d_1),0)\) is first "next \(\varepsilon_0\) admissibles"-stable
  • \(C(\Omega_2+\varphi(C(\Omega_2+C(\Omega_2,0),d_0),d_1),0)\) is first "next \(\omega^\text{CK}_1\) admissibles"-stable
  • \(C(\Omega_2+\varphi(C(\Omega_2+C(\Omega_2+d_1^{d_1^\omega+1},0),d_0),d_1),0)\) is first "next \(C(\Omega_2+d_1^{d_1^\omega+1},0)\) admissibles"-stable
  • \(C(\Omega_2+C(\Omega_2+d_0,d_0),0)\) is fixed point of \(\alpha\mapsto\)"next \(\alpha\) admissibles"-stable (this ordinal is not 1-stable)
  • \(C(\Omega_2+\varphi(C(\Omega_2+d_0,d_0),d_1),0)\) is such \(\alpha\) that is "next \(\alpha\) admissibles"-stable, i.e. first \(\lambda a.C(\Omega_2+a,a)\)-stable
  • \(C(\Omega_2+C(\Omega_2,C(\Omega_2+d_0,d_0)),0)\) is such \(\alpha\) that is "next \(\alpha+1\) admissibles"-stable, i.e. first \(\lambda a.C(\Omega_2,C(\Omega_2+a,a))\)-stable
  • \(C(\Omega_2+\varphi(C(\Omega_2+d_0+1,d_0),d_1),0)\) is such \(\alpha\) that is "next \(\alpha\omega\) admissibles"-stable, i.e. first \(\lambda a.C(\Omega_2+a+1,a)\)-stable
  • \(C(\Omega_2+\varphi(C(\Omega_2+d_02,d_0),d_1),0)\) is such \(\alpha\) that is "next \(\alpha^2\) admissibles"-stable, i.e. first \(\lambda a.C(\Omega_2+a2,a)\)-stable
  • \(C(\Omega_2+\varphi(C(\Omega_2+\varepsilon_{d_0+1},d_0),d_1),0)\) is such \(\alpha\) that is "next \(\varepsilon_{\alpha+1}\) admissibles"-stable, i.e. first \(\lambda a.C(\Omega_2+\varepsilon_{a+1},a)\)-stable
  • \(C(\Omega_2+C(\Omega_2+d_1,d_0),0)\) is such \(\alpha\) that is "next \(\alpha^+\) admissibles"-stable, i.e. first \(\lambda a.C(\Omega_2+C(\Omega_2,a),a)\)-stable
  • \(C(\Omega_2+C(\Omega_2+d_2,d_0),0)\) is such \(\alpha\) that is "next \(\alpha^{++}\) admissibles"-stable, i.e. first \(\lambda a.C(\Omega_2+C(\Omega_2,C(\Omega_2,a)),a)\)-stable
  • \(C(\Omega_2+\varphi(C(\Omega_2+C(\Omega_2+d_0,d_0),d_0),d_1),0)\) is first \(\lambda a.C(\Omega_2+C(\Omega_2+a,a),a)\)-stable
  • \(C(\Omega_2+C(\Omega_22,d_0),0)\) is limit of "next admissible"-stable, "next 'next admissible' many admissibles"-stable, "next 'next "next admissible" many admissibles' many admissibles"-stable, etc.
  • \(C(\Omega_2+C(\Omega_22,d_0)+C(\Omega_2+C(\Omega_22,d_0),d_0)d_1,0)\) is first \(\Pi_2\)-reflecting onto limits of "next admissible"-stable, "next 'next admissible' many admissibles"-stable, "next 'next "next admissible" many admissibles' many admissibles"-stable, etc.
  • \(C(\Omega_2+C(\Omega_22,d_0)+\varepsilon_{C(\Omega_2+C(\Omega_22,d_0),d_0)+d_1},0)\) is first 1-stable onto limits of "next admissible"-stable, "next 'next admissible' many admissibles"-stable, "next 'next "next admissible" many admissibles' many admissibles"-stable, etc.
  • \(C(\Omega_2+C(\Omega_22,d_0)+\varphi(C(\Omega_2+C(\Omega_22,d_0),d_0),1),0)\) is level-2 limit of "next admissible"-stable, "next 'next admissible' many admissibles"-stable, "next 'next "next admissible" many admissibles' many admissibles"-stable, etc.
  • \(C(\Omega_2+C(\Omega_22,d_0)+\varphi(C(\Omega_2+C(\Omega_22,d_0),d_0),d_1),0)\) is first \(\lambda a.C(\Omega_2+d_0,a)\)-stable, i.e. an ordinal stable up to next "admissible or limit of admissibles"-fixed-point, i.e. first "next \(\beta=\omega^\text{CK}_\beta\)"-stable
  • \(C(\Omega_2+C(\Omega_22,d_0)+C(\Omega_2,C(\Omega_2+C(\Omega_22,d_0),d_0)),0)\) is first \(\lambda a.C(\Omega_2,C(\Omega_2+d_0,a))\)-stable, i.e. first "next \(\beta+1\) many admissibles"-stable where \(\beta=\omega^\text{CK}_\beta\)
  • \(C(\Omega_2+C(\Omega_22,d_0)+\varphi(C(\Omega_2+C(\Omega_2+C(\Omega_22,d_0),d_0),C(\Omega_2+C(\Omega_22,d_0),d_0)),d_1),0)\) is first \(\lambda a.C(\Omega_2+C(\Omega_2+d_0,a),C(\Omega_2+d_0,a))\)-stable, i.e. first "next \(\beta2\) many admissibles"-stable where \(\beta=\omega^\text{CK}_\beta\)
  • \(C(\Omega_2+C(\Omega_22,d_0)+\varphi(C(\Omega_2+C(\Omega_22,d_0),C(\Omega_2+C(\Omega_22,d_0),d_0)),d_1),0)\) is first \(\lambda a.C(\Omega_2+d_0,C(\Omega_2+d_0,a))\)-stable, i.e. an ordinal stable up to next next "admissible or limit of admissibles"-fixed-point
  • \(C(\Omega_2+C(\Omega_22,d_0)+\varphi(C(\Omega_2+C(\Omega_22,d_0)+d_0,d_0),d_1),0)\) is first \(\lambda a.C(\Omega_2+d_0+a,a)\)-stable
  • \(C(\Omega_2+C(\Omega_22,d_0)2+\varphi(C(\Omega_2+C(\Omega_22,d_0)2,d_0),d_1),0)\) is first \(\lambda a.C(\Omega_2+d_02,a)\)-stable
  • \(C(\Omega_2+\varepsilon_{C(\Omega_22,d_0)+1}+\varphi(C(\Omega_2+\varepsilon_{C(\Omega_22,d_0)+1},d_0),d_1),0)\) is first \(\lambda a.C(\Omega_2+\varepsilon_{d_0+1},a)\)-stable
  • \(C(\Omega_2+C(\Omega_2,C(\Omega_22,d_0)),0)\) is first \(\lambda a.C(\Omega_2+d_1,a)\)-stable, i.e. first "next recursively inaccessible"-stable

Up to \(\omega\)-ply-stable

Let \(\sigma=\sigma_1=C(\Omega_22,0)\), \(\sigma_{n+1}=C(\Omega_22,\sigma_n)\) and \(\alpha^+=C(\Omega_2,\alpha)\) (next admissible ordinal after \(\alpha\)).

  • \(C(\Omega_2+\sigma_2^+,0)\) is first \(\lambda a.C(\Omega_2+\sigma^+,a)\)-stable, i.e. first "next recursively inaccessible"-stable
  • \(C(\Omega_2+\sigma_2^++\varphi(C(\Omega_2+\sigma_2^+,\sigma),\varepsilon_{\sigma^+2}),0)\) is first \(\lambda a.C(\Omega_2+\sigma^+,a)+1\)-stable
  • \(C(\Omega_2+\sigma_2^++C(\Omega_2,C(\Omega_2+\sigma_2^+,\sigma)),0)\) is first \(\lambda a.C(\Omega_2+\sigma^+,a)^+\)-stable, i.e. first "next admissible after recursively inaccessible"-stable
  • \(C(\Omega_2+\sigma_2^++\sigma_2+\varphi(C(\Omega_2+\sigma_2^++\sigma_2,\sigma),\sigma^+),0)\) is first \(\lambda a.C(\Omega_2+\sigma^++\sigma,a)\)-stable, i.e. first "next recursively inaccessible-fixed-point"-stable
  • \(C(\Omega_2+\sigma_2^+2,0)\) is first \(\lambda a.C(\Omega_2+\sigma^+2,a)\)-stable, i.e. first "next recursively 2-inaccessible"-stable
  • \(C(\Omega_2+\sigma_2^+\sigma_2+\sigma_2^+,0)\) is first \(\lambda a.C(\Omega_2+\sigma^+\sigma+\sigma^+,a)\)-stable, i.e. first "next recursively (1,0)-inaccessible"-stable
  • \(C(\Omega_2+(\sigma_2^+)^2,0)\) is first \(\lambda a.C(\Omega_2+(\sigma^+)^2,a)\)-stable, i.e. first "next recursively Mahlo"-stable
  • \(C(\Omega_2+(\sigma_2^+)^{\sigma_2^+},0)\) is first \(\lambda a.C(\Omega_2+(\sigma^+)^{\sigma^+},a)\)-stable, i.e. first "next \(\Pi_3\)-reflecting"-stable
  • \(C(\Omega_2+\varepsilon_{\sigma_2^++1},0)\) is limit of "next \(\Pi_n\)-reflecting"-stable
  • \(C(\Omega_2+\varepsilon_{\sigma_2^++1}+\varphi(C(\Omega_2+\varepsilon_{\sigma_2^++1},\sigma),\sigma^+),0)\) is first "next limit of \(\Pi_n\)-reflecting"-stable
  • \(C(\Omega_2+\varepsilon_{\sigma_2^++1}+\sigma_2^+,0)\) is first "next admissible limit of \(\Pi_n\)-reflecting"-stable
  • \(C(\Omega_2+\varepsilon_{\sigma_2^++1}2+\varphi(C(\Omega_2+\varepsilon_{\sigma_2^++1}2,\sigma),\sigma^+),0)\) is first "next limit of \(\Pi_n\)-reflecting limit of \(\Pi_n\)-reflecting"-stable
  • \(C(\Omega_2+\varepsilon_{\sigma_2^++1}\sigma_2^+,0)\) is first "next \(\Pi_2\)-reflecting onto limit of \(\Pi_n\)-reflecting"-stable
  • \(C(\Omega_2+\varepsilon_{\sigma_2^++1}^{\sigma_2^+},0)\) is first "next \(\Pi_3\)-reflecting onto limit of \(\Pi_n\)-reflecting"-stable
  • \(C(\Omega_2+\varepsilon_{\sigma_2^++2},0)\) is first "next limit of \(\Pi_n\)-reflecting onto limit of \(\Pi_n\)-reflecting"-stable
  • \(C(\Omega_2+\varepsilon_{\sigma_2^+2},0)\) is first "next 1-stable"-stable, i.e. doubly 1-stable, i.e. first \(\lambda a.C(\Omega_2+\varepsilon_{\sigma^+2},a)\)-stable

"next 1-stable"-stable is equivalent to ("next 1-stable"+1)-stable due to the transitivity of stable chain. Using formulas, \(L_\alpha\prec_{\Sigma_1}L_\beta\land L_\beta\prec_{\Sigma_1}L_{\beta+1}\rightarrow L_\alpha\prec_{\Sigma_1}L_{\beta+1}\), and all these are denoted by a chain of \(L_\alpha\prec_{\Sigma_1}L_\beta\prec_{\Sigma_1}L_{\beta+1}\).

  • \(C(\Omega_2+\varepsilon_{\sigma_2^+2}+\varphi(C(\Omega_2+\varepsilon_{\sigma_2^+2},\sigma),1),0)\) is first doubly 1-stable onto doubly 1-stables
  • \(C(\Omega_2+\varepsilon_{\sigma_2^+2}+\varphi(C(\Omega_2+\varepsilon_{\sigma_2^+2},\sigma),\varepsilon_{\sigma^+2}),0)\) is first \(\lambda a.C(\Omega_2+\varepsilon_{\sigma^+2},a)+2\)-stable
  • \(C(\Omega_2+\varepsilon_{\sigma_2^+2}+\sigma_2^+,0)\) is first \(\lambda a.C(\Omega_2+\varepsilon_{\sigma^+2}+\sigma^+,a)\)-stable, i.e. first "next admissible limit of 1-stables"-stable
  • \(C(\Omega_2+\varepsilon_{\sigma_2^+2}2,0)\) is first \(\lambda a.C(\Omega_2+\varepsilon_{\sigma^+2}2,a)\)-stable, i.e. first "next 1-stable limit of 1-stables"-stable
  • \(C(\Omega_2+\varepsilon_{\sigma_2^+2}\sigma_2^+,0)\) is first \(\lambda a.C(\Omega_2+\varepsilon_{\sigma^+2}\sigma^+,a)\)-stable, i.e. first "next \(\Pi_2\)-reflecting onto 1-stables"-stable
  • \(C(\Omega_2+\varepsilon_{\sigma_2^+3},0)\) is first \(\lambda a.C(\Omega_2+\varepsilon_{\sigma^+3},a)\)-stable, i.e. first "next 1-stable onto 1-stables"-stable
  • \(C(\Omega_2+\varepsilon_{\varepsilon_{\sigma_2^+2}},0)\) is first doubly 2-stable
  • \(C(\Omega_2+\varphi(\sigma,\sigma_2^++1),0)\) is fixed point of \(\alpha\mapsto\)doubly \(\alpha\)-stable
  • \(C(\Omega_2+\varphi(\sigma,\sigma_2^++1)+\varphi(C(\Omega_2+\varphi(\sigma,\sigma_2^++1),\sigma),\sigma^+),0)\) is first \(\lambda a.C(\Omega_2+\varphi(a,\sigma^++1),a)\)-stable
  • \(C(\Omega_2+\varphi(\sigma,\sigma_2^++1)+\sigma_2^+,0)\) is first \(\lambda a.C(\Omega_2+\varphi(a,\sigma^++1)+\sigma^+,a)\)-stable
  • \(C(\Omega_2+\varphi(\sigma,\sigma_2^+2),0)\) is first \(\lambda a.C(\Omega_2+\varphi(a,\sigma^+2),a)\)-stable
  • \(C(\Omega_2+\varphi(\sigma+1,\sigma_2^+2),0)\) is first \(\lambda a.C(\Omega_2+\varphi(a+1,\sigma^+2),a)\)-stable
  • \(C(\Omega_2+\varphi(\sigma^+,\sigma_2^++1),0)\) is first \(\lambda a.C(\Omega_2+\varphi(a^+,\sigma^++1),a)\)-stable
  • \(C(\Omega_2+\varphi(\sigma_2,\sigma_2^++1)+\varphi(C(\Omega_2+\varphi(\sigma_2,\sigma_2^++1),\sigma),\sigma^+),0)\) is first \(\lambda a.C(\Omega_2+\varphi(\sigma,\sigma^++1),a)\)-stable, i.e. first "next fixed point of \(\alpha\mapsto\alpha\)-stable"-stable
  • \(C(\Omega_2+\varphi(\sigma_2,\sigma_2^+2)+\varphi(C(\Omega_2+\varphi(\sigma_2,\sigma_2^+2),\sigma),\sigma^+),0)\) is first \(\lambda a.C(\Omega_2+\varphi(\sigma,\sigma^+2),a)\)-stable, i.e. first "next \(\lambda a.a\)-stable"-stable
  • \(C(\Omega_2+\varphi(\sigma_2^+,1),0)\) is first \(\lambda a.C(\Omega_2+\varphi(\sigma^+,1),a)\)-stable, i.e. first "next 'next admissible'-stable"-stable
  • \(C(\Omega_2+\sigma_2^{++},0)\) is first \(\lambda a.C(\Omega_2+\sigma^{++},a)\)-stable, i.e. first "next 'next next admissible'-stable"-stable
  • \(C(\Omega_2+C(\Omega_2+1,\sigma_2),0)\) is limit of "next 'next n admissibles'-stable"-stable
  • \(C(\Omega_2+C(\Omega_2,C(\Omega_2+1,\sigma_2)),0)\) is first "next 'next \(\omega+1\) admissibles'-stable"-stable
  • \(C(\Omega_2+C(\Omega_2+\sigma,\sigma_2),0)\) is fixed point of \(\alpha\mapsto\)"next 'next \(\alpha\) admissibles'-stable"-stable
  • \(C(\Omega_2+C(\Omega_2+\sigma,\sigma_2)+\varphi(C(\Omega_2+C(\Omega_2+\sigma,\sigma_2),\sigma),\sigma^+),0)\) is first \(\lambda a.C(\Omega_2+C(\Omega_2+a,\sigma),a)\)-stable, i.e. first ordinal \(\alpha\) that is "next 'next \(\alpha\) admissibles'-stable"-stable
  • \(C(\Omega_2+C(\Omega_2+\sigma^+,\sigma_2),0)\) is first \(\lambda a.C(\Omega_2+C(\Omega_2+a^+,\sigma),a)\)-stable
  • \(C(\Omega_2+C(\Omega_2+\sigma_2,\sigma_2)+\varphi(C(\Omega_2+C(\Omega_2+\sigma_2,\sigma_2),\sigma),\sigma^+),0)\) is first \(\lambda a.C(\Omega_2+C(\Omega_2+\sigma,\sigma),a)\)-stable, i.e. first ordinal stable up to a fixed point of \(\alpha\mapsto\)"next \(\alpha\) admissibles"-stable
  • \(C(\Omega_2+\varphi(C(\Omega_2+\sigma_2,\sigma_2),\sigma_2^+),0)\) is first \(\lambda a.C(\Omega_2+\varphi(C(\Omega_2+\sigma,\sigma),\sigma^+),a)\)-stable, i.e. first ordinal stable up to an \(\alpha\) that is "next \(\alpha\) admissibles"-stable
  • \(C(\Omega_2+C(\Omega_2+\sigma_2^+,\sigma_2),0)\) is first \(\lambda a.C(\Omega_2+C(\Omega_2+\sigma^+,\sigma),a)\)-stable, i.e. first "next 'next "next admissible" many admissibles'-stable"-stable
  • \(C(\Omega_2+C(\Omega_2+C(\Omega_2+\sigma_2^+,\sigma_2),\sigma_2),0)\) is first \(\lambda a.C(\Omega_2+C(\Omega_2+C(\Omega_2+\sigma^+,\sigma),\sigma),a)\)-stable, i.e. first "next 'next "next 'next admissible' many admissibles" many admissibles'-stable"-stable
  • \(C(\Omega_2+\sigma_3,0)\) is limit of "next 'next admissible'-stable"-stable, "next 'next "next admissible" many admissibles'-stable"-stable, "next 'next "next 'next admissible' many admissibles" many admissibles'-stable"-stable, etc.
  • \(C(\Omega_2+\sigma_3+\varphi(C(\Omega_2+\sigma_3,\sigma),\sigma^+),0)\) is first \(\lambda a.C(\Omega_2+\sigma_2,a)\)-stable, i.e. first ordinal stable up to next limit of "next admissible"-stable, "next 'next admissible' many admissibles"-stable, "next 'next "next admissible" many admissibles' many admissibles"-stable, etc.
  • \(C(\Omega_2+\sigma_3+\varphi(C(\Omega_2+\sigma_3,\sigma_2),\sigma_2^+),0)\) is first \(\lambda a.C(\Omega_2+\sigma_2+\varphi(C(\Omega_2+\sigma_2,\sigma),\sigma^+),a)\)-stable, i.e. first "next 'next \(\beta=\omega^\text{CK}_\beta\)'-stable"-stable
  • \(C(\Omega_2+\sigma_3^+,0)\) is first \(\lambda a.C(\Omega_2+\sigma_2^+,a)\)-stable, i.e. first "next 'next recursively inaccessible'-stable"-stable
  • \(C(\Omega_2+\varepsilon_{\sigma_3^+2},0)\) is first \(\lambda a.C(\Omega_2+\varepsilon_{\sigma_2^+2},a)\)-stable, i.e. first triply 1-stable
  • \(C(\Omega_2+\varepsilon_{\sigma_4^+2},0)\) is first \(\lambda a.C(\Omega_2+\varepsilon_{\sigma_3^+2},a)\)-stable, i.e. first quadruply 1-stable
  • \(C(\Omega_2+C(\Omega_22+1,0),0)\) is limit of n-ply-stable

For an abstract stable \(f\)-stable, the corresponding ordinal is approximately \(f(\sigma)\), while \(\sigma_n\)'s in \(f\) are substituted into \(\sigma_{n+1}\). So \(C(\Omega_2+C(\Omega_22+1,0),0)\) is first limit of such substitution.

  • \(C(\Omega_2+C(\Omega_22+1,0)+\sigma^+,0)\) is admissible limit of n-ply-stable
  • \(C(\Omega_2+C(\Omega_22+1,0)+C(\Omega_2+C(\Omega_22+1,0),\sigma)\sigma^+,0)\) is \(\Pi_2\)-reflecting onto limits of n-ply-stable
  • \(C(\Omega_2+C(\Omega_22+1,0)+\varepsilon_{C(\Omega_2+C(\Omega_22+1,0),\sigma)+\sigma^+},0)\) is 1-stable onto limits of n-ply-stable
  • \(C(\Omega_2+C(\Omega_22+1,0)+\varphi(C(\Omega_2+C(\Omega_22+1,0),\sigma),1),0)\) is limit of n-ply-stable onto limits of n-ply-stable
  • \(C(\Omega_2+C(\Omega_22+1,0)+\varphi(C(\Omega_2+C(\Omega_22+1,0),\sigma),\sigma^+),0)\) is first "next limit of n-ply-stable"-stable
  • \(C(\Omega_2+C(\Omega_22+1,0)+\sigma_2^+,0)\) is first "next admissible limit of n-ply-stable"-stable
  • \(C(\Omega_2+C(\Omega_22+1,0)+\varepsilon_{C(\Omega_2+C(\Omega_22+1,0),\sigma_2)+\sigma_2^+},0)\) is first "next 1-stable onto limits of n-ply-stable"-stable
  • \(C(\Omega_2+C(\Omega_22+1,0)+\varphi(C(\Omega_2+C(\Omega_22+1,0),\sigma_2),1),0)\) is first "next limit of n-ply-stable onto limits of n-ply-stable"-stable
  • \(C(\Omega_2+C(\Omega_22+1,0)+\varphi(C(\Omega_2+C(\Omega_22+1,0),\sigma_2),\sigma_2^+),0)\) is first "next 'next limit of n-ply-stable'-stable"-stable
  • \(C(\Omega_2+C(\Omega_22+1,0)+\varphi(C(\Omega_2+C(\Omega_22+1,0),\sigma_3),\sigma_3^+),0)\) is first "next 'next "next limit of n-ply-stable"-stable'-stable"-stable
  • \(C(\Omega_2+C(\Omega_22+1,0)2,0)\) is limit of n-ply "limit of n-ply-stable"-stable (2nd limit of "substitution")
  • \(C(\Omega_2+C(\Omega_22+1,0)3,0)\) is limit of n-ply "limit of n-ply 'limit of n-ply-stable'-stable"-stable
  • \(C(\Omega_2+C(\Omega_22+1,0)\sigma^+,0)\) is first \(\omega\)-ply-stable

For an \(\omega\)-ply-stable ordinal \(\alpha\), \(L_\alpha\prec_{\Sigma_1}L_\beta\), where \(\beta\) is still \(\omega\)-ply-stable. So \(\omega\)-ply-stable ordinals make a chain \(\{\alpha_n\}_{n<\omega}\) such that \(L_{\alpha_n}\prec_{\Sigma_1}L_{\alpha_{n+1}}\), with a supremum \(\alpha_\omega=\sup\{\alpha_n|n<\omega\}\), which might not be admissible.

A related concept about \(\omega\)-ply-stable ordinal is nonprojectable ordinal. \(\beta\) is nonprojectable if \(\sup\{\alpha<\beta|L_\alpha\prec_{\Sigma_1}L_\beta\}=\beta\), meaning that there are infinite amount of \(\alpha\)'s which are stable to the same target \(\beta\). Stability has some properties:

  1. If \(\alpha<\beta<\gamma\) and \(L_\alpha\prec_{\Sigma_1}L_\gamma\), then \(L_\alpha\prec_{\Sigma_1}L_\beta\). (Larger stability implies smaller stability)
  2. If \(L_\alpha\prec_{\Sigma_1}L_\beta\) and \(L_\beta\prec_{\Sigma_1}L_\gamma\), then \(L_\alpha\prec_{\Sigma_1}L_\gamma\). ("Stability chain" is transitive)
  3. If a set \(A\) is nonempty and \(\forall\alpha\in A(L_\alpha\prec_{\Sigma_1}L_\beta)\), then \(L_{\sup A}\prec_{\Sigma_1}L_\beta\). ("Stable to target \(\beta\)" is closed)

So the existence of an \(\omega\)-ply-stable ordinal and the existence of a nonprojectable ordinal are equivalent. Moreover, the limit of least \(\omega\) \(\omega\)-ply-stable ordinals is the least nonprojectable ordinal, and every nonprojectable is a limit of \(\omega\)-ply-stable ordinals.

I am not sure about comparisons between "limit of n-ply-stable" to "\(\omega\)-ply-stable", but even if they were equal, the following things still make a comparably strong hierarchy.

Up to \(\omega+1\)-ply 1-stable

Let \(\sigma=\sigma_1=C(\Omega_22,0)\), \(\sigma_{n+1}=C(\Omega_22,\sigma_n)\) and \(\alpha^+=C(\Omega_2,\alpha)\) (next admissible ordinal after \(\alpha\)).

  • \(C(\Omega_2+C(\Omega_22+1,0)\sigma^+,C(\Omega_2+C(\Omega_22+1,0)\sigma^+,0))\) is 2nd \(\omega\)-ply-stable
  • \(C(\Omega_2+C(\Omega_22+1,0)\sigma^++1,0)\) is first nonprojectable
  • \(C(\Omega_2+C(\Omega_22+1,0)\sigma^++1,\alpha)\) is next nonprojectable after \(\alpha\)
  • \(C(\Omega_2+C(\Omega_22+1,0)\sigma^++2,0)\) is first limit of nonprojectables (this ordinal is not nonprojectable)
  • \(C(\Omega_2+C(\Omega_22+1,0)\sigma^++\sigma^+,0)\) is first admissible limit of \(\omega\)-ply-stables
  • \(C(\Omega_2+C(\Omega_22+1,0)\sigma^++C(\Omega_2+C(\Omega_22+1,0)\sigma^+,\sigma),0)\) is first \(\omega\)-ply-stable limit of \(\omega\)-ply-stables (also \(\omega\)-ply-stable limit of nonprojectables) (\(\omega\)-ply-stable limits of \(\omega\)-ply-stables are stable up to different nonprojectables, so \(C(\Omega_2+C(\Omega_22+1,0)\sigma^++C(\Omega_2+C(\Omega_22+1,0)\sigma^+,\sigma)+1,0)\) is not nonprojectable limit of nonprojectables)
  • \(C(\Omega_2+C(\Omega_22+1,0)\sigma^++\varphi(C(\Omega_2+C(\Omega_22+1,0)\sigma^+,\sigma),1),0)\) is first \(\omega\)-ply-stable onto \(\omega\)-ply-stables
  • \(C(\Omega_2+C(\Omega_22+1,0)\sigma^++\varphi(C(\Omega_2+C(\Omega_22+1,0)\sigma^+,\sigma),\varepsilon_{\sigma^+2}),0)\) is first \(\lambda a.C(\Omega_2+C(\Omega_22+1,0)\sigma^++1,a)+1\)-stable, i.e. first ("next nonprojectable"+1)-stable
  • \(C(\Omega_2+C(\Omega_22+1,0)\sigma^++C(\Omega_2+C(\Omega_22+1,0)\sigma^+,\sigma)^+,0)\) is first \(\lambda a.C(\Omega_2+C(\Omega_22+1,0)\sigma^++1,a)^+\)-stable
  • \(C(\Omega_2+C(\Omega_22+1,0)\sigma^++C(\Omega_2+C(\Omega_22+1,0)\sigma^+,C(\Omega_2+C(\Omega_22+1,0)\sigma^+,\sigma)),0)\) is first \(\lambda a.C(\Omega_2+C(\Omega_22+1,0)\sigma^++1,C(\Omega_2+C(\Omega_22+1,0)\sigma^++1,a))\)-stable, i.e. first "next next nonprojectable"-stable
  • \(C(\Omega_2+C(\Omega_22+1,0)\sigma^++\varphi(C(\Omega_2+C(\Omega_22+1,0)\sigma^++1,\sigma),\sigma^+),0)\) is first \(\lambda a.C(\Omega_2+C(\Omega_22+1,0)\sigma^++2,a)\)-stable, i.e. first "next \(\omega\) many nonprojectables"-stable
  • \(C(\Omega_2+C(\Omega_22+1,0)\sigma^++\varphi(C(\Omega_2+C(\Omega_22+1,0)\sigma^++\sigma,\sigma),\sigma^+),0)\) is first \(\lambda a.C(\Omega_2+C(\Omega_22+1,0)\sigma^++a,a)\)-stable, i.e. first \(\alpha\) that is "next \(\alpha\) many nonprojectables"-stable
  • \(C(\Omega_2+C(\Omega_22+1,0)\sigma^++C(\Omega_2+C(\Omega_22+1,0)\sigma^++\sigma^+,\sigma),0)\) is first \(\lambda a.C(\Omega_2+C(\Omega_22+1,0)\sigma^++a^+,a)\)-stable, i.e. first \(\alpha\) that is "next \(\alpha^+\) many nonprojectables"-stable
  • \(C(\Omega_2+C(\Omega_22+1,0)\sigma^++C(\Omega_2+C(\Omega_22+1,0)\sigma^++C(\Omega_2+C(\Omega_22+1,0)\sigma^+,\sigma),\sigma),0)\) is first \(\lambda a.C(\Omega_2+C(\Omega_22+1,0)\sigma^++C(\Omega_2+C(\Omega_22+1,0)\sigma^+,a),a)\)-stable, i.e. first "next 'next \(\omega\)-ply-stable' many nonprojectables"-stable
  • \(C(\Omega_2+C(\Omega_22+1,0)\sigma^++\sigma_2+\varphi(C(\Omega_2+C(\Omega_22+1,0)\sigma^++\sigma_2,\sigma),\sigma^+),0)\) is first \(\lambda a.C(\Omega_2+C(\Omega_22+1,0)\sigma^++\sigma,a)\)-stable, i.e. first "next nonprojectable-fixed-point"-stable
  • \(C(\Omega_2+C(\Omega_22+1,0)\sigma^++\sigma_2^+,0)\) is first \(\lambda a.C(\Omega_2+C(\Omega_22+1,0)\sigma^++\sigma^+,a)\)-stable, i.e. first "next admissible limit of \(\omega\)-ply-stables"-stable
  • \(C(\Omega_2+C(\Omega_22+1,0)\sigma^++\varphi(C(\Omega_2+C(\Omega_22+1,0)\sigma_2^+,\sigma_2),1),0)\) is first \(\lambda a.C(\Omega_2+C(\Omega_22+1,0)\sigma^++\varphi(C(\Omega_2+C(\Omega_22+1,0)\sigma^+,\sigma),1),a)\)-stable, i.e. first "next \(\omega\)-ply-stable onto \(\omega\)-ply-stables"-stable
  • \(C(\Omega_2+C(\Omega_22+1,0)\sigma^++\varphi(C(\Omega_2+C(\Omega_22+1,0)\sigma_2^+,\sigma_2),\varepsilon_{\sigma_2^+2}),0)\) is first \(\lambda a.C(\Omega_2+C(\Omega_22+1,0)\sigma^++\varphi(C(\Omega_2+C(\Omega_22+1,0)\sigma^+,\sigma),\varepsilon_{\sigma^+2}),a)\)-stable, i.e. first "next ('next nonprojectable'+1)-stable"-stable
  • \(C(\Omega_2+C(\Omega_22+1,0)\sigma^++\varphi(C(\Omega_2+C(\Omega_22+1,0)\sigma_3^+,\sigma_3),\varepsilon_{\sigma_3^+2}),0)\) is first \(\lambda a.C(\Omega_2+C(\Omega_22+1,0)\sigma^++\varphi(C(\Omega_2+C(\Omega_22+1,0)\sigma_2^+,\sigma_2),\varepsilon_{\sigma_2^+2}),a)\)-stable, i.e. first "next 'next ("next nonprojectable"+1)-stable'-stable"-stable
  • \(C(\Omega_2+C(\Omega_22+1,0)\sigma^++C(\Omega_22+1,0),0)\) is limit of n-ply ("next nonprojectable"+1)-stable
  • \(C(\Omega_2+C(\Omega_22+1,0)\sigma^+2,0)\) is first "nonprojectable limit of nonprojectables"-stable
  • \(C(\Omega_2+C(\Omega_22+1,0)\sigma^+2+1,0)\) is first nonprojectable limit of nonprojectables, i.e. level-1 nonprojectable limit of nonprojectables
  • \(C(\Omega_2+C(\Omega_22+1,0)\sigma^+2+\sigma^+,0)\) is first admissible limit of nonprojectable limit of nonprojectables
  • \(C(\Omega_2+C(\Omega_22+1,0)\sigma^+2+\sigma_2^+,0)\) is first "next admissible limit of nonprojectable limit of nonprojectables"-stable
  • \(C(\Omega_2+C(\Omega_22+1,0)\sigma^+2+C(\Omega_22+1,0),0)\) is limit of n-ply ("nonprojectable limit of nonprojectables"+1)-stable
  • \(C(\Omega_2+C(\Omega_22+1,0)\sigma^+3,0)\) is first "nonprojectable limit of nonprojectable limits of nonprojectables"-stable
  • \(C(\Omega_2+C(\Omega_22+1,0)\sigma^+3+1,0)\) is first nonprojectable limit of nonprojectable limits of nonprojectables, i.e. level-2 nonprojectable limit of nonprojectables
  • \(C(\Omega_2+C(\Omega_22+1,0)\sigma^+4+1,0)\) is first level-3 nonprojectable limit of nonprojectables
  • \(C(\Omega_2+C(\Omega_22+1,0)\sigma^+\omega,0)\) is limit of level-n nonprojectable limit of nonprojectables
  • \(C(\Omega_2+C(\Omega_22+1,0)\sigma^+\omega+C(\Omega_2+C(\Omega_22+1,0)\sigma^+\omega,\sigma),0)\) is limit of "level-n nonprojectable limit of nonprojectables" limits of level-n nonprojectable limit of nonprojectables
  • \(C(\Omega_2+C(\Omega_22+1,0)\sigma^+\omega+\varphi(C(\Omega_2+C(\Omega_22+1,0)\sigma^+\omega,\sigma),\sigma^+),0)\) is first "next limit of level-n nonprojectable limit of nonprojectables"-stable
  • \(C(\Omega_2+C(\Omega_22+1,0)\sigma^+\omega+\varphi(C(\Omega_2+C(\Omega_22+1,0)\sigma^+\omega,\sigma_2),\sigma_2^+),0)\) is first "next 'next limit of level-n nonprojectable limit of nonprojectables'-stable"-stable
  • \(C(\Omega_2+C(\Omega_22+1,0)\sigma^+\omega+C(\Omega_22+1,0),0)\) is limit of n-ply "limit of level-n nonprojectable limit of nonprojectables"-stable
  • \(C(\Omega_2+C(\Omega_22+1,0)\sigma^+\omega+C(\Omega_22+1,0)\sigma^+,0)\) is "level-\(\omega\) nonprojectable limit of nonprojectables"-stable
  • \(C(\Omega_2+C(\Omega_22+1,0)\sigma^+\omega+C(\Omega_22+1,0)\sigma^++1,0)\) is level-\(\omega\) nonprojectable limit of nonprojectables
  • \(C(\Omega_2+C(\Omega_22+1,0)\sigma^+\sigma,0)\) is fixed point of \(\alpha\mapsto\)level-\(\alpha\) nonprojectable limit of nonprojectables
  • \(C(\Omega_2+C(\Omega_22+1,0)\sigma^+\sigma+\varphi(C(\Omega_2+C(\Omega_22+1,0)\sigma^+\sigma,\sigma),\sigma^+),0)\) is first \(\alpha\) that is "next level-\(\alpha\) nonprojectable limit of nonprojectables"-stable
  • \(C(\Omega_2+C(\Omega_22+1,0)\sigma^+\sigma+\varphi(C(\Omega_2+C(\Omega_22+1,0)\sigma^+\sigma,\sigma_2),\sigma_2^+),0)\) is first \(\alpha\) that is "next 'next level-\(\alpha\) nonprojectable limit of nonprojectables'-stable"-stable
  • \(C(\Omega_2+C(\Omega_22+1,0)\sigma^+\sigma+C(\Omega_22+1,0),0)\) is limit of \(\alpha\) that is n-ply "next level-\(\alpha\) nonprojectable limit of nonprojectables"-stable
  • \(C(\Omega_2+C(\Omega_22+1,0)\sigma^+\sigma+C(\Omega_22+1,0)\sigma^+,0)\) is first \(\alpha\) that is "next level-\(\alpha+1\) nonprojectable limit of nonprojectables"-stable
  • \(C(\Omega_2+C(\Omega_22+1,0)(\sigma^+)^2,0)\) is first \(\alpha\) that is "next level-\(\alpha^+\) nonprojectable limit of nonprojectables"-stable
  • \(C(\Omega_2+C(\Omega_22+1,0)C(\Omega_2+C(\Omega_22+1,0)(\sigma^+)^2,\sigma),0)\) is first \(\alpha\) that is "next level-\(C(\Omega_2+C(\Omega_22+1,0)\sigma^+\alpha^+,\alpha)\) nonprojectable limit of nonprojectables"-stable
  • \(C(\Omega_2+C(\Omega_22+1,0)\sigma_2+\varphi(C(\Omega_2+C(\Omega_22+1,0)\sigma_2,\sigma),\sigma^+),0)\) is first "next fixed point of \(\alpha\mapsto\)level-\(\alpha\) nonprojectable limit of nonprojectables"-stable
  • \(C(\Omega_2+C(\Omega_22+1,0)\sigma_2+\varphi(C(\Omega_2+C(\Omega_22+1,0)\sigma_2,\sigma_2),\sigma_2^+),0)\) is first "next \(\alpha\) that is level-\(\alpha\) nonprojectable limit of nonprojectables"-stable
  • \(C(\Omega_2+C(\Omega_22+1,0)\sigma_2^+,0)\) is first "next \(\alpha\) that is level-\(\alpha^+\) nonprojectable limit of nonprojectables"-stable
  • \(C(\Omega_2+C(\Omega_22+1,0)\sigma_3^+,0)\) is first "next 'next \(\alpha\) that is level-\(\alpha^+\) nonprojectable limit of nonprojectables'-stable"-stable
  • \(C(\Omega_2+C(\Omega_22+1,0)^2,0)\) is limit of n-ply "\(\alpha\) that is level-\(\alpha^+\) nonprojectable limit of nonprojectables"-stable
  • \(C(\Omega_2+C(\Omega_22+1,0)^2\sigma^+,0)\) is first "\(\alpha=\)level-\(\alpha\) nonprojectable limit of nonprojectables"-stable
  • \(C(\Omega_2+C(\Omega_22+1,0)^2\sigma^++1,0)\) is first \(\alpha=\)level-\(\alpha\) nonprojectable limit of nonprojectables, i.e. first level-(1,0) nonprojectable limit of nonprojectables
  • \(C(\Omega_2+C(\Omega_22+1,0)^2\sigma^++C(\Omega_22+1,0)\sigma^++1,0)\) is first level-(1,1) nonprojectable limit of nonprojectables
  • \(C(\Omega_2+C(\Omega_22+1,0)^2\sigma^+2+1,0)\) is first level-(2,0) nonprojectable limit of nonprojectables
  • \(C(\Omega_2+C(\Omega_22+1,0)^3\sigma^++1,0)\) is first level-(1,0,0) nonprojectable limit of nonprojectables

I am not sure about size of "nonprojectable limit of nonprojectables" ordinals, but even if this concept is removed, the following things still make a comparably strong hierarchy.

  • \(C(\Omega_2+C(\Omega_22+1,0)^++1,0)\) is first nonprojectable and also admissible (since it is limit of \(\omega\)-ply-stable ordinals, it is also recursively inaccessible)
  • \(C(\Omega_2+C(\Omega_22+1,0)^++C(\Omega_22+1,0)\sigma^++1,0)\) is first nonprojectable limit of "nonprojectable and admissible" ordinals
  • \(C(\Omega_2+C(\Omega_22+1,0)^+2+1,0)\) is first "nonprojectable and admissible" limit of "nonprojectable and admissible" ordinals
  • \(C(\Omega_2+(C(\Omega_22+1,0)^+)^2+1,0)\) is first nonprojectable and also recursively Mahlo
  • \(C(\Omega_2+(C(\Omega_22+1,0)^+)^2+C(\Omega_22+1,0)^++1,0)\) is first "nonprojectable and admissible" limit of "nonprojectable and recursively Mahlo" ordinals
  • \(C(\Omega_2+(C(\Omega_22+1,0)^+)^22+1,0)\) is first "nonprojectable and recursively Mahlo" limit of "nonprojectable and recursively Mahlo" ordinals
  • \(C(\Omega_2+(C(\Omega_22+1,0)^+)^3+1,0)\) is first nonprojectable and also recursively 2-Mahlo
  • \(C(\Omega_2+(C(\Omega_22+1,0)^+)^\sigma+\varphi(C(\Omega_2+(C(\Omega_22+1,0)^+)^\sigma,\sigma),\sigma^+),0)\) is first \(\alpha\) that is "nonprojectable and also recursively \(\alpha\)-Mahlo"-stable
  • \(C(\Omega_2+(C(\Omega_22+1,0)^+)^{\sigma^+},0)\) is first \(\alpha\) that is "next nonprojectable and also recursively \(\alpha^+\)-Mahlo"-stable
  • \(C(\Omega_2+(C(\Omega_22+1,0)^+)^{\sigma_2^+},0)\) is first "next \(\alpha\) that is 'next nonprojectable and also recursively \(\alpha^+\)-Mahlo'-stable"-stable
  • \(C(\Omega_2+(C(\Omega_22+1,0)^+)^{C(\Omega_22+1,0)},0)\) is limit of n-ply "next \(\alpha\) that is 'next nonprojectable and also recursively \(\alpha^+\)-Mahlo'-stable"-stable
  • \(C(\Omega_2+(C(\Omega_22+1,0)^+)^{C(\Omega_22+1,0)}\sigma^++1,0)\) is nonprojectable and also fixed point of \(\alpha\mapsto\)recursively \(\alpha\)-Mahlo
  • \(C(\Omega_2+(C(\Omega_22+1,0)^+)^{C(\Omega_22+1,0)+1}+1,0)\) is first \(\alpha\) that is nonprojectable and also recursively \(\alpha\)-Mahlo, i.e. nonprojectable and also recursively (1,0)-Mahlo
  • \(C(\Omega_2+(C(\Omega_22+1,0)^+)^{C(\Omega_22+1,0)^+}+1,0)\) is first nonprojectable and also \(\Pi_3\)-reflecting
  • \(C(\Omega_2+(C(\Omega_22+1,0)^+)^{(C(\Omega_22+1,0)^+)^{C(\Omega_22+1,0)^+}}+1,0)\) is first nonprojectable and also \(\Pi_4\)-reflecting
  • \(C(\Omega_2+\varepsilon_{C(\Omega_22+1,0)^++1},0)\) is limit of nonprojectable and also \(\Pi_n\)-reflecting
  • \(C(\Omega_2+\varepsilon_{C(\Omega_22+1,0)^++1}+\sigma^+,0)\) is first admissible limit of nonprojectable and also \(\Pi_n\)-reflecting
  • \(C(\Omega_2+\varepsilon_{C(\Omega_22+1,0)^++1}+\sigma_2^+,0)\) is first "admissible limit of nonprojectable and also \(\Pi_n\)-reflecting"-stable
  • \(C(\Omega_2+\varepsilon_{C(\Omega_22+1,0)^++1}+C(\Omega_22+1,0)\sigma^++1,0)\) is first nonprojectable limit of nonprojectable and also \(\Pi_n\)-reflecting
  • \(C(\Omega_2+\varepsilon_{C(\Omega_22+1,0)^++1}2,0)\) is limit of "nonprojectable and \(\Pi_n\)-reflecting" limit of nonprojectable and also \(\Pi_n\)-reflecting
  • \(C(\Omega_2+\varepsilon_{C(\Omega_22+1,0)^++1}C(\Omega_22+1,0)^++1,0)\) is first nonprojectable and also \(\Pi_2\)-reflecting onto limits of "nonprojectable and \(\Pi_n\)-reflecting"
  • \(C(\Omega_2+\varepsilon_{C(\Omega_22+1,0)^++1}^2,0)\) is limit of nonprojectable, \(\Pi_n\)-reflecting, and \(\Pi_2\)-reflecting onto limits of "nonprojectable and \(\Pi_n\)-reflecting"
  • \(C(\Omega_2+\varepsilon_{C(\Omega_22+1,0)^++1}^{C(\Omega_22+1,0)^+}+1,0)\) is first nonprojectable and also \(\Pi_3\)-reflecting onto limits of "nonprojectable and \(\Pi_n\)-reflecting"
  • \(C(\Omega_2+\varepsilon_{C(\Omega_22+1,0)^++2}+1,0)\) is limit of nonprojectable and \(\Pi_n\)-reflecting onto limits of "nonprojectable and \(\Pi_n\)-reflecting"
  • \(C(\Omega_2+\varepsilon_{C(\Omega_22+1,0)^+2},0)\) is first \(\omega+1\)-ply 1-stable
  • \(C(\Omega_2+\varepsilon_{C(\Omega_22+1,0)^+2}+1,0)\) is first nonprojectable and 1-stable

Up to \(\alpha\mapsto\alpha\)-ply-stable

Let \(\sigma=\sigma_1=C(\Omega_22,0)\), \(\sigma_{n+1}=C(\Omega_22,\sigma_n)\); \(\alpha^+=C(\Omega_2,\alpha)\) (next admissible ordinal after \(\alpha\)); \(C^0(\alpha,\beta)=\beta\) and \(C^{n+1}(\alpha,\beta)=C(\alpha,C^n(\alpha,\beta))\).

  • \(C(\Omega_2+\varepsilon_{\varepsilon_{C(\Omega_22+1,0)^+2}}+1,0)\) is first nonprojectable and 2-stable
  • \(C(\Omega_2+\zeta_{C(\Omega_22+1,0)^+2}+1,0)\) is first nonprojectable and \(\omega\)-stable
  • \(C(\Omega_2+\varphi(3,C(\Omega_22+1,0)^+2)+1,0)\) is first nonprojectable and \(\omega^2\)-stable
  • \(C(\Omega_2+\varphi(\sigma,C(\Omega_22+1,0)^++1),0)\) is fixed point of \(\alpha\mapsto\omega+1\)-ply \(\alpha\)-stable
  • \(C(\Omega_2+\varphi(\sigma,C(\Omega_22+1,0)^++1)+\varphi(C(\Omega_2+\varphi(\sigma,C(\Omega_22+1,0)^++1),\sigma),\sigma^+),0)\) is first \(\alpha\) that is \(\omega+1\)-ply \(\alpha\)-stable
  • \(C(\Omega_2+\varphi(\sigma,C(\Omega_22+1,0)^++1)+\varphi(C(\Omega_2+\varphi(\sigma,C(\Omega_22+1,0)^++1),\sigma_2),\sigma_2^+),0)\) is first "next \(\alpha\) that is \(\omega+1\)-ply \(\alpha\)-stable"-stable
  • \(C(\Omega_2+\varphi(\sigma,C(\Omega_22+1,0)^++1)+C(\Omega_22+1,0),0)\) is limit of n-ply "next \(\alpha\) that is \(\omega+1\)-ply \(\alpha\)-stable"-stable
  • \(C(\Omega_2+\varphi(\sigma,C(\Omega_22+1,0)^++1)+C(\Omega_22+1,0)\sigma^+,0)\) is first \(\alpha\) stable up to a nonprojectable and limit of "nonprojectable and \(\alpha\)-stable" ordinals
  • \(C(\Omega_2+\varphi(\sigma,C(\Omega_22+1,0)^++1)+C(\Omega_22+1,0)^+,0)\) is first \(\alpha\) stable up to a nonprojectable and admissible limit of "nonprojectable and \(\alpha\)-stable" ordinals
  • \(C(\Omega_2+\varphi(\sigma,C(\Omega_22+1,0)^++1)+\varepsilon_{C(\Omega_22+1,0)^+2},0)\) is first \(\alpha\) stable up to a nonprojectable and 1-stable limit of "nonprojectable and \(\alpha\)-stable" ordinals
  • \(C(\Omega_2+\varphi(\sigma,C(\Omega_22+1,0)^++1)2,0)\) is fixed point of \(\beta\mapsto\alpha\) stable up to a nonprojectable and \(\beta\)-stable limit of "nonprojectable and \(\alpha\)-stable" ordinals
  • \(C(\Omega_2+\varphi(\sigma,C(\Omega_22+1,0)^++1)C(\Omega_22+1,0)^+,0)\) is first \(\alpha\) stable up to a nonprojectable and \(\Pi_2\)-reflecting on "nonprojectable and \(\alpha\)-stable" ordinals
  • \(C(\Omega_2+\varphi(\sigma,C(\Omega_22+1,0)^++2),0)\) is fixed point of \(\beta\mapsto\alpha\) stable up to a nonprojectable and \(\beta\)-stable on "nonprojectable and \(\alpha\)-stable" ordinals
  • \(C(\Omega_2+\varphi(\sigma,C(\Omega_22+1,0)^+2),0)\) is first \(\alpha\) stable up to a nonprojectable ordinal that is \(\alpha\)-stable on \(\alpha\)-stables
  • \(C(\Omega_2+\varphi(\sigma,\varepsilon_{C(\Omega_22+1,0)^+2}),0)\) is first \(\alpha\) that is \(\omega+1\)-ply \(\alpha+1\)-stable
  • \(C(\Omega_2+\varphi(\sigma^+,C(\Omega_22+1,0)^++1),0)\) is first \(\alpha\) that is \(\omega+1\)-ply \(\alpha^+\)-stable
  • \(C(\Omega_2+\varphi(C(\Omega_2+C(\Omega_22+1,0)\sigma^+,\sigma),C(\Omega_22+1,0)^++1),0)\) is first \(\alpha\) that is \(\omega+1\)-ply \(\beta\)-stable, where \(\beta\) is next \(\omega\)-ply-stable after \(\alpha\)
  • \(C(\Omega_2+\varphi(C(\Omega_2+\varphi(\sigma^+,C(\Omega_22+1,0)^++1),\sigma),C(\Omega_22+1,0)^++1),0)\) is first \(\alpha\) that is \(\omega+1\)-ply \(\beta\)-stable, where \(\beta\) is next \(\omega+1\)-ply \(\beta^+\)-stable after \(\alpha\)
  • \(C(\Omega_2+\varphi(\sigma_2,C(\Omega_22+1,0)^++1),0)\) is fixed point of \(\alpha\mapsto\omega+1\)-ply "next \(\omega+1\)-ply \(\alpha\)-stable"-stable
  • \(C(\Omega_2+\varphi(\sigma_2,C(\Omega_22+1,0)^++1)+\varphi(C(\Omega_2+\varphi(\sigma_2,C(\Omega_22+1,0)^++1),\sigma_2),\sigma_2^+),0)\) is first \(\omega+1\)-ply "next \(\alpha\) that is \(\omega+1\)-ply \(\alpha\)-stable"-stable
  • \(C(\Omega_2+\varphi(\sigma_3,C(\Omega_22+1,0)^++1)+\varphi(C(\Omega_2+\varphi(\sigma_3,C(\Omega_22+1,0)^++1),\sigma_3),\sigma_3^+),0)\) is first \(\omega+1\)-ply "next \(\omega+1\)-ply 'next \(\alpha\) that is \(\omega+1\)-ply \(\alpha\)-stable'-stable"-stable
  • \(C(\Omega_2+\varphi(C(\Omega_22+1,0)\sigma^+,C(\Omega_22+1,0)^++1),0)\) is first \(\omega+1\)-ply \(\lambda a.a\)-stable
  • \(C(\Omega_2+\varphi(C(\Omega_22+1,0)\sigma^+,C(\Omega_22+1,0)^++1)+1,0)\) is first nonprojectable and \(\lambda a.a\)-stable
  • \(C(\Omega_2+\varphi(C(\Omega_22+1,0)^+,1)+1,0)\) is first nonprojectable and \(\lambda a.a^+\)-stable
  • \(C(\Omega_2+C(\Omega_22+1,0)^{++}+1,0)\) is first nonprojectable and \(\lambda a.a^{++}\)-stable
  • \(C(\Omega_2+C(\Omega_2+\sigma,C(\Omega_22+1,0)),0)\) is fixed point of \(\alpha\mapsto\omega+1\)-ply "next \(\alpha\) many admissibles"-stable
  • \(C(\Omega_2+C(\Omega_2+C(\Omega_22+1,0)\sigma^+,C(\Omega_22+1,0))+1,0)\) is first nonprojectable \(\alpha\) that is "next \(\alpha\) many admissibles"-stable, or nonprojectable and \(\lambda a.C(\Omega+a,a)\)-stable
  • \(C(\Omega_2+C(\Omega_2+C(\Omega_22+1,0)^+,C(\Omega_22+1,0))+1,0)\) is first nonprojectable and "next 'next admissible' many admissibles"-stable, or nonprojectable and \(\lambda a.C(\Omega+a^+,a)\)-stable
  • \(C(\Omega_2+C(\Omega_2+C(\Omega_2+C(\Omega_22+1,0)^+,C(\Omega_22+1,0)),C(\Omega_22+1,0))+1,0)\) is first nonprojectable and "next 'next "next admissible" many admissibles' many admissibles"-stable, or nonprojectable and \(\lambda a.C(\Omega+C(\Omega+a^+,a),a)\)-stable
  • \(C(\Omega_2+C(\Omega_22,C(\Omega_22+1,0)),0)\) is limit of "nonprojectable and \(\alpha\mapsto\)'next \(\alpha\) many admissibles'-stable"
  • \(C(\Omega_2+C(\Omega_22,C(\Omega_22+1,0))+\) \(\varphi(C(\Omega_2+C(\Omega_22,C(\Omega_22+1,0)),C(\Omega_22+1,0)),C(\Omega_22+1,0)^+)+1,0)\) is first nonprojectable and \(\lambda a.C(\Omega_2+\sigma,a)\)-stable
  • \(C(\Omega_2+C(\Omega_22,C(\Omega_22+1,0))^++1,0)\) is first nonprojectable and \(\lambda a.C(\Omega_2+\sigma^+,a)\)-stable
  • \(C(\Omega_2+C^2(\Omega_22,C(\Omega_22+1,0))+\) \(\varphi(C(\Omega_2+C^2(\Omega_22,C(\Omega_22+1,0)),C(\Omega_22,C(\Omega_22+1,0))),C(\Omega_22,C(\Omega_22+1,0))^+),0)\) is first nonprojectable and \(\lambda a.C(\Omega_2+\sigma_2,a)\)-stable
  • \(C(\Omega_2+C^2(\Omega_22,C(\Omega_22+1,0))^++1,0)\) is first nonprojectable and \(\lambda a.C(\Omega_2+\sigma_2^+,a)\)-stable
  • \(C(\Omega_2+C^2(\Omega_22+1,0),0)\) is limit of \(\omega+n\)-ply-stable
  • \(C(\Omega_2+C^2(\Omega_22+1,0)+\sigma^+,0)\) is admissible limit of \(\omega+n\)-ply-stable
  • \(C(\Omega_2+C^2(\Omega_22+1,0)+\sigma_2^+,0)\) is "admissible limit of \(\omega+n\)-ply-stable"-stable
  • \(C(\Omega_2+C^2(\Omega_22+1,0)+C(\Omega_22+1,0)\sigma^++1,0)\) is nonprojectable limit of n-ply-stable
  • \(C(\Omega_2+C^2(\Omega_22+1,0)+C(\Omega_22+1,0)^++1,0)\) is nonprojectable and admissible limit of n-ply-stable
  • \(C(\Omega_2+C^2(\Omega_22+1,0)+C(\Omega_22,C(\Omega_22+1,0))^++1,0)\) is nonprojectable and "admissible limit of n-ply-stable"-stable
  • \(C(\Omega_2+C^2(\Omega_22+1,0)2,0)\) is level-2 limit of \(\omega+n\)-ply-stable
  • \(C(\Omega_2+C^2(\Omega_22+1,0)\sigma^+,0)\) is first \(\omega2\)-ply-stable (also, the beginning of a stable chain of length \(\omega2\))
  • \(C^2(\Omega_2+C^2(\Omega_22+1,0)\sigma^++1,0)\) is first nonprojectable at the end of a stable chain of length \(\omega2\)
  • \(C(\Omega_2+\varepsilon_{C^2(\Omega_22+1,0)^+2},0)\) is first \(\omega2+1\)-ply 1-stable
  • \(C(\Omega_2+C^3(\Omega_22+1,0)\sigma^+,0)\) is first \(\omega3\)-ply-stable
  • \(C(\Omega_2+C(\Omega_22+2,0),0)\) is limit of \(\omega n\)-ply-stable
  • \(C(\Omega_2+C(\Omega_22+2,0)2,0)\) is limit of \(\omega n\)-ply "limit of \(\omega n\)-ply-stable"-stable
  • \(C(\Omega_2+C(\Omega_22+2,0)\sigma^+,0)\) is first \(\omega^2\)-ply-stable
  • \(C(\Omega_2+C(\Omega_22+\sigma,0),0)\) is fixed point of \(\alpha\mapsto\alpha\)-ply-stable
  • \(C(\Omega_2+C(\Omega_22+\sigma,0)\sigma^+,0)\) is first \(\alpha\) that is \(\alpha\)-ply-stable, or \(\lambda a.C(\Omega_2+C(\Omega_22+a,0),a)\)-stable

Up to \(\alpha\)-ply-stable with target \(\alpha\)

Let \(\sigma=\sigma_1=C(\Omega_22,0)\), \(\sigma_{n+1}=C(\Omega_22,\sigma_n)\), and \(\alpha^+=C(\Omega_2,\alpha)\) (next admissible ordinal after \(\alpha\)).

  • \(C(\Omega_2+\varepsilon_{C(\Omega_22+\sigma,0)^+2},0)\) is first \(\alpha\) that is \(\alpha+1\)-ply 1-stable, or \(\lambda a.C(\Omega_2+\varepsilon_{C(\Omega_22+a,0)^+2},a)\)-stable
  • \(C(\Omega_2+C(\Omega_22+\sigma+1,0)\sigma^+,0)\) is first \(\alpha\) that is \(\alpha\omega\)-ply-stable, or \(\lambda a.C(\Omega_2+C(\Omega_22+a+1,0),a)\)-stable
  • \(C(\Omega_2+C(\Omega_22+\sigma^+,0),0)\) is first \(\alpha\) that is \(\alpha^+\)-ply-stable, or \(\lambda a.C(\Omega_2+C(\Omega_22+a^+,0),a)\)-stable
  • \(C(\Omega_2+C(\Omega_22+\sigma_2,0)\sigma^+,0)\) is first "next \(\alpha\) that is \(\alpha\)-ply-stable"-stable, or \(\lambda a.C(\Omega_2+C(\Omega_22+\sigma,0),a)\)-stable
  • \(C(\Omega_2+C(\Omega_22+\sigma_3,0)\sigma^+,0)\) is first "next 'next \(\alpha\) that is \(\alpha\)-ply-stable'-stable"-stable, or \(\lambda a.C(\Omega_2+C(\Omega_22+\sigma_2,0),a)\)-stable
  • \(C(\Omega_2+C(\Omega_22+C(\Omega_22+1,0),0),0)\) is limit of n-ply "next \(\alpha\) that is \(\alpha\)-ply-stable"-stable
  • \(C(\Omega_2+C(\Omega_22+C(\Omega_22+1,0),0)\sigma^+,0)\) is first \(\omega\)-ply "\(\alpha\) that is \(\alpha\)-ply-stable"-stable
  • \(C(\Omega_2+C(\Omega_22+C(\Omega_22+1,0),0)\sigma^++1,0)\) is first nonprojectable \(\alpha\) that is \(\alpha\)-ply-stable
  • \(C(\Omega_2+C(\Omega_22+C(\Omega_22+2,0),0)\sigma^+,0)\) is first \(\omega^2\)-ply "\(\alpha\) that is \(\alpha\)-ply-stable"-stable
  • \(C(\Omega_2+C(\Omega_22+C(\Omega_22+\sigma,0),0),0)\) is fixed point of \(\beta\mapsto\beta\)-ply "\(\alpha\) that is \(\alpha\)-ply-stable"-stable
  • \(C(\Omega_2+C(\Omega_22+C(\Omega_22+\sigma,0),0)\sigma^+,0)\) is first \(\beta\) that is \(\beta\)-ply "\(\alpha\) that is \(\alpha\)-ply-stable"-stable, or \(\lambda a.C(\Omega_2+C(\Omega_22+C(\Omega_22+a,0),0),a)\)-stable
  • \(C(\Omega_2+C(\Omega_22+C(\Omega_22+\sigma^+,0),0),0)\) is first \(\beta\) that is \(\beta^+\)-ply "\(\alpha\) that is \(\alpha\)-ply-stable"-stable, or \(\lambda a.C(\Omega_2+C(\Omega_22+C(\Omega_22+a^+,0),0),a)\)-stable
  • \(C(\Omega_2+C(\Omega_22+C(\Omega_22+\sigma_2^+,0),0),0)\) is first "next \(\beta\) that is \(\beta^+\)-ply '\(\alpha\) that is \(\alpha\)-ply-stable'-stable"-stable, or \(\lambda a.C(\Omega_2+C(\Omega_22+C(\Omega_22+\sigma^+,0),0),a)\)-stable
  • \(C(\Omega_2+C(\Omega_22+C(\Omega_22+C(\Omega_22+1,0),0),0)\sigma^+,0)\) is first \(\omega\)-ply "next \(\beta\) that is \(\beta\)-ply '\(\alpha\) that is \(\alpha\)-ply-stable'-stable"-stable
  • \(C(\Omega_2+C(\Omega_22+C(\Omega_22+C(\Omega_22+\sigma,0),0),0)\sigma^+,0)\) is first \(\alpha_0\) that is \(\alpha_0\)-ply-stable up to another \(\alpha_1\) that is \(\alpha_1\)-ply-stable up to another \(\alpha_2\) that is \(\alpha_2\)-ply-stable, or \(\lambda a.C(\Omega_2+C(\Omega_22+C(\Omega_22+C(\Omega_22+a,0),0),0),a)\)-stable
  • \(C(\Omega_2+C(\Omega_23,0),0)\) is limit of "\(\alpha_0\) that is \(\alpha_0\)-ply-stable up to another \(\alpha_1\) that is \(\alpha_1\)-ply-stable up to another ... \(\alpha_n\) that is \(\alpha_n\)-ply-stable" (or "\(\alpha_0\) that is \(\alpha_n\)-ply-stable" for short)

Notice that \(C(\Omega_2+C(\Omega_22+C(\Omega_23,0),0),0)\) is not standard, nor \(C(\Omega_2+C(\Omega_23,0)+C(\Omega_22+C(\Omega_23,0),0),0)\). The term for \(C(\Omega_2+C(\Omega_23,0)+C(\Omega_22+C(\Omega_22+C(\Omega_22+\cdots,0),0),0),0)\) should be \(C(\Omega_2+C(\Omega_23,0)2,0)\). At this point, \(C(\Omega_23,0)\) seems to work as the fixed point of \(\alpha\mapsto C(\Omega_22+\alpha,0)\).

  • \(C(\Omega_2+C(\Omega_23,0)+\sigma^+,0)\) is admissible limit of "\(\alpha_0\) that is \(\alpha_n\)-ply-stable"
  • \(C(\Omega_2+C(\Omega_23,0)+\sigma_2^+,0)\) is "next admissible limit of '\(\alpha_0\) that is \(\alpha_n\)-ply-stable'"-stable
  • \(C(\Omega_2+C(\Omega_23,0)+C(\Omega_22+1,0)\sigma^+,0)\) is \(\omega\)-ply "limit of \(\alpha_0\) that is \(\alpha_n\)-ply-stable"-stable
  • \(C(\Omega_2+C(\Omega_23,0)+C(\Omega_22+\sigma,0)\sigma^+,0)\) is \(\beta\) that is \(\beta\)-ply "limit of \(\alpha_0\) that is \(\alpha_n\)-ply-stable"-stable
  • \(C(\Omega_2+C(\Omega_23,0)+C(\Omega_22+C(\Omega_22+\sigma,0),0)\sigma^+,0)\) is \(\beta_0\) that is \(\beta_0\)-ply "\(\beta_1\) that is \(\beta_1\)-ply 'limit of \(\alpha_0\) that is \(\alpha_n\)-ply-stable'-stable"-stable
  • \(C(\Omega_2+C(\Omega_23,0)2,0)\) is limit of \(\beta_0\) that is \(\beta_n\)-ply "limit of \(\alpha_0\) that is \(\alpha_n\)-ply-stable"-stable

\(C(\Omega_2+C(\Omega_23,0)\sigma^+,0)\) is first \(\alpha_0\) stable up to \(\alpha_\omega\)-ply, which means a set \(\{\alpha_n|n<\omega\}\) such that \(\alpha_n<\alpha_{n+1}\) and \(\alpha_n\) is \(\alpha_n\)-ply \(\alpha_{n+1}\)-stable for all \(n<\omega\). Then \(\alpha_0\) is \(\alpha_\omega=\sup\{\alpha_n|n<\omega\}=\alpha_0+\alpha_1+\cdots+\alpha_n+\cdots\)-ply-stable. For all \(\gamma<\alpha_\omega\), there is \(\alpha_n>\gamma\) such that \(\alpha_0\) is \(\alpha_n\)-stable, so \(\alpha_0\) is actually \(\gamma\)-stable for all \(\gamma<\alpha_\omega\), then \(\alpha_0\) is \(\alpha_\omega\)-stable. As a result, \(\alpha_0\) is \(\alpha_\omega\)-ply-stable up to a target \(\alpha_\omega\).

Up to a \(\Pi_2\)-reflecting-target

Let \(\sigma=\sigma_1=C(\Omega_22,0)\), \(\sigma_{n+1}=C(\Omega_22,\sigma_n)\), \(\rho=\rho_1=C(\Omega_23,0)\), \(\rho_{n+1}=C(\Omega_23,\rho_n)\), and \(\alpha^+=C(\Omega_2,\alpha)\) (next admissible ordinal after \(\alpha\)).

  • \(C(\Omega_2+\rho\sigma^+,0)\) is first \(\alpha_0\) that is \(\alpha_\omega\)-ply-stable
  • \(C(\Omega_2+\rho\sigma^++C(\Omega_2+\rho\sigma^+,0),0)\) is first such \(\alpha_1\) in the above \(\{\alpha_n|n<\omega\}\) ("\(\alpha_1\)-target" for short), which is again another "\(\alpha_0\) that is \(\alpha_\omega\)-ply-stable" (but it share the same \(\alpha_\omega\))
  • \(C(\Omega_2+\rho\sigma^++\sigma,0)\) is first \(\alpha_\omega\)-target
  • \(C(\Omega_2+\rho\sigma^++\sigma^+,0)\) is first admissible limit of \(\alpha_\omega\)-targets
  • \(C(\Omega_2+\rho\sigma^++\sigma_2^+,0)\) is first "next admissible limit of \(\alpha_\omega\)-targets"-stable
  • \(C(\Omega_2+\rho\sigma^++C(\Omega_22+1,0)\sigma^+,0)\) is first \(\omega\)-ply "next admissible limit of \(\alpha_\omega\)-targets"-stable
  • \(C(\Omega_2+\rho\sigma^++C(\Omega_22+\sigma,0)\sigma^+,0)\) is first \(\alpha\) that is \(\alpha\)-ply "next admissible limit of \(\alpha_\omega\)-targets"-stable
  • \(C(\Omega_2+\rho\sigma^++C(\Omega_22+C(\Omega_22+\sigma,0),0)\sigma^+,0)\) is first \(\alpha\) that is \(\alpha\)-ply "next \(\beta\) that is \(\beta\)-ply 'next admissible limit of \(\alpha_\omega\)-targets'-stable"-stable
  • \(C(\Omega_2+\rho\sigma^+2,0)\) is first "\(\alpha_\omega\)-target limit of \(\alpha_\omega\)-targets"-stable
  • \(C(\Omega_2+\rho\sigma^+2+\sigma,0)\) is first \(\alpha_\omega\)-target limit of \(\alpha_\omega\)-targets
  • \(C(\Omega_2+\rho\sigma^+3+\sigma,0)\) is first level-2 \(\alpha_\omega\)-target limit of \(\alpha_\omega\)-targets
  • \(C(\Omega_2+\rho^2\sigma^++\sigma,0)\) is first level-(1,0) \(\alpha_\omega\)-target limit of \(\alpha_\omega\)-targets
  • \(C(\Omega_2+\rho^++\sigma,0)\) is first \(\alpha_\omega\)-target and admissible
  • \(C(\Omega_2+(\rho^+)^2+\sigma,0)\) is first \(\alpha_\omega\)-target and recursively Mahlo
  • \(C(\Omega_2+(\rho^+)^{\rho^+}+\sigma,0)\) is first \(\alpha_\omega\)-target and \(\Pi_3\)-reflecting
  • \(C(\Omega_2+\varepsilon_{\rho^+2}+\sigma,0)\) is first \(\alpha_\omega\)-target and 1-stable
  • \(C(\Omega_2+\rho^{++}+\sigma,0)\) is first \(\alpha_\omega\)-target and \(\lambda a.a^{++}\)-stable
  • \(C(\Omega_2+C(\Omega_22+1,\rho)\sigma^+,0)\) is first \(\alpha_0\) that is \(\alpha_\omega+\omega\)-ply-stable.

Note that the notion of \(\beta\)-target means a stable chain of length \(\beta\), but a \(\beta\)-target can be larger than \(\beta\). e.g. \(\omega\)-target is identical to nonprojectable. However, a \(\alpha_\omega\)-target equals \(\alpha_\omega\).

  • \(C(\Omega_2+C(\Omega_22+1,\rho)\sigma^++\sigma,0)\) is first \(\alpha_\omega\)-target and \(\omega\)-ply-stable
  • \(C(\Omega_2+C(\Omega_22+\sigma,\rho)\sigma^+,0)\) is first \(\alpha_0\) that is \(\alpha_\omega+\alpha_0\)-ply-stable
  • \(C(\Omega_2+C(\Omega_22+\sigma,\rho)\sigma^++\sigma,0)\) is first \(\alpha_\omega\)-target and \(\alpha_0\)-ply-stable
  • \(C(\Omega_2+C(\Omega_22+C(\Omega_22+\sigma,0),\rho)\sigma^+,0)\) is first \(\alpha_0\) that is \(\alpha_\omega+\alpha_1\)-ply-stable
  • \(C(\Omega_2+\rho_2,0)\) is limit of "\(\alpha_0\) that is \(\alpha_\omega+\alpha_n\)-ply-stable"
  • \(C(\Omega_2+\rho_2+C(\Omega_22+\sigma,0)\sigma^+,0)\) is first \(\beta\) that is \(\beta\)-ply "limit of \(\alpha_0\) that is \(\alpha_\omega+\alpha_n\)-ply-stable"-stable
  • \(C(\Omega_2+\rho_2+\rho\sigma^+,0)\) is first \(\beta_0\) that is \(\beta_\omega\)-ply "limit of \(\alpha_0\) that is \(\alpha_\omega+\alpha_n\)-ply-stable"-stable
  • \(C(\Omega_2+\rho_22,0)\) is limit of "\(\beta_0\) that is \(\beta_\omega+\beta_n\)-ply 'limit of \(\alpha_0\) that is \(\alpha_\omega+\alpha_n\)-ply-stable'-stable"
  • \(C(\Omega_2+\rho_2\sigma^+,0)\) is first \(\alpha_0\) that is \(\alpha_\omega2\)-ply-stable, or \(\alpha_0\) that is \(\alpha_{\omega+1}\)-ply-stable
  • \(C(\Omega_2+\rho_2\sigma^++\sigma,0)\) is first \(\alpha_\omega\)-target that is \(\alpha_\omega\)-ply-stable
  • \(C(\Omega_2+\varepsilon_{\rho_2^+2},0)\) is first \(\alpha_0\) that is \(\alpha_\omega2+1\)-ply 1-stable
  • \(C(\Omega_2+C(\Omega_22+\sigma,\rho_2)\sigma^+,0)\) is first \(\alpha_0\) that is \(\alpha_\omega2+\alpha_0\)-ply-stable
  • \(C(\Omega_2+C(\Omega_22+C(\Omega_22+\sigma,0),\rho_2)\sigma^+,0)\) is first \(\alpha_0\) that is \(\alpha_\omega2+\alpha_1\)-ply-stable
  • \(C(\Omega_2+\rho_3\sigma^+,0)\) is first \(\alpha_0\) that is \(\alpha_\omega3\)-ply-stable
  • \(C(\Omega_2+C(\Omega_23+1,0)\sigma^+,0)\) is first \(\alpha_0\) that is \(\alpha_\omega\omega\)-ply-stable, or \(\alpha_0\) that is \(\alpha_{\omega+1}+\alpha_\omega\omega\)-ply-stable
  • \(C(\Omega_2+C(\Omega_23+1,0)\sigma^++\sigma,0)\) is first \(\alpha_\omega\)-target that is \(\alpha_\omega\omega\)-ply-stable
  • \(C(\Omega_2+C(\Omega_23+1,0)\sigma^++C(\Omega_2+C(\Omega_23+1,0)\sigma^++\sigma,0)+1,\) \(C(\Omega_2+C(\Omega_23+1,0)\sigma^++\sigma,0))\) is first \(\alpha_\omega\omega\)-target
  • \(C(\Omega_2+C(\Omega_23+\sigma,0)\sigma^+,0)\) is first \(\alpha_0\) that is \(\alpha_\omega\alpha_0\)-ply-stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+\sigma,0),0)\sigma^+,0)\) is first \(\alpha_0\) that is \(\alpha_\omega\alpha_1\)-ply-stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+\rho,0),0)\sigma^+,0)\) is first \(\alpha_0\) that is \(\alpha_\omega^2\)-ply-stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+\rho,0)2,0)\sigma^+,0)\) is first \(\alpha_0\) that is \(\alpha_\omega^3\)-ply-stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+\rho,0)^+,0)+\sigma,0)\) is first \(\alpha_\omega\)-target \(\beta\) that is \(\beta^+\)-ply-stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_2+C(\Omega_22,C(\Omega_22+\rho,0)),C(\Omega_22+\rho,0)),0)\sigma^++\sigma,0)\) is first \(\alpha_\omega\)-target \(\beta\) that is \(C(\Omega_2+\sigma,\beta)\)-ply-stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_2+C(\Omega_22+C(\Omega_22+\rho,0),C(\Omega_22+\rho,0))\) \(,C(\Omega_22+\rho,0)),0)\sigma^++\sigma,0)\) is first \(\alpha_\omega\)-target \(\beta\) that is \(C(\Omega_2+C(\Omega_22+\beta,0),\beta)\)-ply-stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_2+C(\Omega_22+C(\Omega_22,C(\Omega_22+\rho,0)),C(\Omega_22+\rho,0))\) \(,C(\Omega_22+\rho,0)),0)\sigma^++\sigma,0)\) is first \(\alpha_\omega\)-target \(\beta\) that is \(C(\Omega_2+C(\Omega_22+\sigma,0),\beta)\)-ply-stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_2+\rho,C(\Omega_22+\rho,0)),0)\sigma^++\sigma,0)\) is first \(\alpha_\omega\)-target \(\beta\) that is \(C(\Omega_2+\rho,\beta)\)-ply-stable, i.e. as a limit of "\(\beta_0\) that is \(\beta_n\)-ply-stable"
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_2+\rho,C(\Omega_22+\rho,0))+\sigma^+,0),0)\) is first \(\alpha_0\) that is \(\alpha_{\omega2}\)-ply-stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_2+\rho,C(\Omega_22+\rho,C(\Omega_22+\rho,0)))+\sigma^+,0),0)\) is first \(\alpha_0\) that is \(\alpha_{\omega3}\)-ply-stable. See the lower but analogous comparisons for erratic expressions between the previous line and here.
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+\rho+1,0),0)\sigma^+,0)\) is first \(\alpha_0\) that is \(\alpha_{\omega^2}\)-ply-stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+\rho+1,0),0)\sigma^++\sigma+1,0)\) is first \(\alpha_{\omega^2}\)-target
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+\rho+2,0),0)\sigma^+,0)\) is first \(\alpha_0\) that is \(\alpha_{\omega^3}\)-ply-stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+\rho+2,0),0)\sigma^++\sigma+2,0)\) is first \(\alpha_{\omega^3}\)-target
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+\rho+\sigma,0),0)\sigma^+,0)\) is first \(\alpha_0\) that is \(\alpha_{\alpha_0}\)-ply-stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+\rho+C(\Omega_22+\sigma,0),0),0)\sigma^+,0)\) is first \(\alpha_0\) that is \(\alpha_{\alpha_1}\)-ply-stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+\rho+C(\Omega_22+\rho,0),0),0)\sigma^+,0)\) is first \(\alpha_0\) that is \(\alpha_{\alpha_\omega}\)-ply-stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+\rho+C(\Omega_22+\rho+C(\Omega_22+\rho,0),0),0),0)\sigma^+,0)\) is first \(\alpha_0\) that is \(\alpha_{\alpha_{\alpha_\omega}}\)-ply-stable

Now we start to use another notion of -ply-stable. If a stable chain begins with C(Ω2+a,b) (since it is smaller than the least stable ordinal \(\sigma\), but it is admissible, we have \(a<\Omega_2\)), then we use the end of the chain (in terms of a, b and C) to denote the length of this chain.

Fro example, the least ordinal C(Ω2+a,b) stable up to a nonprojectable is C(Ω2+a+1,b)-stable, with chain length \(\omega\). The next one is C(Ω2+a,C(Ω2+a,b)), the next is C(Ω2+a,C(Ω2+a,C(Ω2+a,b))), and so on. The least nonprojectable is thus C(Ω2+a+1,b), mark the end of the chain.

Then C(Ω2+a+2,b)-stable is identical to \(\omega^2\)-ply-stable, C(Ω2+a+C(Ω2+a,b),b)-stable is identical to \(\alpha_1\)-ply-stable, \(C(\Omega_2+a+\sigma,b)\)-stable is identical to \(\alpha_\omega\)-ply-stable, \(C(\Omega_2+a+\sigma+1,b)\)-stable is identical to \(\alpha_{\omega^2}\)-ply-stable, \(C(\Omega_2+a+\sigma+C(\Omega_2+a,b),b)\)-stable is identical to \(\alpha_{\alpha_0}\)-ply-stable, \(C(\Omega_2+a+\sigma+C(\Omega_2+a+\sigma,b),b)\)-stable is identical to \(\alpha_{\alpha_\omega}\)-ply-stable, etc.

  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+\rho2,0),0)\sigma^+,0)\) is first C(Ω2+a,b) that is \(C(\Omega_2+a+\sigma2,b)\)-stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+\rho2+1,0),0)\sigma^+,0)\) is first C(Ω2+a,b) that is \(C(\Omega_2+a+\sigma2+1,b)\)-stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+\rho3,0),0)\sigma^+,0)\) is first C(Ω2+a,b) that is \(C(\Omega_2+a+\sigma3,b)\)-stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+\rho^2,0),0)\sigma^+,0)\) is first C(Ω2+a,b) that is \(C(\Omega_2+a+\sigma^2,b)\)-stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+\rho^\rho,0),0)\sigma^+,0)\) is first C(Ω2+a,b) that is \(C(\Omega_2+a+\sigma^\sigma,b)\)-stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+\rho^+,0),0),0)\) is first C(Ω2+a,b) that is \(C(\Omega_2+a+\sigma^+,b)\)-stable

\(C(\Omega_2+C(\Omega_23+C(\Omega_22+\rho^+,0),0)+\sigma^+,0)\) is thus the end of the stable chain. It seems to be an admissible limit of the chained ordinals, but this concept is too weak, since every nonprojectable and 1-stable already has this property. Instead, we need a \(\Pi_2\)-reflecting. This ordinal is thus first \(\alpha\) that is \(\Pi_2\)-reflecting onto \(\{\beta<\alpha|L_\beta\prec_{\Sigma_1}L_\alpha\}\). Also it is first \(\alpha\) that \(L_\alpha\cap\mathcal P(\omega)\) is a model of \(\Delta^1_3\text{-CA}_0+\text{BI}\). Also it is first \(\Delta_2\)-gap ordinal.

Up to a 1[2]stable

In the following comparisons, I'll neglect unconfirm differences between "limit of n-ply-stable" and "\(\omega\)-ply-stable", between "limit point of \(\alpha\mapsto\alpha\)-ply-stable" and "\(\alpha\) that is \(\alpha\)-ply-stable", etc.

Let \(\alpha^+=C(\Omega_2,\alpha)\) (next admissible ordinal after \(\alpha\)), \(\alpha^*=C(\Omega_22,\alpha)\), and \(\rho=C(\Omega_23,0)\). So the previous \(\sigma=0^*\).

  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+\rho^+,0),0)+0^{*+},0)\) is first \(\alpha\) that is \(\Pi_2\)-reflecting onto \(\{\beta<\alpha|L_\beta\prec_{\Sigma_1}L_\alpha\}\)
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+\rho^+,0),0)+0^{*+}+1,0)\) is limit of \(\alpha\)'s that are \(\Pi_2\)-reflecting on \(\{\beta<\alpha|L_\beta\prec_{\Sigma_1}L_\alpha\}\)
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+\rho^+,0),0)+0^{*+}2,0)\) is first admissible limit of \(\alpha\)'s that are \(\Pi_2\)-reflecting on \(\{\beta<\alpha|L_\beta\prec_{\Sigma_1}L_\alpha\}\)
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+\rho^+,0),0)+\) \(C(\Omega_2+C(\Omega_23+C(\Omega_22+\rho^+,0),0),0^*),0)\) is first C(Ω2+a,b) that is \(C(\Omega_2+a+0^{*+},b)\)-stable and also limit of \(\alpha\)'s that are \(\Pi_2\)-reflecting onto \(\{\beta<\alpha|L_\beta\prec_{\Sigma_1}L_\alpha\}\)
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+\rho^+,0),0)+\) \(C(\Omega_2+C(\Omega_23+C(\Omega_22+\rho^+,0),0),0^*)+0^{*+},0)\) is first \(\alpha\) that is \(\Pi_2\)-reflecting onto \(\{\beta<\alpha|L_\beta\prec_{\Sigma_1}L_\alpha\}\), and also limit of \(\gamma\)'s that are \(\Pi_2\)-reflecting onto \(\{\beta<\gamma|L_\beta\prec_{\Sigma_1}L_\gamma\}\)
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+\rho^+,0),0)+\) \(C(\Omega_2+C(\Omega_23+C(\Omega_22+\rho^+,0),0),0^*)0^{*+},0)\) is first \(\Pi_2\)-reflecting onto \(\alpha\)'s that are \(\Pi_2\)-reflecting onto \(\{\beta<\alpha|L_\beta\prec_{\Sigma_1}L_\alpha\}\)
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+\rho^+,0),0)+\) \(C(\Omega_2+C(\Omega_23+C(\Omega_22+\rho^+,0),0),0^*)^{0^{*+}},0)\) is first \(\Pi_3\)-reflecting onto C(Ω2+a,b)'s that are \(C(\Omega_2+a+0^{*+},b)\)-stable, i.e. \(\Pi_3\)-reflecting onto \(\alpha\)'s that are \(\Pi_2\)-reflecting onto \(\{\beta<\alpha|L_\beta\prec_{\Sigma_1}L_\alpha\}\)
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+\rho^+,0),0)+\) \(\varphi(C(\Omega_2+C(\Omega_23+C(\Omega_22+\rho^+,0),0),0^*),\varepsilon_{0^{*+}2}),0)\) is first \(\lambda a.C(\Omega_2+C(\Omega_23+C(\Omega_22+\rho^+,0),0)+0^{*+},a)+1\)-stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+\rho^+,0),0)+0^{**+},0)\) is first \(\lambda a.C(\Omega_2+C(\Omega_23+C(\Omega_22+\rho^+,0),0)+0^{*+}2,a)\)-stable, i.e. "next admissible limit of \(\alpha\)'s that are \(\Pi_2\)-reflecting onto \(\{\beta<\alpha|L_\beta\prec_{\Sigma_1}L_\alpha\}\)"-stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+\rho^+,0),0)+C(\Omega_22+1,0),0)\) is first \(\omega\)-ply "(\(\alpha\) that is \(\Pi_2\)-reflecting onto \(\{\beta<\alpha|L_\beta\prec_{\Sigma_1}L_\alpha\}\))+1"-stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+\rho^+,0),0)+\rho,0)\) is first "next \(\alpha_0\) that is \(\alpha_\omega\)-ply-stable"-ply "(\(\alpha\) that is \(\Pi_2\)-reflecting onto \(\{\beta<\alpha|L_\beta\prec_{\Sigma_1}L_\alpha\}\))+1"-stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+\rho^+,0),0)+C(\Omega_23+C(\Omega_22+\rho+1,0),0),0)\) is first "next \(\alpha_0\) that is \(\alpha_{\omega^2}\)-ply-stable"-ply "(\(\alpha\) that is \(\Pi_2\)-reflecting onto \(\{\beta<\alpha|L_\beta\prec_{\Sigma_1}L_\alpha\}\))+1"-stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+\rho^+,0),0)2+0^{*+},0)\) is first \(\alpha\) that is \(\Pi_2\)-reflecting onto \(\{\beta<\alpha|L_\beta\prec_{\Sigma_1}L_\alpha\}\) and also limit of \(\gamma\)'s that are \(\Pi_2\)-reflecting onto \(\{\beta<\gamma|L_\beta\prec_{\Sigma_1}L_\gamma\}\)
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+\rho^+,0),0)^++0^{*+},0)\) is first \(\alpha\) that is \(\Pi_2\)-reflecting onto \(\{\beta<\alpha|L_\beta\prec_{\Sigma_1}L_\alpha\}\) and also \(\Pi_3\)-reflecting
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+\rho^+,0),0)^+2+0^{*+},0)\) is first \(\alpha\) that is \(\Pi_2\)-reflecting onto \(\{\beta<\alpha|L_\beta\prec_{\Sigma_1}L_\alpha\}\), and also \(\Pi_3\)-reflecting, and also limit of \(\gamma\)'s that are \(\Pi_2\)-reflecting onto \(\{\beta<\gamma|L_\beta\prec_{\Sigma_1}L_\gamma\}\) and also \(\Pi_3\)-reflecting
  • \(C(\Omega_2+(C(\Omega_23+C(\Omega_22+\rho^+,0),0)^+)^2+0^{*+},0)\) is first \(\alpha\) that is \(\Pi_2\)-reflecting onto \(\{\beta<\alpha|L_\beta\prec_{\Sigma_1}L_\alpha\}\), and also \(\Pi_3\)-reflecting, and also \(\Pi_2\)-reflecting onto \(\gamma\)'s that are \(\Pi_2\)-reflecting onto \(\{\beta<\gamma|L_\beta\prec_{\Sigma_1}L_\gamma\}\) and also \(\Pi_3\)-reflecting
  • \(C(\Omega_2+(C(\Omega_23+C(\Omega_22+\rho^+,0),0)^+)^{C(\Omega_23+C(\Omega_22+\rho^+,0),0)^+}+0^{*+},0)\) is first \(\alpha\) that is \(\Pi_2\)-reflecting onto \(\{\beta<\alpha|L_\beta\prec_{\Sigma_1}L_\alpha\}\), and also \(\Pi_3\)-reflecting onto \(\Pi_3\)-reflectings
  • \(C(\Omega_2+\varepsilon_{C(\Omega_23+C(\Omega_22+\rho^+,0),0)^+2}+0^{*+},0)\) is first \(\alpha\) that is \(\Pi_2\)-reflecting onto \(\{\beta<\alpha|L_\beta\prec_{\Sigma_1}L_\alpha\}\), and also 1-stable
  • \(C(\Omega_2+C(\Omega_22+1,C(\Omega_23+C(\Omega_22+\rho^+,0),0))+0^{*+},0)\) is first \(\alpha\) that is \(\Pi_2\)-reflecting onto \(\{\beta<\alpha|L_\beta\prec_{\Sigma_1}L_\alpha\}\), and also \(\omega\)-ply-stable
  • \(C(\Omega_2+C(\Omega_23,C(\Omega_23+C(\Omega_22+\rho^+,0),0))+0^{*+},0)\) is first \(\alpha\) that is \(\Pi_2\)-reflecting onto \(\{\beta<\alpha|L_\beta\prec_{\Sigma_1}L_\alpha\}\), and also "next \(\alpha_\omega\)-target after \(\min\{\beta<\alpha|L_\beta\prec_{\Sigma_1}L_\alpha\}\)"-ply-stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+\rho^+,0),C(\Omega_23+C(\Omega_22+\rho^+,0),0))+0^{*+},0)\) is first \(\alpha\) that is \(\Pi_2\)-reflecting onto \(\{\beta<\alpha|L_\beta\prec_{\Sigma_1}L_\alpha\}\), and also "next \(\gamma\) that is \(\Pi_2\)-reflecting onto \(\{\beta<\gamma|L_\beta\prec_{\Sigma_1}L_\gamma\}\) after \(\min\{\beta<\alpha|L_\beta\prec_{\Sigma_1}L_\alpha\}\)"-ply-stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+\rho^+,0)+1,0)+0^{*+},0)\) is first \(\alpha\) that is \(\Pi_2\)-reflecting onto \(\{\beta<\alpha|L_\beta\prec_{\Sigma_1}L_\alpha\}\), and also "next \(\omega\) many \(\gamma\)'s that are \(\Pi_2\)-reflecting onto \(\{\beta<\gamma|L_\beta\prec_{\Sigma_1}L_\gamma\}\) after \(\min\{\beta<\alpha|L_\beta\prec_{\Sigma_1}L_\alpha\}\)"-ply-stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+\rho^+,0)+\) \(C(\Omega_22+C(C(\Omega_23+C(\Omega_22+\rho^+,0),0),\rho),0),0)+0^{*+},0)\) is first \(\alpha\) that is \(\Pi_2\)-reflecting onto \(\{\beta<\alpha|L_\beta\prec_{\Sigma_1}L_\alpha\}\), and also "next 'next \(\gamma\) that is \(\Pi_2\)-reflecting onto \(\{\beta<\gamma|L_\beta\prec_{\Sigma_1}L_\gamma\}\) after \(\min\{\beta<\alpha|L_\beta\prec_{\Sigma_1}L_\alpha\}\)' many \(\gamma\)'s that are \(\Pi_2\)-reflecting onto \(\{\beta<\gamma|L_\beta\prec_{\Sigma_1}L_\gamma\}\) after \(\min\{\beta<\alpha|L_\beta\prec_{\Sigma_1}L_\alpha\}\)"-ply-stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+\rho^+,0)2,0)+0^{*+},0)\) is first \(\alpha\) that is \(\Pi_2\)-reflecting onto \(\{\beta<\alpha|L_\beta\prec_{\Sigma_1}L_\alpha\}\), and also \(\alpha\)-ply-stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+\rho^+,0)3,0)+0^{*+},0)\) is first \(\alpha\) that is \(\Pi_2\)-reflecting onto \(\{\beta<\alpha|L_\beta\prec_{\Sigma_1}L_\alpha\}\), and also \(\alpha2\)-ply-stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+\rho^+,0)^+,0)+0^{*+},0)\) is first \(\alpha\) that is \(\Pi_2\)-reflecting onto \(\{\beta<\alpha|L_\beta\prec_{\Sigma_1}L_\alpha\}\), and also \(\alpha^+\)-ply-stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_2+\rho,C(\Omega_22+\rho^+,0)),0)+0^{*+},0)\) is first \(\alpha\) that is \(\Pi_2\)-reflecting onto \(\{\beta<\alpha|L_\beta\prec_{\Sigma_1}L_\alpha\}\), and also "next \(\gamma_0\) that is \(\gamma_\omega\)-ply-stable after \(\alpha\)"-ply-stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+\rho^+,C(\Omega_22+\rho^+,0)),0),0)\) is first "next next \(\alpha\)'s that are \(\Pi_2\)-reflecting onto \(\{\beta<\alpha|L_\beta\prec_{\Sigma_1}L_\alpha\}\)"-stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+\rho^++1,0),0),0)\) is first "next \(\omega\) many \(\alpha\)'s that are \(\Pi_2\)-reflecting onto \(\{\beta<\alpha|L_\beta\prec_{\Sigma_1}L_\alpha\}\)"-stable, i.e. C(Ω2+a,b) that is \(C(\Omega_2+a+0^{*+}+1,b)\)-stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+\rho^++1,0),0)+0^{*+}+1,0)\) is first \(\alpha\) that is limit of \(\Pi_2\)-reflectings onto \(\{\beta<\alpha|L_\beta\prec_{\Sigma_1}L_\alpha\}\)
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+\rho^+2,0),0),0)\) is first C(Ω2+a,b) that is \(C(\Omega_2+a+0^{*+}2,b)\)-stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+\rho^+2,0),0)+0^{*+}2,0)\) is first \(\alpha\) that is \(\Pi_2\)-reflecting onto \(\Pi_2\)-reflectings onto \(\{\beta<\alpha|L_\beta\prec_{\Sigma_1}L_\alpha\}\)
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+(\rho^+)^2,0),0)+(0^{*+})^2,0)\) is first \(\alpha\) that is \(\Pi_3\)-reflectings onto \(\{\beta<\alpha|L_\beta\prec_{\Sigma_1}L_\alpha\}\)
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+\varepsilon_{\rho^+2},0),0)+\varepsilon_{0^{*+}2},0)\) is first \(\alpha\) that is 1-stable onto \(\{\beta<\alpha|L_\beta\prec_{\Sigma_1}L_\alpha\}\)

Beyond 1[2]stable

Define another notion from reflecting-onto or stable-onto: \(\alpha\) is \(\beta\)-\(\Pi_n[\eta]\)reflecting if for all \(\eta'<\eta\), \(\alpha\) is \(\beta\)-\(\Pi_n\)-reflecting onto \(\{\gamma<\alpha|\exists\alpha'\le\alpha(\gamma\text{ is }\alpha'[\eta']\text{stable}\land\gamma+\alpha'=\alpha)\}\); an ordinal is \(\beta[\eta]\)stable if it is \(\beta'\)-\(\Pi_n[\eta]\)reflecting for all \(\beta'<\beta\) and \(n<\omega\). So every ordinal is automatically \(\beta\)-\(\Pi_n[0]\)reflecting and \(\beta[0]\)stable for all \(\beta\) and n; \(\beta\)-\(\Pi_n[1]\)reflecting and \(\beta[1]\)stable are, respectively, the normal \(\beta\)-\(\Pi_n\)-reflecting and \(\beta\)-stable. \(\beta\)[n]stable may relate to \((\beta,n)\)-stable, but more precise correspondence is unclear. "\((\beta,2)\)-stable for some \(\beta\)" seems stronger than 1[2]stable, and the least ordinal with such property is the \(\zeta\) in ITTM (the supremum of all eventually writable ordinals).

Let \(\alpha^+=C(\Omega_2,\alpha)\) (next admissible ordinal after \(\alpha\)), \(\alpha^*=C(\Omega_22,\alpha)\), and \(\alpha^{+_n}=C(\Omega_2n,\alpha)\). So the previous \(\sigma=0^*\) and \(\rho=0^{+_3}\).

  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+\varepsilon_{0^{+_3+}2},0),0)+\varepsilon_{0^{*+}2},0)\) is first 1[2]stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+\varphi(0^*,0^{+_3+}2),0),0),0)\) is first C(Ω2+a,b) that is \(C(\Omega_2+a+\varphi(C(\Omega_2+a,b),0^{*+}2),b)\)-stable, i.e. \(\alpha\) that is "\(\alpha\)[2]stable"-stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+\varphi(0^{+_3},0^{+_3+}2),0),0),0)\) is first C(Ω2+a,b) that is \(C(\Omega_2+a+\varphi(0^*,0^{*+}2),b)\)-stable, i.e. "\(\alpha\) that is \(\alpha\)[2]stable"-stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+\varphi(0^{+_3},0^{+_3+}2),0),0)+\varphi(0^*,0^{*+}2),0)\) is first \(\alpha\) that is \(\alpha\)[2]stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+0^{+_3++},0),0)+0^{*++},0)\) is first "next next admissible"[2]stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+C(\Omega_2+0^*,0^{+_3}),0),0),0)\) is first C(Ω2+a,b) that is \(C(\Omega_2+a+C(\Omega_2+C(\Omega_2+a,b),0^*),b)\)-stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+C(\Omega_2+0^{+_3},0^{+_3}),0),0),0)\) is first C(Ω2+a,b) that is \(C(\Omega_2+a+C(\Omega_2+0^*,0^*),b)\)-stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+C(\Omega_2+0^{+_3},0^{+_3}),0),0)+C(\Omega_2+0^*,0^*),0)\) is first \(\alpha\) that is "next \(\alpha\) many admissibles"[2]stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+C(\Omega_2+0^{+_3+},0^{+_3}),0),0)+C(\Omega_2+0^{*+},0^*),0)\) is first "next 'next admissible' many admissibles"[2]stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+C(\Omega_2+C(\Omega_2+0^{+_3+},0^{+_3}),0^{+_3}),0),0)\) \(+C(\Omega_2+C(\Omega_2+0^{*+},0^*),0^*),0)\) is first "next 'next "next admissible" many admissibles' many admissibles"[2]stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+C(\Omega_2+0^{+_3*},0^{+_3}),0),0)+C(\Omega_2+0^{**},0^*),0)\) is first "next admissible-fixed-point"[2]stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+C(\Omega_2+0^{+_3*+},0^{+_3}),0),0)+C(\Omega_2+0^{**+},0^*),0)\) is first "next recursively inaccessible"[2]stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+C(\Omega_2+\varepsilon_{0^{+_3*+}2},0^{+_3}),0),0)+C(\Omega_2+\varepsilon_{0^{**+}2},0^*),0)\) is first "next 1-stable"[2]stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+C(\Omega_2+\varepsilon_{0^{+_3**+}2},0^{+_3}),0),0)+C(\Omega_2+\varepsilon_{0^{***+}2},0^*),0)\) is first "next doubly 1-stable"[2]stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+C(\Omega_2+C(\Omega_22+1,0^{+_3}),0^{+_3}),0),0)\) \(+C(\Omega_2+C(\Omega_22+1,0),0^*),0)\) is first "next nonprojectable"[2]stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+C(\Omega_2+C(\Omega_22+0^{+_3},0^{+_3}),0^{+_3}),0),0)\) \(+C(\Omega_2+C(\Omega_22+0^*,0),0^*),0)\) is first \(\alpha\) that is "next \(\alpha\)-ply-stable"[2]stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+\) \(C(\Omega_2+C(\Omega_22+C(\Omega_2+C(\Omega_22+1,0^{+_3}),0^{+_3}),0^{+_3}),0^{+_3}),0),0)\) \(+C(\Omega_2+C(\Omega_22+C(\Omega_2+C(\Omega_22+1,0),0^*),0),0^*),0)\) is first \(\alpha\) that is "next 'next nonprojectable after \(\alpha\)'-ply-stable"[2]stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+C(\Omega_2+0^{+_3+_3},0^{+_3}),0),0)\) \(+C(\Omega_2+C(\Omega_22+0^{**},0),0^*),0)\) is first "next \(\alpha\) that is \(\alpha\)-ply-stable"[2]stable. Now TON becomes erratic.
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+C(\Omega_2+\varepsilon_{0^{+_3+_3+}2},0^{+_3}),0),0)\) \(+C(\Omega_2+\varepsilon_{C(\Omega_22+0^{**},0)^+2},0^*),0)\) is first "next \(\alpha\) that is \(\alpha+1\)-ply 1-stable"[2]stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+C(\Omega_2+C(\Omega_22+1,0^{+_3+_3}),0^{+_3}),0),0)\) \(+C(\Omega_2+C(\Omega_22+1,C(\Omega_22+0^{**},0)),0^*),0)\) is first "next \(\alpha\) that is \(\alpha+\omega\)-ply-stable"[2]stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+C(\Omega_2+0^{+_3+_3+_3},0^{+_3}),0),0)\) \(+C(\Omega_2+C(\Omega_22+0^{**},C(\Omega_22+0^{**},0)),0^*),0)\) is first "next \(\alpha\) that is \(\alpha2\)-ply-stable"[2]stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+C(\Omega_2+C(\Omega_23+1,0),0^{+_3}),0),0)\) \(+C(\Omega_2+C(\Omega_22+0^{**}+1,0),0^*),0)\) is first "next \(\alpha\) that is \(\alpha\omega\)-ply-stable"[2]stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+0^{+_3*},0),0)+C(\Omega_2+C(\Omega_22+0^{**}+0^*,0),0^*),0)\) is first \(\alpha\) that is "next \(\beta\) that is \(\beta\alpha\)-ply-stable"[2]stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+0^{+_3*}+C(\Omega_2+C(\Omega_22+0^{+_3*+},0^{+_3}),0^{+_3}),0),0)\) \(+C(\Omega_2+C(\Omega_22+0^{**+},0),0^*),0)\) is first "next \(\alpha\) that is \(\alpha^+\)-ply-stable"[2]stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+0^{+_3*}+C(\Omega_2+0^{+_3+_3},0^{+_3}),0),0)\) \(+C(\Omega_2+C(\Omega_22+0^{***},0),0^*),0)\) is first "next '\(\alpha\) that is \(\alpha\)-ply-stable'-stable"[2]stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+0^{+_3**}+C(\Omega_2+0^{+_3+_3},0^{+_3}),0),0)\) \(+C(\Omega_2+C(\Omega_22+0^{****},0),0^*),0)\) is first "next doubly '\(\alpha\) that is \(\alpha\)-ply-stable'-stable"[2]stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+C(\Omega_22+1,0^{+_3}),0),0)\) \(+C(\Omega_2+C(\Omega_22+C(\Omega_22+1,0),0),0^*),0)\) is first "next \(\omega\)-ply '\(\alpha\) that is \(\alpha\)-ply-stable'-stable"[2]stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+C(\Omega_22+0^{+_3},0^{+_3}),0),0)\) \(+C(\Omega_2+C(\Omega_22+C(\Omega_22+0^*,0),0),0^*),0)\) is first \(\alpha\) that is "next \(\alpha\)-ply '\(\beta\) that is \(\beta\)-ply-stable'-stable"[2]stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+C(\Omega_22+C(\Omega_22+0^{+_3},0^{+_3}),0^{+_3}),0),0)\) \(+C(\Omega_2+C(\Omega_22+C(\Omega_22+C(\Omega_22+0^*,0),0),0),0^*),0)\) is first \(\alpha\) that is "next \(\alpha\)-ply '\(\alpha_1\) that is \(\alpha_1\)-ply "\(\alpha_2\) that is \(\alpha_2\)-ply-stable"-stable'-stable"[2]stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+0^{+_3+_3},0),0),0)\) is first C(Ω2+a,b) that is \(C(\Omega_2+a+C(\Omega_2+0^{+_3},0^*),b)\)-stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+0^{+_3+_3},0),0)+C(\Omega_2+0^{+_3},0^*),0)\) is first "next \(\alpha_0\) that is \(\alpha_\omega\)-ply-stable"[2]stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+\varepsilon_{0^{+_3+_3+}2},0),0),0)\) is first C(Ω2+a,b) that is \(C(\Omega_2+a+C(\Omega_2+C(\Omega_23+C(\Omega_22+\varepsilon_{0^{+_3+}2},0),0),0^*),b)\)-stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+\varepsilon_{0^{+_3+_3+}2},0),0)\) \(+C(\Omega_2+C(\Omega_23+C(\Omega_22+\varepsilon_{0^{+_3+}2},0),0),0^*),0)\) is first "next 1[2]stable"[2]stable, i.e. doubly 1[2]stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+\varepsilon_{0^{+_3+_3+_3+}2},0),0)\) \(+C(\Omega_2+C(\Omega_23+C(\Omega_22+\varepsilon_{0^{+_3+_3+}2},0),0),0^*),0)\) is first triply 1[2]stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+C(\Omega_23+1,0),0),0),0)\) is first "\(\omega\)-ply[2]stable"-stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+C(\Omega_23+1,0),0),0)+\) \(C(\Omega_2+C(\Omega_23+C(\Omega_22+C(\Omega_23+1,0),0),0),0^*),0)\) is first \(\omega\)-ply[2]stable
  • \(C(\Omega_2+C(\Omega_23+C(\Omega_22+C(\Omega_23+1,0),0),0)+\) \(C(\Omega_2+C(\Omega_23+C(\Omega_22+C(\Omega_23+1,0),0),0),0^*)+1,0)\) is first 0-\(\Pi_1\)[3]reflecting, i.e. the \(\alpha=\sup\{\beta<\alpha|\beta\text{ is }\alpha\text{-stable onto }\{\gamma<\beta|L_\gamma\prec_{\Sigma_1}L_\beta\}\}\)

I guessed that the \(\beta[n]\)stable is approximately but weaker than \((\beta,n)\)-stable. As a result, the 0-\(\Pi_1\)[3]reflecting is also the \(\alpha\) that \(L_\alpha\models\text{KP}+\Sigma_2\text{-Separation}\) and \(L_\alpha\cap\mathcal P(\omega)\models\Pi^1_3\text{-CA}+\text{BI}\). The "\(\omega\)-ply[2]stable"-stable ordinal should be the \(\alpha\) that \(L_\alpha\cap\mathcal P(\omega)\models\Pi^1_3\text{-CA}_0\).

  • \(C(\Omega_2+C(\Omega_23+0^{+_3},0)+C(\Omega_2+C(\Omega_23+0^{+_3},0),0^*),0)\) is first \(\alpha\) that is \(\alpha\)-ply[2]stable
  • \(C(\Omega_2+0^{+_4}+C(\Omega_2+0^{+_4},0^*),0)\) is first \(\alpha_0\) that is \(\alpha_\omega\)-ply[2]stable, which means a set \(\{\alpha_n|n<\omega\}\) such that \(\alpha_n<\alpha_{n+1}\) and \(\alpha_n\) is \(\alpha_n\)-ply \(\alpha_{n+1}\)[2]stable for all \(n<\omega\).
  • \(C(\Omega_2+0^{+_4}+C(\Omega_2+0^{+_4},0^*)+0^*,0)\) is first \(\alpha_\omega\)-target for [2]stable
  • \(C(\Omega_2+C(\Omega_24+C(\Omega_23+0^{+_4+},0),0),0)\) is first "0-\(\Pi_2\)[3]reflecting"-stable
  • \(C(\Omega_2+C(\Omega_24+C(\Omega_23+0^{+_4+},0),0)+\) \(C(\Omega_2+C(\Omega_24+C(\Omega_23+0^{+_4+},0),0),0^*),0)\) is first "0-\(\Pi_2\)[3]reflecting"[2]stable
  • \(C(\Omega_2+C(\Omega_24+C(\Omega_23+0^{+_4+},0),0)+\) \(C(\Omega_2+C(\Omega_24+C(\Omega_23+0^{+_4+},0),0),0^*)+0^{*+},0)\) is first 0-\(\Pi_2\)[3]reflecting. This ordinal is the \(\alpha\) that \(L_\alpha\cap\mathcal P(\omega)\models\Delta^1_4\text{-CA}+\text{BI}\). Also, it is first \(\Delta_3\)-gap ordinal.
  • \(C(\Omega_2+C(\Omega_24+C(\Omega_23+\varepsilon_{0^{+_4+}2},0),0)+\) \(C(\Omega_2+C(\Omega_24+C(\Omega_23+\varepsilon_{0^{+_4+}2},0),0),0^*)+\varepsilon_{0^{*+}2},0)\) is first 1[3]stable
  • \(C(\Omega_2+C(\Omega_25+C(\Omega_24+0^{+_5+},0),0)+\) \(C(\Omega_2+C(\Omega_25+C(\Omega_24+0^{+_5+},0),0),0^*)2+0^{*+},0)\) is first 0-\(\Pi_2\)[4]reflecting, a.k.a. first \(\Delta_4\)-gap ordinal
  • \(C(\Omega_2+C(\Omega_25+C(\Omega_24+\varepsilon_{0^{+_5+}2},0),0)+\) \(C(\Omega_2+C(\Omega_25+C(\Omega_24+\varepsilon_{0^{+_5+}2},0),0),0^*)2+\varepsilon_{0^{*+}2},0)\) is first 1[4]stable
  • \(C(\Omega_2+C(\Omega_2\omega,0),0)\) is first ordinal stable up to a gap ordinal
  • \(C(\Omega_2+C(\Omega_2\omega,0)\omega,0)\) is first gap ordinal, a.k.a. first 1[ω]stable ordinal, a.k.a. the \(\alpha\) that \(L_\alpha\cap\mathcal P(\omega)\) is a model of second-order arithmetic.

Details under incomputable stability

From the above comparisons, it seems that some expressions in Taranovsky's ordinal notation with a \(C(\Omega_2+C(\Omega_22+1,0),0)\) reaches \(\Pi^1_2\text{-CA}_0\). The least computable ordinal with occurence of a \(C(\Omega_2+C(\Omega_22+1,0),0)\) is \(C(C(\Omega_22+C(\Omega_2+C(\Omega_22+1,0),0),0),0)\), so a question naturally comes: does \(C(C(\Omega_22+C(\Omega_2+C(\Omega_22+1,0),0),0),0)\) reach \(\Pi^1_2\text{-CA}_0\)?

Interpretation 1

To get it, let's focus on the first incomputable stability - "next admissible"-stable ordinals. Inside an ordinal collapsing function, it's "large enough", and works as a "diagonalizer" over the expression \(f(\alpha)\) in "\(\alpha\) is \(f(\alpha)\)-stable". For example, the least collapsing of "next admissible"-stable should be the limit of \(\theta(\)1-stable\(,0)\), \(\theta(\lambda a.\theta(1\)-stable\(,a)\)-stable\(,0)\), \(\theta(\lambda a.\theta(\lambda b.\theta(1\)-stable\(,b)\)-stable\(,a)\)-stable\(,0)\), etc.

When we talk about \(\lambda a.\Gamma_{a+1}\)-stable ordinals, the "\(a\)" is so large that we need larger ordinal (\(a^+\)) for collapsing such as \(\lambda a.\theta(a^+,a)\)-stable or \(\lambda a.\psi_{a^+}((a^+)^{a^+})\)-stable. The notation \(\lambda a.\theta(\Omega,a)\)-stable just leads to \(\lambda a.\varphi(\Omega,a+1)\)-stable (where \(\Omega\) means admissible, \(\Omega_{a+n}=a^{\overbrace{++\cdots+}^n}\)). In this sense, the "1-stable" in \(\lambda a.\theta(1\)-stable\(,a)\)-stable actually means the next 1-stable ordinal after \(a\).

Let \(d_0=C(\Omega_22,0)\) and \(d_{n+1}=C(\Omega_2,d_n)\).

  • \(C(\Omega_2+d_1^{d_1^{C(d_1,d_0)}+1},0)\) is first \(\lambda a.\varepsilon_{a+1}\)-stable, or first \(\lambda a.\theta(1,a)\)-stable
  • \(C(\Omega_2+d_1^{d_1^{C(d_1d_0,d_0)}+1},0)\) is first \(\lambda a.\varphi(a,1)\)-stable, or first \(\lambda a.\theta(a,a)\)-stable
  • \(C(\Omega_2+d_1^{d_1^{C(d_1^2,d_0)}+1},0)\) is first \(\lambda a.\Gamma_{a+1}\)-stable, or first \(\lambda a.\theta(a^+,a)\)-stable
  • \(C(\Omega_2+d_1^{d_1^{C(d_2,d_0)}+1},0)\) is first \(\lambda a.\theta(a^{++},a)\)-stable
  • \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_2+1,d_0),d_0)}+1},0)\) is first \(\lambda a.\theta(\Omega_{a+\omega},a)\)-stable
  • \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_2+d_0,d_0),d_0)}+1},0)\) is first \(\lambda a.\theta(\Omega_{a2},a)\)-stable
  • \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_2+d_1,d_0),d_0)}+1},0)\) is first \(\lambda a.\psi_{\Omega_{a+1}}(\Omega_{\Omega_{a+1}})\)-stable
  • \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_22,d_0),d_0)}+1},0)\) is first \(\lambda a.\psi_{\Omega_{a+1}}(\Omega_{\Omega_{a+1}^{\Omega_{a+1}^{\Omega_{a+1}}}})\)-stable
  • \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+1,0),d_0)}+1},0)\) is first \(\lambda a.\psi_{\Omega_{a+1}}(\Omega_{\Omega_{a+1}^{\Omega_{a+1}^{\Omega_{a+1}}}\omega})\)-stable
  • \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(\Omega_2+d_0,0),0),d_0)}+1},0)\) is first \(\lambda a.\psi_{\Omega_{a+1}}(\Omega_{\Omega_{a+1}^{\Omega_{a+1}^{\Omega_{a+1}}}\psi_I(0)})\)-stable
  • \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(\Omega_2+d_0+1,0),0),d_0)}+1},0)\) is first \(\lambda a.\psi_{\Omega_{a+1}}(\Omega_{\Omega_{a+1}^{\Omega_{a+1}^{\Omega_{a+1}}}\psi_I(\omega)})\)-stable
  • \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(\Omega_2+d_1,0),0),d_0)}+1},0)\) is first \(\lambda a.\psi_{\Omega_{a+1}}(\Omega_{\Omega_{a+1}^{\Omega_{a+1}^{\Omega_{a+1}}}I})\)-stable
  • \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(\Omega_2+d_1+1,0),0),d_0)}+1},0)\) is first \(\lambda a.\psi_{\Omega_{a+1}}(\Omega_{\Omega_{a+1}^{\Omega_{a+1}^{\Omega_{a+1}}}I_\omega})\)-stable
  • \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(\Omega_2+d_1^{d_1}+1,0),0),d_0)}+1},0)\) is first \(\lambda a.\psi_{\Omega_{a+1}}(\Omega_{\Omega_{a+1}^{\Omega_{a+1}^{\Omega_{a+1}}}K_\omega})\)-stable
  • \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(\Omega_2+d_1^{d_1^\omega+1},0),0),d_0)}+1},0)\) is first \(\lambda a.\psi_{\Omega_{a+1}}(\Omega_{\Omega_{a+1}^{\Omega_{a+1}^{\Omega_{a+1}}}\text{first 1-stable}})\)-stable
  • \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(\Omega_2+d_1^{d_1^\omega+1},0),0),d_0)}+1},0),0),d_0)}+1},0)\) is first \(\lambda a.\psi_{\Omega_{a+1}}(\Omega_{\Omega_{a+1}^{\Omega_{a+1}^{\Omega_{a+1}}}\text{first }\lambda b.\psi_{\Omega_{b+1}}(\Omega_{\Omega_{b+1}^{\Omega_{b+1}^{\Omega_{b+1}}}\text{first 1-stable}})\text{-stable}})\)-stable
  • \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(\Omega_2+d_1^{d_1^{d_1}},0),0),d_0)}+1},0)\) is first \(\lambda a.\psi_{\Omega_{a+1}}(\Omega_{\Omega_{a+1}^{\Omega_{a+1}^{\Omega_{a+1}}}a})\)-stable
  • \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(\Omega_2+d_1^{d_1^{d_1}},0)2,0),d_0)}+1},0)\) is first \(\lambda a.\psi_{\Omega_{a+1}}(\Omega_{\Omega_{a+1}^{\Omega_{a+1}^{\Omega_{a+1}}}a^2})\)-stable
  • \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(C(\Omega_2,C(\Omega_2+d_1^{d_1^{d_1}},0)),C(\Omega_2+d_1^{d_1^{d_1}},0)),0),d_0)}+1},0)\) is first \(\lambda a.\psi_{\Omega_{a+1}}(\Omega_{\Omega_{a+1}^{\Omega_{a+1}^{\Omega_{a+1}}}\varepsilon_{a+1}})\)-stable
  • \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(d_0,C(\Omega_2+d_1^{d_1^{d_1}},0)),0),d_0)}+1},0)\) is first \(\lambda a.\psi_{\Omega_{a+1}}(\psi_{I_{a+1}}(0))\)-stable
  • \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(\Omega_2+d_1,C(\Omega_2+d_1^{d_1^{d_1}},0)),0),d_0)}+1},0)\) is first \(\lambda a.\psi_{\Omega_{a+1}}(I_{a+1})\)-stable
  • \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(\Omega_2+d_1^{d_1},C(\Omega_2+d_1^{d_1^{d_1}},0)),0),d_0)}+1},0)\) is first \(\lambda a.\psi_{\Omega_{a+1}}(K_{a+1})\)-stable
  • \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(\Omega_2+d_1^{d_1^\omega+1},C(\Omega_2+d_1^{d_1^{d_1}},0)),0),d_0)}+1},0)\) is first \(\lambda a.\psi_{\Omega_{a+1}}(\text{next }1\text{-stable after }a)\)-stable
  • \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(\Omega_2+d_1^{d_1^{C(\Omega_2+d_1^{d_1^{d_1}},0)}+1},C(\Omega_2+d_1^{d_1^{d_1}},0)),0),d_0)}+1},0)\) is first \(\lambda a.\psi_{\Omega_{a+1}}(\text{next }\lambda b.a\text{-stable after }a)\)-stable
  • \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(\Omega_2+d_1^{d_1^{d_0}+1},C(\Omega_2+d_1^{d_1^{d_1}},0)),0),d_0)}+1},0)\) is first \(\lambda a.\psi_{\Omega_{a+1}}(\text{next }\lambda b.b\text{-stable after }a)\)-stable
  • \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+1,0),d_0)}+1},C(\Omega_2+d_1^{d_1^{d_1}},0)),0),d_0)}+1},0)\) is first \(\lambda a.\psi_{\Omega_{a+1}}(\text{next }\lambda b.\psi_{\Omega_{b+1}}(\Omega_{\Omega_{b+1}^{\Omega_{b+1}^{\Omega_{b+1}}}\omega})\text{-stable after }a)\)-stable
  • \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(\Omega_2+d_1^{d_1^{d_1}},0),0),d_0)}+1},C(\Omega_2+d_1^{d_1^{d_1}},0)),0),d_0)}+1},0)\) is first \(\lambda a.\psi_{\Omega_{a+1}}(\text{next }\lambda b.\psi_{\Omega_{b+1}}(\Omega_{\Omega_{b+1}^{\Omega_{b+1}^{\Omega_{b+1}}}a})\text{-stable after }a)\)-stable
  • \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(\Omega_2+d_1^{d_1^\omega+1},C(\Omega_2+d_1^{d_1^{d_1}},0)),0),d_0)}+1},C(\Omega_2+d_1^{d_1^{d_1}},0)),0),d_0)}+1},0)\) is first \(\lambda a.\psi_{\Omega_{a+1}}(\text{next }\lambda b.\psi_{\Omega_{b+1}}(\Omega_{\Omega_{b+1}^{\Omega_{b+1}^{\Omega_{b+1}}}\text{next }1\text{-stable after }a})\text{-stable after }a)\)-stable
  • \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+1,0),d_0)}+1},C(\Omega_2+d_1^{d_1^{d_1}},0)),0),d_0)}+1},C(\Omega_2+d_1^{d_1^{d_1}},0)),0),d_0)}+1},0)\) is first \(\lambda a.\psi_{\Omega_{a+1}}(\text{next }\lambda b.\psi_{\Omega_{b+1}}(\Omega_{\Omega_{b+1}^{\Omega_{b+1}^{\Omega_{b+1}}}\text{next }\lambda c.\psi_{\Omega_{c+1}}(\Omega_{\Omega_{c+1}^{\Omega_{c+1}^{\Omega_{c+1}}}\omega})\text{-stable after }a})\text{-stable after }a)\)-stable
  • \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(\Omega_2+d_1^{d_1^{d_1}},C(\Omega_2+d_1^{d_1^{d_1}},0)),0),d_0)}+1},0)\) is first \(\lambda a.\psi_{\Omega_{a+1}}(\text{next }\lambda b.\psi_{\Omega_{b+1}}(\Omega_{\Omega_{b+1}^{\Omega_{b+1}^{\Omega_{b+1}}}b})\text{-stable after }a)\)-stable
  • \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(\Omega_2+d_1^{d_1^{d_1}},C(\Omega_2+d_1^{d_1^{d_1}},0))+C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(\Omega_2+d_1^{d_1^{d_1}},C(\Omega_2+d_1^{d_1^{d_1}},0)),0),d_0)+\omega}+1},C(\Omega_2+d_1^{d_1^{d_1}},0)),0),d_0)}+1},0)\) is first \(\lambda a.\psi_{\Omega_{a+1}}(\text{next }\lambda b.\psi_{\Omega_{b+1}}(\Omega_{\Omega_{b+1}^{\Omega_{b+1}^{\Omega_{b+1}}}b})+1\text{-stable after }a)\)-stable
  • \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(\Omega_2+d_1^{d_1^{d_1}},C(\Omega_2+d_1^{d_1^{d_1}},0))2,0),d_0)}+1},0)\) is first \(\lambda a.\psi_{\Omega_{a+1}}(\text{next }\lambda b.\psi_{\Omega_{b+1}}(\Omega_{\Omega_{b+1}^{\Omega_{b+1}^{\Omega_{b+1}}}b^2})\text{-stable after }a)\)-stable
  • \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(d_0,C(\Omega_2+d_1^{d_1^{d_1}},C(\Omega_2+d_1^{d_1^{d_1}},0))),0),d_0)}+1},0)\) is first \(\lambda a.\psi_{\Omega_{a+1}}(\text{next }\lambda b.\psi_{\Omega_{b+1}}(\psi_{I_{b+1}}(0))\text{-stable after }a)\)-stable
  • \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(\Omega_2+d_1,C(\Omega_2+d_1^{d_1^{d_1}},C(\Omega_2+d_1^{d_1^{d_1}},0))),0),d_0)}+1},0)\) is first \(\lambda a.\psi_{\Omega_{a+1}}(\text{next }\lambda b.\psi_{\Omega_{b+1}}(I_{b+1})\text{-stable after }a)\)-stable
  • \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(\Omega_2+d_1^{d_1^\omega+1},C(\Omega_2+d_1^{d_1^{d_1}},C(\Omega_2+d_1^{d_1^{d_1}},0))),0),d_0)}+1},0)\) is first \(\lambda a.\psi_{\Omega_{a+1}}(\text{next }\lambda b.\psi_{\Omega_{b+1}}(\text{next }1\text{-stable after }b)\text{-stable after }a)\)-stable
  • \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(\Omega_2+d_1^{d_1^{d_1}},C(\Omega_2+d_1^{d_1^{d_1}},C(\Omega_2+d_1^{d_1^{d_1}},0))),0),d_0)}+1},0)\) is first \(\lambda a.\psi_{\Omega_{a+1}}(\text{next }\lambda b.\psi_{\Omega_{b+1}}(\text{next }\lambda c.\psi_{\Omega_{c+1}}(\Omega_{\Omega_{c+1}^{\Omega_{c+1}^{\Omega_{c+1}}}c})\text{-stable after }b)\text{-stable after }a)\)-stable
  • \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(\Omega_2+d_1^{d_1^{d_1}}+1,0),0),d_0)}+1},0)\) is first \(\lambda a.\psi_{\Omega_{a+1}}(\lambda b.\Omega_{b+1}\text{-stable})\)-stable
  • \(C(C(\Omega_22+C(\Omega_2+d_1^{d_1^{d_1}}+1,0),0),0)=\psi(\lambda a.\Omega_{a+1}\text{-stable})\)

\(C(\Omega_2+d_1^{d_1^{d_1}},0)\) thus works as the bound varible in abstraction-stable, \(C(\Omega_2+d_1^{d_1^{d_1}},C(\Omega_2+d_1^{d_1^{d_1}},0))\) works as the inner bound varible in \(\lambda a.\psi_{\Omega_{a+1}}(\text{abstraction-stable})\)-stable, and so on.

Due to the inconvenience applying the diagonalizer behavior in Taranovsky's ordinal notation, I next compare stable ordinals with my array notation. Let separator ■ = {1{1,,2,,}2,,}

  • {1{1,,2{2,,}2,,}2} corresponds with 1-stable
  • {1{1,,2{1{1■2}2,,}2,,}2} corresponds with \(\lambda a.a\)-stable
  • {1{1,,2{2{1■2}2,,}2,,}2} corresponds with \(\lambda a.a\omega\)-stable
  • {1{1,,2{1{1■2}3,,}2,,}2} corresponds with \(\lambda a.a^2\)-stable
  • {1{1,,2{1{1■2}1{1■2}2,,}2,,}2} corresponds with \(\lambda a.a^a\)-stable
  • {1{1,,2{1{2■2}2,,}2,,}2} corresponds with \(\lambda a.a^{a^\omega}\)-stable
  • {1{1,,2{1{1{1■2}2■2}2,,}2,,}2} corresponds with \(\lambda a.a^{a^a}\)-stable
  • {1{1,,2{1{1,,2■2}2,,}2,,}2} corresponds with \(\lambda a.\varepsilon_{a+1}\)-stable
  • {1{1,,2{1{1,,3■2}2,,}2,,}2} corresponds with \(\lambda a.\psi_{\Omega_{a+1}}(\Omega_{a+2})\)-stable
  • {1{1,,2{1{1,,1{1■2}2■2}2,,}2,,}2} corresponds with \(\lambda a.\psi_{\Omega_{a+1}}(\Omega_{a2})\)-stable
  • {1{1,,2{1{1,,1{1,,2■2}2■2}2,,}2,,}2} corresponds with \(\lambda a.\psi_{\Omega_{a+1}}(\Omega_{\Omega_{a+1}})\)-stable
  • {1{1,,2{1{1,,1{1,,1{1■2}2■2}2■2}2,,}2,,}2} corresponds with \(\lambda a.\psi_{\Omega_{a+1}}(\Omega_{\Omega_{a2}})\)-stable
  • {1{1,,2{1{1,,1,,2■2}2,,}2,,}2} corresponds with \(\lambda a.\psi_{\Omega_{a+1}}(\psi_{I_{a+1}}(0))\)-stable
  • {1{1,,2{1{1{1,,2{2,,}2,,}2■2}2,,}2,,}2} corresponds with \(\lambda a.\psi_{\Omega_{a+1}}(\text{next }1\text{-stable after }a)\)-stable
  • {1{1,,2{1{1{1,,2{1{1■2}2,,}2,,}2■2}2,,}2,,}2} corresponds with \(\lambda a.\psi_{\Omega_{a+1}}(\text{next }\lambda b.a\text{-stable after }a)\)-stable
  • {1{1,,2{1{1■3}2,,}2,,}2} corresponds with \(\lambda a.\psi_{\Omega_{a+1}}(\text{next }\lambda b.b\text{-stable after }a)\)-stable
  • {1{1,,2{1{1■4}2,,}2,,}2} corresponds with \(\lambda a.\psi_{\Omega_{a+1}}(\text{next }\lambda b.\psi_{\Omega_{b+1}}(\text{next }\lambda c.c\text{-stable after }b)\text{-stable after }a)\)-stable
  • s(n,n{1■1,2}2) has growth rate \(\psi(\lambda a.\Omega_{a+1}\text{-stable})\)
  • s(n,n{1{1{1,,2{1{1■1,2}2,,}2,,}2}2{1,,2{1{1■1,2}2,,}2,,}2}2) has growth rate \(\psi(\lambda a.\Omega_{a+1}\text{-stable}+\lambda a.\psi_{\Omega_{a+1}}(\lambda b.\Omega_{b+1}\text{-stable})\text{-stable})\)
  • {1{1,,2{1{1■1,2}2,,}2,,}2} corresponds with \(\lambda a.\psi_{\Omega_{a+1}}(\lambda b.\Omega_{b+1}\text{-stable})\)-stable
  • {1{1,,2{1{1{1{1,,2{1{1■1,2}2,,}2,,}2■2}2{1,,2{1{1■1,2}2,,}2,,}2■2}2,,}2,,}2} corresponds with \(\lambda a.\psi_{\Omega_{a+1}}(\lambda b.\Omega_{b+1}\text{-stable}+\lambda b.\psi_{\Omega_{b+1}}(\lambda c.\Omega_{c+1}\text{-stable})\text{-stable})\)-stable
  • s(n,n{1{1■1,2}2■1,2}2) has growth rate \(\psi(\lambda a.\Omega_{a+1}\text{-stable}\cdot2)\)
  • s(n,n{1,,2■1,2}2) has growth rate \(\psi(\text{next admissible after }\lambda a.\Omega_{a+1}\text{-stable})\)
  • s(n,n{1,,1{1,,1,,2■1,2}1{1,,1,,2■1,2}2■1,2}2) has growth rate \(\psi(\text{next recursively inaccessible after }\lambda a.\Omega_{a+1}\text{-stable})\)
  • s(n,n{1{1,,2,,}2■1,2}2) has growth rate \(\psi(\text{next recursively Mahlo after }\lambda a.\Omega_{a+1}\text{-stable})\)
  • s(n,n{1{1,,1,,2,,}2■1,2}2) has growth rate \(\psi(\text{next }\Pi_3\text{-reflecting after }\lambda a.\Omega_{a+1}\text{-stable})\)
  • s(n,n{1{1{2,,}2,,}2■1,2}2) has growth rate \(\psi(\text{next }1\text{-stable after }\lambda a.\Omega_{a+1}\text{-stable})\)
  • s(n,n{1■2,2}2) has growth rate \(\psi(\text{next }\lambda a.a\text{-stable after }\lambda a.\Omega_{a+1}\text{-stable})\)
  • s(n,n{1■3,2}2) has growth rate \(\psi(\text{next }\lambda a.\psi_{\Omega_{a+1}}(\text{next }\lambda b.b\text{-stable after }a)\text{-stable after }\lambda a.\Omega_{a+1}\text{-stable})\)
  • s(n,n{1■4,2}2) has growth rate \(\psi(\text{next }\lambda a.\psi_{\Omega_{a+1}}(\text{next }\lambda b.\psi_{\Omega_{b+1}}(\text{next }\lambda c.c\text{-stable after }b)\text{-stable after }a)\text{-stable after }\lambda a.\Omega_{a+1}\text{-stable})\)
  • s(n,n{1■1,3}2) has growth rate \(\psi(\text{next }\lambda a.\psi_{\Omega_{a+1}}(\lambda b.\Omega_{b+1}\text{-stable})\text{-stable after }\lambda a.\Omega_{a+1}\text{-stable})\)
  • s(n,n{1■2,3}2) has growth rate \(\psi(\text{next }\lambda a.\psi_{\Omega_{a+1}}(\text{next }\lambda b.b\text{-stable after }\lambda b.\Omega_{b+1}\text{-stable})\text{-stable after }\lambda a.\Omega_{a+1}\text{-stable})\)
  • s(n,n{1■1,4}2) has growth rate \(\psi(\text{next }\lambda a.\psi_{\Omega_{a+1}}(\text{next }\lambda b.\psi_{\Omega_{b+1}}(\lambda c.\Omega_{c+1}\text{-stable})\text{-stable after }\lambda b.\Omega_{b+1}\text{-stable})\text{-stable after }\lambda a.\Omega_{a+1}\text{-stable})\)
  • s(n,n{1■1,1,2}2) has growth rate \(\psi(2\text{nd }\lambda a.\Omega_{a+1}\text{-stable})\)
  • s(n,n{1■1,1,1,2}2) has growth rate \(\psi(3\text{rd }\lambda a.\Omega_{a+1}\text{-stable})\)
  • s(n,n{1■1`2}2) has growth rate \(\psi(\Omega\text{th }\lambda a.\Omega_{a+1}\text{-stable})\)
  • s(n,n{1■1{1,,1,,2}2}2) has growth rate \(\psi(I\text{th }\lambda a.\Omega_{a+1}\text{-stable})\)
  • s(n,n{1■1{1{1,,2{2,,}2,,}2}2}2) has growth rate \(\psi(1\text{-stable th }\lambda a.\Omega_{a+1}\text{-stable})\)
  • s(n,n{1■1{1{1,,2{1{1■2}2,,}2,,}2}2}2) has growth rate \(\psi(\lambda a.a\text{-stable th }\lambda a.\Omega_{a+1}\text{-stable})\)
  • s(n,n{1■1{1{1,,2{1{1■1,2}2,,}2,,}2}2}2) has growth rate \(\psi(\lambda a.\psi_{\Omega_{a+1}}(\lambda b.\Omega_{b+1}\text{-stable})\text{-stable th }\lambda a.\Omega_{a+1}\text{-stable})\)
  • s(n,n{1■1{1{1,,2{1{1■1{1{1,,2{1{1■1,2}2,,}2,,}2}2}2,,}2,,}2}2}2) has growth rate \(\psi(\lambda a.\psi_{\Omega_{a+1}}(\lambda b.\psi_{\Omega_{b+1}}(\lambda c.\Omega_{c+1}\text{-stable})\text{-stable th }\lambda b.\Omega_{b+1}\text{-stable})\text{-stable th }\lambda a.\Omega_{a+1}\text{-stable})\)
  • s(n,n{1■1{1■2}2}2) has growth rate \(\psi(\lambda a.\Omega_{a+1}\text{-stable th }\lambda a.\Omega_{a+1}\text{-stable})\)
  • s(n,n{1■1,2{1■2}2}2) has growth rate \(\psi(\text{next }\lambda a.\psi_{\Omega_{a+1}}(\lambda b.\Omega_{b+1}\text{-stable})\text{-stable after }(\lambda a.\Omega_{a+1}\text{-stable th }\lambda a.\Omega_{a+1}\text{-stable}))\)
  • s(n,n{1■1{1{1,,2{1{1■1{1■2}2}2,,}2,,}2}2{1■2}2}2) has growth rate \(\psi(\text{next }\lambda a.\psi_{\Omega_{a+1}}(\lambda b.\Omega_{b+1}\text{-stable th }\lambda b.\Omega_{b+1}\text{-stable})\text{-stable after }(\lambda a.\Omega_{a+1}\text{-stable th }\lambda a.\Omega_{a+1}\text{-stable}))\)
  • s(n,n{1■1{1■2}3}2) has growth rate \(\psi(\lambda a.\Omega_{a+1}\text{-stable}+1\text{th }\lambda a.\Omega_{a+1}\text{-stable})\)
  • s(n,n{1■1{1■2}1{1■2}2}2) has growth rate \(\psi(\lambda a.\Omega_{a+1}\text{-stable}\cdot2\text{th }\lambda a.\Omega_{a+1}\text{-stable})\)
  • s(n,n{1■1{1■2}2{1■2}2}2) has growth rate \(\psi(\lambda a.\Omega_{a+1}\text{-stable}\cdot2+1\text{th }\lambda a.\Omega_{a+1}\text{-stable})\)
  • s(n,n{1■1{1■2}1{1■2}1{1■2}2}2) has growth rate \(\psi((\lambda a.\Omega_{a+1}\text{-stable})^2\text{th }\lambda a.\Omega_{a+1}\text{-stable})\)
  • s(n,n{1■1{2■2}2}2) has growth rate \(\psi((\lambda a.\Omega_{a+1}\text{-stable})^\omega\text{th }\lambda a.\Omega_{a+1}\text{-stable})\)
  • s(n,n{1■1{1,,2■2}2}2) has growth rate \(\psi((\text{next admissible after }\lambda a.\Omega_{a+1}\text{-stable})\text{th }\lambda a.\Omega_{a+1}\text{-stable})\)
  • s(n,n{1■1{1{1,,2{1{1■3}2,,}2,,}2■2}2}2) has growth rate \(\psi((\text{next }\lambda a.a\text{-stable after }\lambda a.\Omega_{a+1}\text{-stable})\text{th }\lambda a.\Omega_{a+1}\text{-stable})\)
  • s(n,n{1■1{1{1,,2{1{1■1,2}2,,}2,,}2■2}2}2) has growth rate \(\psi((\text{next }\lambda a.\psi_{\Omega_{a+1}}(\lambda b.\Omega_{b+1}\text{-stable})\text{-stable after }\lambda a.\Omega_{a+1}\text{-stable})\text{th }\lambda a.\Omega_{a+1}\text{-stable})\)
  • s(n,n{1■1{1■3}2}2) has growth rate \(\psi((2\text{nd }\lambda a.\Omega_{a+1}\text{-stable})\text{th }\lambda a.\Omega_{a+1}\text{-stable})\)
  • s(n,n{1■1{1■4}2}2) has growth rate \(\psi((3\text{rd }\lambda a.\Omega_{a+1}\text{-stable})\text{th }\lambda a.\Omega_{a+1}\text{-stable})\)
  • s(n,n{1■1{1■1{1{1,,2{1{1■2}2,,}2,,}2}2}2}2) has growth rate \(\psi((\lambda a.a\text{-stable th }\lambda a.\Omega_{a+1}\text{-stable})\text{th }\lambda a.\Omega_{a+1}\text{-stable})\)
  • s(n,n{1■1{1■1{1■2}2}2}2) has growth rate \(\psi((\lambda a.\Omega_{a+1}\text{-stable th }\lambda a.\Omega_{a+1}\text{-stable})\text{th }\lambda a.\Omega_{a+1}\text{-stable})\)
  • s(n,n{1■1,,2}2) has growth rate \(\psi(\lambda a.\Omega_{a+1}\text{-stable-fixed-point})\)
  • s(n,n{1■1{1■1,,2}1{1■1,,2}2}2) has growth rate \(\psi(\text{admissible limit of }\lambda a.\Omega_{a+1}\text{-stable})\)
  • s(n,n{1■1{1,,2,,}2}2) has growth rate \(\psi(\text{recursively Mahlo limit of }\lambda a.\Omega_{a+1}\text{-stable})\)
  • s(n,n{1■1{1,,2{2,,}2,,}2}2) has growth rate \(\psi(1\text{-stable limit of }\lambda a.\Omega_{a+1}\text{-stable})\)
  • s(n,n{1■1{1,,2{1{1■2}2,,}2,,}2}2) has growth rate \(\psi(\lambda a.\text{(first }\lambda b.\Omega_{b+1}\text{-stable)-stable limit of }\lambda a.\Omega_{a+1}\text{-stable})\)
  • s(n,n{1■1{1,,2{1{1■1■2}2,,}2,,}2}2) has growth rate \(\psi(\lambda a.a\text{-stable limit of }\lambda a.\Omega_{a+1}\text{-stable})\)
  • s(n,n{1■1{1,,2{1{1■2■2}2,,}2,,}2}2) has growth rate \(\psi(\lambda a.\psi_{\Omega_{a+1}}(\lambda b.b\text{-stable})\text{-stable limit of }\lambda a.\Omega_{a+1}\text{-stable})\)
  • s(n,n{1■1,2■2}2) has growth rate \(\psi(\lambda a.\psi_{\Omega_{a+1}}(\lambda b.\Omega_{b+1}\text{-stable})\text{-stable limit of }\lambda a.\Omega_{a+1}\text{-stable})\)
  • s(n,n{1■1{1,,2{1{1■1■3}2,,}2,,}2■2}2) has growth rate \(\psi(\lambda a.\psi_{\Omega_{a+1}}(\lambda b.b\text{-stable limit of }\lambda b.\Omega_{b+1}\text{-stable})\text{-stable limit of }\lambda a.\Omega_{a+1}\text{-stable})\)
  • s(n,n{1■1■1,2}2) has growth rate \(\psi(\lambda a.\Omega_{a+1}\text{-stable limit of }\lambda a.\Omega_{a+1}\text{-stable})\)
  • s(n,n{1■1■1■1,2}2) has growth rate \(\psi(\lambda a.\Omega_{a+1}\text{-stable limit of }\lambda a.\Omega_{a+1}\text{-stable limit of }\lambda a.\Omega_{a+1}\text{-stable})\)
  • s(n,n{1{1,,2{1,,2,,}2,,}2}2) has growth rate \(\psi(\Pi_2\text{-reflecting onto }\lambda a.\Omega_{a+1}\text{-stables})\)
  • s(n,n{1{1,,2{2,,}2{1,,2,,}2,,}2}2) has growth rate \(\psi(1\text{-stable onto }\lambda a.\Omega_{a+1}\text{-stables})\)
  • s(n,n{1{1,,2{1{1{1{1,,2,,}3,,}2}2,,}2{1,,2,,}2,,}2}2) has growth rate \(\psi(\lambda a.a\text{-stable onto }\lambda a.\Omega_{a+1}\text{-stables})\)
  • s(n,n{1{1{1,,2,,}3,,}1,2}2) has growth rate \(\psi(\lambda a.\Omega_{a+1}\text{-stable onto }\lambda a.\Omega_{a+1}\text{-stables})\)
  • s(n,n{1{1,,2{1,,2,,}1{2,,}2,,}2}2) has growth rate \(\psi(\lambda a.\Omega_{a+1}+1\text{-stable})\)
  • s(n,n{1{1,,2{1,,2,,}1{1{1{1{1,,2,,}1{1,,2,,}2,,}2}2,,}2,,}2}2) has growth rate \(\psi(\lambda a.\Omega_{a+1}+a\text{-stable})\)
  • s(n,n{1{1{1,,2,,}1{1,,2,,}2,,}1,2}2) has growth rate \(\psi(\lambda a.\Omega_{a+1}2\text{-stable})\)
  • s(n,n{1{1,,2{2,,2,,}2,,}2}2) has growth rate \(\psi(\lambda a.\Omega_{a+1}\omega\text{-stable})\)
  • s(n,n{1{1{1,,3,,}2,,}1,2}2) has growth rate \(\psi(\lambda a.\Omega_{a+1}^2\text{-stable})\)
  • s(n,n{1{1{1,,1,,2,,}2,,}1,2}2) has growth rate \(\psi(\lambda a.\Omega_{a+1}^{\Omega_{a+1}}\text{-stable})\)
  • s(n,n{1{1{1,,2,,2,,}2,,}1,2}2) has growth rate \(\psi(\lambda a.\Omega_{a+1}^{\Omega_{a+1}+1}\text{-stable})\)
  • s(n,n{1{1{1,,1,,3,,}2,,}1,2}2) has growth rate \(\psi(\lambda a.\Omega_{a+1}^{\Omega_{a+1}2}\text{-stable})\)
  • s(n,n{1{1{1,,1,,1,,2,,}2,,}1,2}2) has growth rate \(\psi(\lambda a.\Omega_{a+1}^{\Omega_{a+1}^2}\text{-stable})\)
  • s(n,n{1{1,,2{1{2,,}2,,}2,,}2}2) has growth rate \(\psi(\lambda a.\Omega_{a+1}^{\Omega_{a+1}^\omega}\text{-stable})\)
  • s(n,n{1{1{1{1,,2,,}2,,}2,,}1,2}2) has growth rate \(\psi(\lambda a.\Omega_{a+1}^{\Omega_{a+1}^{\Omega_{a+1}}}\text{-stable})\)
  • s(n,n{1{1{1{1,,1,,2,,}2,,}2,,}1,2}2) has growth rate \(\psi(\lambda a.\Omega_{a+1}^{\Omega_{a+1}^{\Omega_{a+1}^{\Omega_{a+1}}}}\text{-stable})\)
  • s(n,n{1{1{1{1{1,,2,,}2,,}2,,}2,,}1,2}2) has growth rate \(\psi(\lambda a.\Omega_{a+1}^{\Omega_{a+1}^{\Omega_{a+1}^{\Omega_{a+1}^{\Omega_{a+1}}}}}\text{-stable})\)
  • s(n,n{1{1,,2`,,2,,}2}2) has growth rate \(\psi(\lambda a.\varepsilon_{\Omega_{a+1}+1}\text{-stable})\)
  • s(n,n{1{1,,2{1,,`1,2,,}2,,}2}2) has growth rate \(\psi(\lambda a.\psi_{\Omega_{a+2}}(\Omega_{a+\omega})\text{-stable})\)
  • s(n,n{1{1,,`1,,2,,}2}2) has growth rate \(\psi(\text{fixed point of }\alpha\mapsto\lambda a.\psi_{\Omega_{a+2}}(\Omega_{a+\alpha})\text{-stable})\)
  • s(n,n{1{1,,2{1,,`1{1{1,,`1,,2,,}2}2,,}2,,}2}2) has growth rate \(\psi(\lambda a.\psi_{\Omega_{a+2}}(\Omega_{a2})\text{-stable})\)
  • s(n,n{1{1,,`1,,2,,}1,2}2) has growth rate \(\psi(\lambda a.\psi_{\Omega_{a+2}}(\Omega_{\Omega_{a+1}})\text{-stable})\)
  • s(n,n{1{1,,`1{1,,`1,,2,,}2,,}1,2}2) has growth rate \(\psi(\lambda a.\psi_{\Omega_{a+2}}(\Omega_{\psi_{\Omega_{a+2}}(\Omega_{\Omega_{a+1}})})\text{-stable})\)
  • s(n,n{1{1,,2{1,,`1`,,2,,}2,,}2}2) has growth rate \(\psi(\lambda a.\psi_{\Omega_{a+2}}(\Omega_{\Omega_{a+2}})\text{-stable})\)

Up to here, the computable part of TON still works well, except some minor difference (\(C(\Omega_2+d_1^{d_1^{d_1}},0)\) corresponds with \(C(\Omega_2+d_1^{d_1^{d_1}}+1,0)\) somewhere in computable expressions). Beyond that, things suddenly go wrong.

  • s(n,n{1{1,,2{1,,`1{1,,`3,,}2,,}2,,}2}2) has growth rate \(\psi(\lambda a.\psi_{\Omega_{a+2}}(\Omega_{\Omega_{a+3}})\text{-stable})\)
  • s(n,n{1{1,,`1{1,,`1,,2,,}2,,}1,2}2) has growth rate \(\psi(\lambda a.\psi_{\Omega_{a+2}}(\Omega_{\Omega_{\Omega_{a+1}}})\text{-stable})\)
  • s(n,n{1{1,,2{1,,`1{1,,`1`,,2,,}2,,}2,,}2}2) has growth rate \(\psi(\lambda a.\psi_{\Omega_{a+2}}(\Omega_{\Omega_{\Omega_{a+2}}})\text{-stable})\)
  • s(n,n{1{1,,`1{1,,`1{1,,`1,,2,,}2,,}2,,}1,2}2) has growth rate \(\psi(\lambda a.\psi_{\Omega_{a+2}}(\Omega_{\Omega_{\Omega_{\Omega_{a+1}}}})\text{-stable})\)
  • s(n,n{1{1,,2{1,,`1,,`2,,}2,,}2}2) has growth rate \(\psi(\lambda a.\psi_{\Omega_{a+2}}(\psi_{I_{a+1}}(0))\text{-stable})\)
  • s(n,n{1{1,,2{1,,`1{1,,`1,,`2,,}3,,}2,,}2}2) has growth rate \(\psi(\lambda a.\psi_{\Omega_{a+2}}(\psi_{I_{a+1}}(1))\text{-stable})\)

Back to TON, \(C(C(\Omega_22+C(\Omega_2+C(\Omega_22+1,0),0),0),0)\) only corresponds with s(n,n{1{1,,`1,2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2), so it does not reach \(\Pi^1_2\text{-CA}_0\).

Interpretation 2

If the structure of "KP + large ordinal axioms" cannot use abstraction-stable ordinals, then the need of larger ordinals will come earlier, and the structure will be weaker than the former interpretation. For example, let \(S\) be a large ordinal such that the collapsing of it results computably-stable ordinals, such as

  • \(\psi_S(0)\) is first 1-stable
  • \(\psi_S(1)\) is first 2-stable
  • \(\psi_S(\Omega)\) is first \(\Omega\)-stable (\(\Omega_\alpha\) means \(\omega^\text{CK}_\alpha\))
  • \(\psi_S(\psi_S(0))\) is first \(\psi_S(0)\)-stable
  • \(\psi_S(S)\) is first \(\lambda a.a\)-stable
  • \(\psi_S(S+1)\) is first \(\lambda a.a+1\)-stable
  • \(\psi_S(S2)\) is first \(\lambda a.a2\)-stable
  • \(\psi_S(S^S)\) is first \(\lambda a.a^a\)-stable
  • \(\psi_S(\varepsilon_{S+1})=\psi_S(\Omega_{S+1})\) is first \(\lambda a.\varepsilon_{a+1}\)-stable or \(\lambda a.\psi_{\Omega_{a+1}}(\Omega_{a+1})\)-stable

The minimum requirement for \(S\) is the ordinal larger than all computably-stable ordinals, so \(S\) should be first "next admissible"-stable ordinal. Similarly, the collapsing of first "next next admissible"-stable (or \(\lambda a.\Omega_{a+2}\)-stable) ordinal will result "computable over next admissible"-stable ordinals.

Let \(d_0=C(\Omega_22,0)\) and \(d_{n+1}=C(\Omega_2,d_n)\).

  • \(C(\Omega_2+d_1^{d_1^{C(d_1,d_0)}+1},0)\) is first \(\lambda a.\varepsilon_{a+1}\)-stable
  • \(C(C(\Omega_22+C(\Omega_2+d_1^{d_1^{C(d_1,d_0)}},0),0),0)=\psi(\varepsilon_{S+1})\) (\(S\) is first \(\lambda a.\Omega_{a+1}\)-stable ordinal)
  • \(C(C(\Omega_22+C(\Omega_2+d_1^{d_1^{C(C(\Omega_2+1,d_0),d_0)}},0),0),0)=\psi(\Omega_{S+\omega})\)
  • \(C(C(\Omega_22+C(\Omega_2+d_1^{d_1^{C(C(\Omega_2+d_0,d_0),d_0)}},0),0),0)=\psi(\Omega_{S2})\)
  • \(C(C(\Omega_22+C(\Omega_2+d_1^{d_1^{C(C(\Omega_2+d_1,d_0),d_0)}},0),0),0)=\psi(\Omega_{\Omega_{S+1}})\)
  • \(C(C(\Omega_22+C(\Omega_2+d_1^{d_1^{C(C(\Omega_2+d_1^{d_1},d_0),d_0)}},0),0),0)=\psi(\Omega_{\Omega_{S+1}^{\Omega_{S+1}^{\Omega_{S+1}}}})\)
  • \(C(C(\Omega_22+C(\Omega_2+d_1^{d_1^{C(C(\Omega_22,d_0),d_0)}},0),0),0)=\psi(\Omega_{\Omega_{S+1}^{\Omega_{S+1}^{\Omega_{S+1}^{\Omega_{S+1}}}}})\)
  • \(C(C(\Omega_22+C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+1,0),d_0)}},0),0),0)=\psi(\Omega_{\Omega_{S+1}^{\Omega_{S+1}^{\Omega_{S+1}^{\Omega_{S+1}}}}\omega})\)
  • \(C(C(\Omega_22+C(\Omega_2+d_1^{d_1^{d_1}},0),0),0)=\psi(\Omega_{\Omega_{S+1}^{\Omega_{S+1}^{\Omega_{S+1}^{\Omega_{S+1}}}}S})\)
  • \(C(C(\Omega_22+C(\Omega_2+d_1^{d_1^{d_1}},0)2,0),0)=\psi(\Omega_{\Omega_{S+1}^{\Omega_{S+1}^{\Omega_{S+1}^{\Omega_{S+1}}}}S^2})\)
  • \(C(C(\Omega_22+C(C(\Omega_2,C(\Omega_2+d_1^{d_1^{d_1}},0)),C(\Omega_2+d_1^{d_1^{d_1}},0)),0),0)=\psi(\Omega_{\Omega_{S+1}^{\Omega_{S+1}^{\Omega_{S+1}^{\Omega_{S+1}}}}\varepsilon_{S+1}})\)
  • \(C(C(\Omega_22+C(C(\Omega_2+C(\Omega_2,C(\Omega_2+d_1^{d_1^{d_1}},0)),C(\Omega_2+d_1^{d_1^{d_1}},0)),C(\Omega_2+d_1^{d_1^{d_1}},0)),0),0)\\ =\psi(\Omega_{\Omega_{S+1}^{\Omega_{S+1}^{\Omega_{S+1}^{\Omega_{S+1}}}}\psi_{\Omega_{S+1}}(\Omega_{\Omega_{S+1}})})\)
  • \(C(C(\Omega_22+C(C(\Omega_2+C(\Omega_2+1,C(\Omega_2+d_1^{d_1^{d_1}},0)),C(\Omega_2+d_1^{d_1^{d_1}},0)),C(\Omega_2+d_1^{d_1^{d_1}},0)),0),0)\\ =\psi(\Omega_{\Omega_{S+\omega}})\)
  • \(C(C(\Omega_22+C(d_0,C(\Omega_2+d_1^{d_1^{d_1}},0)),0),0)=\psi(\psi_{I_{S+1}}(0))\)
  • \(C(C(\Omega_22+C(\Omega_2+d_1+1,C(\Omega_2+d_1^{d_1^{d_1}},0)),0),0)=\psi(I_{S+\omega})\)
  • \(C(C(\Omega_22+C(\Omega_2+d_1^{d_1}+1,C(\Omega_2+d_1^{d_1^{d_1}},0)),0),0)=\psi(K_{S+\omega})\)
  • \(C(C(\Omega_22+C(\Omega_2+d_1^{d_1^{d_0}},C(\Omega_2+d_1^{d_1^{d_1}},0)),0),0)=\psi(2\text{nd }\lambda a.\Omega_{a+1}\text{-stable})\)
  • \(C(C(\Omega_22+C(\Omega_2+d_1^{d_1^{d_1}}+1,0),0),0)=\psi(\omega\text{th }\lambda a.\Omega_{a+1}\text{-stable})\)
  • \(C(C(\Omega_22+C(\Omega_2+d_1^{d_1^{d_1}}+d_0,0),0),0)=\psi(\lambda a.\Omega_{a+1}\text{-stable-fixed-point})\)
  • \(C(C(\Omega_22+C(\Omega_2+d_1^{d_1^{d_1}+1},0),0),0)=\psi(\Pi_2\text{-reflecting onto }\lambda a.\Omega_{a+1}\text{-stables})\)
  • \(C(C(\Omega_22+C(\Omega_2+d_1^{d_1^{d_1}+d_1},0),0),0)=\psi(\Pi_3\text{-reflecting onto }\lambda a.\Omega_{a+1}\text{-stables})\)
  • \(C(C(\Omega_22+C(\Omega_2+d_1^{d_1^{d_1}+d_1^\omega},0),0),0)=\psi(1\text{-stable onto }\lambda a.\Omega_{a+1}\text{-stables})\)
  • \(C(C(\Omega_22+C(\Omega_2+d_1^{d_1^{d_1}2},0),0),0)=\psi(\lambda a.\Omega_{a+1}\text{-stable onto }\lambda a.\Omega_{a+1}\text{-stables})\)
  • \(C(C(\Omega_22+C(\Omega_2+d_1^{d_1^{d_1+\omega}},0),0),0)=\psi(\lambda a.\Omega_{a+1}+1\text{-stable})=\psi(\psi_{S_2}(1))\) (\(S_2\) is first \(\lambda a.\Omega_{a+2}\)-stable ordinal)
  • \(C(C(\Omega_22+C(\Omega_2+d_1^{d_1^{d_1+d_0}},0),0),0)=\psi(S_2)\)
  • \(C(C(\Omega_22+C(\Omega_2+d_1^{d_1^{d_12+\omega}},0),0),0)=\psi(S_2+\psi_{S_2}(S_2+1))\)
  • \(C(C(\Omega_22+C(\Omega_2+d_1^{d_1^{d_12+d_0}},0),0),0)=\psi(S_22)\)
  • \(C(C(\Omega_22+C(\Omega_2+d_1^{d_1^{d_1\omega}},0),0),0)=\psi(S_2\omega)\)
  • \(C(C(\Omega_22+C(\Omega_2+d_1^{d_1^{d_1d_0}},0),0),0)=\psi(S_2^2)\)
  • \(C(C(\Omega_22+C(\Omega_2+d_1^{d_1^{d_1^\omega}},0),0),0)=\psi(S_2^\omega)\)
  • \(C(C(\Omega_22+C(\Omega_2+C(d_2,d_1),0),0),0)=\psi(\varepsilon_{S_2+1})\)
  • \(C(C(\Omega_22+C(\Omega_2+C(C(\Omega_2+1,d_0),d_1),0),0),0)=\psi(\Omega_{S_2+\omega})\)
  • \(C(C(\Omega_22+C(\Omega_2+C(C(\Omega_2+d_0,d_0),d_1),0),0),0)=\psi(\Omega_{S_22})\)
  • \(C(C(\Omega_22+C(\Omega_2+C(C(\Omega_2+d_1,d_0),d_1)^{d_1^\omega},0),0),0)=\psi(\Omega_{S_22}+\psi_{S_2}(\Omega_{S_22}+1))\)
  • \(C(C(\Omega_22+C(\Omega_2+C(C(\Omega_2+d_1+1,d_0),d_1),0),0),0)=\psi(\Omega_{S_2\omega})\)
  • \(C(C(\Omega_22+C(\Omega_2+C(C(\Omega_2+C(C(\Omega_2+1,d_0),d_1),d_0),d_1),0),0),0)=\psi(\Omega_{\psi_{\Omega_{S_2+1}}(\Omega_{S_2+\omega})})\)
  • \(C(C(\Omega_22+C(\Omega_2+C(C(\Omega_22,d_0),d_1),0),0),0)=\psi(\Omega_{\Omega_{S_2+1}})\)
  • \(C(C(\Omega_22+C(\Omega_2+C(C(\Omega_22+1,0),d_1),0),0),0)=\psi(\Omega_{\Omega_{S_2+1}\omega})\)
  • \(C(C(\Omega_22+C(\Omega_2+d_2,0),0),0)=\psi(\Omega_{\Omega_{S_2+1}S_2})\)

Beyond that, TON is difficult to compare, so the comparisons below again use my array notation.

  • s(n,n{1`,,2,,}2) has growth rate \(\psi(\varepsilon_{S_2+1})\)
  • s(n,n{1`,,1,,2,,}2) has growth rate \(\psi(\varepsilon_{S_22})\)
  • s(n,n{1,,`3,,}2) has growth rate \(\psi(\varepsilon_{\Omega_{S_2+1}})\)
  • s(n,n{1,,`1,2,,}2) has growth rate \(\psi(\Omega_{S_2+\omega})\)
  • s(n,n{1,,`1,,2,,}2) has growth rate \(\psi(\Omega_{S_22})\)
  • s(n,n{1,,`1`,,2,,}2) has growth rate \(\psi(\Omega_{\Omega_{S_2+1}})\)
  • s(n,n{1,,`1{1,,`1,,2,,}2,,}2) has growth rate \(\psi(\Omega_{\Omega_{S_22}})\)
  • s(n,n{1,,`1,,`2,,}2) has growth rate \(\psi(\psi_{I_{S_2+1}}(0))\)
  • s(n,n{1,,`1{1,,`1,,`2,,}3,,}2) has growth rate \(\psi(\psi_{I_{S_2+1}}(1))\)

Back to TON, \(C(C(\Omega_22+C(\Omega_2+C(\Omega_22+1,0),0),0),0)\) only corresponds with s(n,n{1{1,,`1,2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2), so it does not reach \(\Pi^1_2\text{-CA}_0\).

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