The Buchholz's psi-functions are a hierarchy of single-argument ordinal functions \(\psi_\nu(\alpha)\) introduced by Wilfried Buchholz in 1986. These functions are a simplified version of the \(\theta\)-functions, but nevertheless have the same strength as those.
Definition
\(C_\nu^0(\alpha) = \Omega_\nu\),
\(C_\nu^{n+1}(\alpha) = C_\nu^n(\alpha) \cup \{\gamma | P(\gamma) \subseteq C_\nu^n(\alpha)\} \cup \{\psi_\mu(\xi) | \xi \in \alpha \cap C_\nu^n(\alpha) \wedge \xi \in C_\mu(\xi) \wedge \mu \leq \omega\}\),
\(C_\nu(\alpha) = \bigcup_{n < \omega} C_\nu^n (\alpha)\),
\( \psi_\nu(\alpha) = \min\{\gamma | \gamma \not\in C_\nu(\alpha)\}\),
where
\(\Omega_\nu=\left\{\begin{array}{lcr} 1\text{ if }\nu=0\\ \aleph_\nu\text{ if }\nu>0\\ \end{array}\right.\)
and \(P(\gamma)=\{\gamma_1,...,\gamma_k\}\) is the set of additive principal numbers in form \(\omega^\xi\),
\(P=\{\alpha\in On: 0<\alpha \wedge \forall \xi, \eta < \alpha (\xi+\eta < \alpha)\}=\{\omega^\xi: \xi \in On\}\),
the sum of which gives this ordinal \(\gamma\):
\(\gamma=\gamma_1+\cdots+\gamma_k\) and \(\gamma_1\geq\cdots\geq\gamma_k\).
Thus \(C_\nu(\alpha)\) denotes the set of all ordinals which can be generated from ordinals \(<\aleph_\nu\) by the functions + (addition) and \(\psi_{\mu\le\omega}(\xi<\alpha)\).
Properties
Buchholz showed following properties of this functions:
\(\psi_\nu(0)=\Omega_\nu\),
\(\psi_\nu(\alpha)\in P\),
\(\psi_\nu(\alpha+1)=\text{min}\{\gamma\in P: \psi_\nu(\alpha)<\gamma\}\),
\(\Omega_\nu\le\psi_\nu(\alpha)<\Omega_{\nu+1} \),
\(\psi_0(\alpha)=\omega^\alpha \text{ if }\alpha<\varepsilon_0\),
\(\psi_\nu(\alpha)=\omega^{\Omega_\nu+\alpha} \text{ if }\alpha<\varepsilon_{\Omega_\nu+1}\),
\(\theta(\varepsilon_{\Omega_\nu+1},0)=\psi(\varepsilon_{\Omega_\nu+1})\) for \(0<\nu\le\omega\).
Extension
Let me rewrite Buchholz's definition as follows:
\(C_\nu^0(\alpha) = \{\beta|\beta<\Omega_\nu\}\),
\(C_\nu^{n+1}(\alpha) = \{\beta+\gamma,\psi_\mu(\eta)|\mu,\beta, \gamma,\eta\in C_{\nu}^n(\alpha)\wedge\eta<\alpha\}\),
\(C_\nu(\alpha) = \bigcup_{n < \omega} C_\nu^n (\alpha)\),
\(\psi_\nu(\alpha) = \min\{\gamma | \gamma \not\in C_\nu(\alpha)\}\),
where
\(\Omega_\nu=\left\{\begin{array}{lcr} 1\text{ if }\nu=0\\ \text{smallest ordinal with cardinality}\aleph_\nu \text{ if }\nu>0\\ \end{array}\right.\)
There is only one little detail difference with original Buchholz definition: ordinal \(\mu\) is not limited by \(\omega\), now ordinal \(\mu\) belong to previous set \(C_n\). For example if \(C_0^0(1)=\{0\}\) then \(C_0^1(1)=\{0,\psi_0(0)=1\}\) and \(C_0^2(1)=\{0,...,\psi_1(0)=\Omega\}\) and \(C_0^3(1)=\{0,...,\psi_\Omega(0)=\Omega_\Omega\}\) and so on.
Limit of this notation is omega fixed point \(\psi(\Omega_{\Omega_{\Omega_{...}}})\).
Explanation
\(C_0^0(\alpha)=\{0\} =\{\beta:\beta<1\}\),
\(C_0(0)=\{0\}\) (since no functions \(\psi(\eta<0)\) and 0+0=0).
Then \(\psi_0(0)=1\).
\(C_0(1)\) includes \(\psi_0(0)=1\) and all possible sums of natural numbers:
\(C_0(1)=\{0,1,2,...,\text{googol}, ...,\text{TREE(googol)},...\}\).
Then \(\psi_0(1)=\omega\) - first transfinite ordinal, which is greater than all natural numbers by its definition.
\(C_0(2)\) includes \(\psi_0(0)=1, \psi_0(0)=\omega\) and all possible sums of them.
Then \(\psi_0(2)=\omega^2\).
For \(C_0(\omega)\) we have set \(C_0(\omega)=\{0,\psi(0)=1,...,\psi(1)=\omega,...,\psi(2)=\omega^2,...,\psi(3)=\omega^3,...\}\).
Then \(\psi_0(\omega)=\omega^\omega\).
For \(C_0(\Omega)\) we have set \(C_0(\Omega)=\{0,\psi(0)=1,...,\psi(1)=\omega,...,\psi(\omega)=\omega^\omega,...,\psi(\omega^\omega)=\omega^{\omega^\omega},...\}\).
Then \(\psi_0(\Omega)=\varepsilon_0\).
For \(C_0(\Omega+1)\) we have set \(C_0(\Omega)=\{0,1,...,\psi_0(\Omega)=\varepsilon_0,...,\varepsilon_0+\varepsilon_0,...\psi_1(0)=\Omega,...\}\).
Then \(\psi_0(\Omega+1)=\varepsilon_0\omega=\omega^{\varepsilon_0+1}\).
\(\psi_0(\Omega2)=\varepsilon_1\),
\(\psi_0(\Omega^2)=\zeta_0\),
\(\varphi(\alpha,1+\beta)=\psi_0(\Omega^\alpha\beta)\),
\(\psi_0(\Omega^\Omega)=\Gamma_0=\theta(\Omega,0)\), using Feferman theta-function,
\(\psi_0(\Omega^{\Omega^\Omega})\) is large Veblen ordinal,
\(\psi_0(\Omega\uparrow\uparrow\omega)=\psi_0(\varepsilon_{\Omega+1})=\theta(\varepsilon_{\Omega+1},0)\).
Okay, now let's research how \(\psi_1\) works:
\(C_1^0(\alpha)=\{\beta:\beta<\Omega_1\}=\{0,\psi(0)=1,2,...\text{googol},...,\psi_0(1)=\omega,...,\psi_0(\Omega)=\varepsilon_0,...\)
\(...,\psi_0(\Omega^\Omega)=\Gamma_0,...,\psi(\Omega^{\Omega^\Omega+\Omega^2}),...\}\) i.e. includes all countable ordinals.
\(C_1(\alpha)\) includes all possible sums of all countable ordinals. Then
\(\psi_1(0)=\Omega_1\) first uncountable ordinal which is greater than all countable ordinal by its definition i.e. smallest number with cardinality \(\aleph_1\).
\(C_1(1)=\{0,...,\psi_0(0)=\omega,...,\psi_1(0)=\Omega,...,\Omega+\omega,...,\Omega+\Omega,...\}\)
Then \(\psi_1(1)=\Omega\omega=\omega^{\Omega+1}\).
Then \(\psi_1(2)=\Omega\omega^2=\omega^{\Omega+2}\),
\(\psi_1(\psi_0(\Omega))=\Omega\varepsilon_0=\omega^{\Omega+\varepsilon_0}\),
\(\psi_1(\psi_0(\Omega^\Omega))=\Omega\Gamma_0=\omega^{\Omega+\Gamma_0}\),
\(\psi_1(\psi_1(0))=\psi_1(\Omega)=\Omega^2=\omega^{\Omega+\Omega}\),
\(\psi_1(\psi_1(\psi_1(0)))=\omega^{\Omega+\omega^{\Omega+\Omega}}=\omega^{\Omega\cdot\Omega}=(\omega^{\Omega})^\Omega=\Omega^\Omega\),
\(\psi_1^5(0)=\Omega^{\Omega^\Omega}\),
\(\psi_1(\Omega_2)=\psi_1^\omega(0)=\Omega\uparrow\uparrow\omega=\varepsilon_{\Omega+1}\).
For case \(\psi(\Omega_2)\) the set \(C_0(\Omega_2)\) includes functions \(\psi_0\) with all arguments less than \(\Omega_2\) i.e. such arguments as \(0, \psi_1(0), \psi_1(\psi_1(0)), \psi_1^3(0),..., \psi_1^\omega(0)\)
and then \(\psi_0(\Omega_2)=\psi_0(\psi_1(\Omega_2))=\psi_0(\varepsilon_{\Omega+1})\).
In general case: \(\psi_0(\Omega_{\nu+1})=\psi_0(\psi_\nu(\Omega_{\nu+1}))=\psi_0(\varepsilon_{\Omega_\nu+1})=\theta(\varepsilon_{\Omega_\nu+1},0)\).
We also can write:
\(\theta(\Omega_\nu\uparrow\uparrow(k),0)=\psi_0(\Omega_\nu\uparrow\uparrow(k+1))\) (at least for \(1\le k<\omega; 1\le\nu<\omega\) it must be true).
Normal form
Previously we wrote in section "Definition" \(\Omega_\nu=1\) if \(\nu=0\), but now, not changing the definition of \(\psi\)-function, let's define for sections "Normal form" and "Fundamental sequences" \(\Omega_0=\omega\) for unification of NF and FS-rules for all cases \(\nu\geq 0\). Then \(\text{cof}(\Omega_\nu)= \Omega_\nu\) (and also let's note that \(\text{cof}(s)= 1\), where \(s\) is a successor ordinal, \(\text{cof}(0)= 0\) and \(\Omega_\nu^0=1\)).
The ordinal \(\alpha\) is an additive principal number (\(\alpha\in P\)) if \(\beta+\gamma<\alpha\) for all \(\beta,\gamma<\alpha\)
1) If \(\alpha\notin P\) (i.e. \(\alpha\) is not additive principal number) then normal form for \(\alpha:\)
\(\alpha=\alpha_1+\cdots+\alpha_k\) where \(\alpha_1,...,\alpha_k\in P\) and \(\alpha>\alpha_1\geq\cdots\geq\alpha_k\) and each \(\alpha_i\) also is written in normal form, \(i \in \{1,...,k\}\).
2) If \(\alpha\in P\) and \(\alpha=\Omega_\nu^\beta \gamma\) then normal form for \(\alpha:\)
\(\alpha=\Omega_\nu^\beta \gamma\) where \(\beta<\Omega_\nu^\beta\wedge\gamma<\Omega_\nu\wedge\gamma\in P\) and \(\nu, \beta, \gamma\) also are written in normal form.
3) If \(\alpha\in P\) and \(\alpha=\psi_\nu(\beta)\) then normal form for \(\alpha:\)
\(\alpha=\psi_\nu(\beta)\) and \(\nu, \beta\) also are written in normal form.
If \(\alpha\notin P\) is an ordinal, such that \(\Omega_\nu<\alpha<\varepsilon_{\Omega_\nu+1}\) (between \(\omega\) and \(\varepsilon_0\) if \(\nu=0\)) then normal form for \(\alpha\):
\(\Omega_\nu^{\beta_1}\gamma_1+\Omega_\nu^{\beta_2}\gamma_2+\cdots+\Omega_\nu^{\beta_k}\gamma_k\), where
- \(\alpha>\Omega_\nu^{\beta_1}\gamma_1\geq \cdots \geq \Omega_\nu^{\beta_k}\gamma_k\),
- \(\text{cof}(\beta_m)\le\Omega_\nu\wedge\beta_m\geq 0\wedge\text{cof}(\gamma_m)<\Omega_\nu\wedge\gamma_m<\Omega_\nu\wedge\gamma_m\in P\) for \(1\le m \le k\),
and Cantor normal form is partial case when \(\nu=0\):
\(\omega^{\beta_1}+\cdots+\omega^{\beta_k}\), where \(\beta_1\geq\cdots\geq\beta_k\) and each \(\beta_m\) can be successor or limit ordinal with cofinality 1 or \(\omega\).
Fundamental sequences
The fundamental sequence for an ordinal with cofinality \(\Omega_\nu\) is a distinguished strictly increasing sequence with length \(\Omega_\nu\), which has the ordinal as its limit. Let \(\alpha\in s\) denotes \(\alpha\) is a successor ordinal and \(\alpha\in L_\nu\) denotes \(\alpha\) is a limit ordinal with cofinality \(\Omega_\nu\) (with length of fundamental sequence \(\Omega_\nu\)). For example, \(\alpha\in L_0\) denotes a limit countable ordinal with cofnality \(\Omega_0=\omega\).
For ordinals, written in mentioned normal form, fundamental sequences are defined as follows:
1) If \(\alpha=\alpha_1+\alpha_2+\cdots+\alpha_k\), where \(\alpha_1\geq\alpha_2\geq\cdots\geq\alpha_k\) then \(\text{cof}(\alpha)=\text{cof}(\alpha_k)\) and \(\alpha[\eta]=\alpha_1+\alpha_2+\cdots+(\alpha_k[\eta])\),
2) if \(\alpha=\psi_\nu(\beta+1)\) then \(\text{cof}(\alpha)=\omega\) and \(\alpha[n]=\psi_\nu(\beta)\cdot n\), and note that \(\psi_0(0)=1\) and \(\psi_\nu(0)=\Omega_\nu\) for \(\nu>0\),
3) if \(\alpha=\psi_\nu(\beta)\) and \(\beta\in L_{\mu\le\nu}\) then \(\text{cof}(\alpha)=\text{cof}(\beta)\) and \(\alpha[\eta]=\psi_\nu(\beta[\eta])\),
4) if \(\alpha=\psi_\nu(\beta)\) and \(\beta\in L_{\mu+1>\nu}\) then \(\text{cof}(\alpha)=\omega\) and \(\left\{\begin{array}{lcr} \alpha[\eta]=\psi_\nu(\beta[\gamma[\eta]])\\ \gamma[0]=\Omega_\mu \text{ if }\mu\geq 1\\ \gamma[0]=0\text{ if }\mu=0\\ \gamma[\eta+1]=\psi_\mu(\beta[\gamma[\eta]])\\ \end{array}\right.,\)
5) if \(\alpha=\Omega_\nu^\beta \) and \(\beta\in s\) then \(\text{cof}(\alpha)=\Omega_\nu\) and \(\alpha[\eta]=\Omega_\nu^{\beta-1}\cdot \eta\),
6) if \(\alpha=\Omega_\nu^\beta \gamma\) and \(\gamma \in L_{\mu<\nu}\) then \(\text{cof}(\alpha)=\text{cof}(\gamma)\) and \(\alpha[\eta]=\Omega_\nu^\beta (\gamma[\eta])\),
7) if \(\alpha=\Omega_\nu^\beta\) and \(\beta \in L_{\mu\le\nu}\) then \(\text{cof}(\alpha)=\text{cof}(\beta)\) and \(\alpha[\eta]=\Omega_\nu^{\beta[\eta]}\),
8) if \(\alpha=\Omega_{\mu+1}\) then \(\text{cof}(\alpha)=\Omega_{\mu+1}\) and \(\alpha[\eta]=\eta\) (as well as \(\omega[n]=n\)),
9) if \(\alpha=\Omega_{\beta}\) and \(\beta\in L_{\mu}\) then \(\text{cof}(\alpha)=\text{cof}(\beta)\) and \(\alpha[\eta]=\Omega_{\beta[\eta]}\).
Rules 1-9 assign FS for each limit ordinal up to omega fixed point \(\psi(\Omega_{\Omega_{\Omega_{...}}})\), where \(\psi\) without lower-line index denotes \(\psi_0\). After the definition of fundamental sequences up to the omega fixed point we can use this notation for fast-growing hierarchy.
Detailed example for illustrating working of rules for fundamental sequences
Let's define \(\psi_I(0)[0]=0\) and \(\psi_I(0)[n+1]=\Omega_{\psi_I(0)[n]}\)
Let us consider the following example and find \(\psi(\psi_I(0))[2]\) where \(\psi\) denotes \(\psi_0\)
Since \(\text{cof}(\psi_0)=\Omega_0=\omega\) as well as \(\text{cof}(\psi_I(0))=\omega\) consequently use rule 3
\(\psi(\psi_I(0))[2]=\psi(\psi_I(0)[2])=\psi(\Omega_{\Omega_0})=\psi(\Omega_\omega)\)
\(\Omega_\omega[\eta]=\Omega_{\omega[\eta]}\) (rule 9)
Thus, \(\text{cof}(\Omega_\omega)=\omega\)
and, consequently, use rules 3 and 9.
\(\psi(\Omega_\omega)[2]=\psi(\Omega_\omega[2])=\psi(\Omega_{\omega[2]})=\psi(\Omega_2)\)
\(\text{cof}(\Omega_2)=\Omega_2>\text{cof}(\psi_0)=\omega\)
and, сonsequently, use rule 4 and 8, taking into account
\(\psi_\nu(\alpha)=\omega^{\Omega_\nu+\alpha}\) for \(\nu>0\) and \(\alpha<\varepsilon_{\Omega_nu+1}\);
\(\psi_0(\alpha)=\omega^{\alpha}\) for \(\alpha<\varepsilon_0\)
Then, according rules 4 and 8
\(\psi(\Omega_2)[2]=\psi(\Omega_2[\gamma[2]])=\psi(\gamma[2])\) where
\(\gamma[0]=\Omega_1=\Omega\)
\(\gamma[1]=\psi_1(\Omega_2[\Omega])=\psi_1(\Omega)=\omega^{\Omega+\Omega}=\Omega^2\)
\(\gamma[2]=\psi_1(\Omega_2[\Omega^2])=\psi_1(\Omega^2)=\omega^{\Omega+\Omega\cdot\Omega}=\omega^{\Omega\cdot\Omega}=\Omega^\Omega\)
\(\psi(\Omega_2)[2]=\psi(\Omega^\Omega)\)
Next step:
\(\Omega^\Omega[\eta]=\Omega^{\Omega[\eta]}\) (rule 7)
\(\text{cof}(\Omega^\Omega)=\Omega>\text{cof}(\psi_0)=\omega\)
and, сonsequently, again use rule 4 and 8
\(\psi(\Omega^\Omega)[2]=\psi(\Omega^{\gamma[2]})\)
where
\(\gamma[0]=0\)
\(\gamma[1]=\psi(\Omega^\Omega[0])=\psi(\Omega^{\Omega[0]})=\psi(\Omega^0)=\psi(1)=\omega\)
\(\gamma[2]=\psi(\Omega^\Omega[\omega])=\psi(\Omega^{\Omega[\omega]})=\psi(\Omega^\omega)\)
\(\psi(\Omega^\Omega)[2]=\psi(\Omega^{\psi(\Omega^\omega)})\)
Next step:
According rules 3 and 7
\( \omega=\text{cof}(\Omega^\omega)=\text{cof}(\psi(\Omega^\omega))=\text{cof}(\Omega^{\psi(\Omega^\omega)})=\text{cof}(\psi(\Omega^{\psi(\Omega^\omega)}))\)
and, сonsequently,
\(\psi(\Omega^{\psi(\Omega^\omega)})[2]=\psi(\Omega^{\psi(\Omega^{\omega[2]})})=\psi(\Omega^{\psi(\Omega^2)})\)
Next step:
\(\Omega^2[\eta]=\Omega\cdot\eta\) and \(\text{cof}(\Omega^2)=\Omega\) (rule 5)
\(\text{cof}(\psi(\Omega^{\psi(\Omega^2)}))=\text{cof}(\Omega^{\psi(\Omega^2)})=\text{cof}(\psi(\Omega^2))=\omega\) (rules 3 and 7)
and, сonsequently,
\(\psi(\Omega^{\psi(\Omega^2)})[2]=\psi(\Omega^{\psi(\Omega^2)[2]})=\)
\(=\psi(\Omega^{\psi(\Omega\cdot\gamma[2])})\)
where
\(\gamma[0]=0\)
\(\gamma[1]=\psi(\Omega\cdot 0)=\psi(0)=\omega^0=1\)
\(\gamma[2]=\psi(\Omega\cdot 1)=\psi(\Omega)\)
Thus \(\psi(\Omega^{\psi(\Omega^2)})[2]=\psi(\Omega^{\psi(\Omega\cdot\psi(\Omega))})\)
Next step:
According rules 3, 6 and 7
\(\omega=\text{cof}(\psi(\Omega))=\text{cof}(\Omega\cdot\psi(\Omega))=\text{cof}(\psi(\Omega\cdot\psi(\Omega)))=\)
\(=\text{cof}(\Omega^{\psi(\Omega\cdot\psi(\Omega))})=\text{cof}(\psi(\Omega^{\psi(\Omega\cdot\psi(\Omega))}))\)
and, сonsequently,
\(\psi(\Omega^{\psi(\Omega\cdot\psi(\Omega))})[2]=\psi(\Omega^{\psi(\Omega\cdot\psi(\Omega)[2])})\)
and \(\psi(\Omega)[2]=\psi(\Omega[\gamma[2]])=\psi(\Omega[\omega])=\psi(\omega)=\omega^\omega\)
where \(\gamma[0]=0\)
\(\gamma[1]=\psi(\Omega[0])=\psi(0)=1\)
\(\gamma[2]=\psi(\Omega[1])=\psi(1)=\omega\)
That is why \(\psi(\Omega^{\psi(\Omega\cdot\psi(\Omega)[2])})=\psi(\Omega^{\psi(\Omega\cdot\omega^\omega)})\)
Next step:
\( \psi(\Omega^{\psi(\Omega\cdot\omega^\omega)})[2]=\psi(\Omega^{\psi(\Omega\cdot\omega^2)})\)
\(\psi(\Omega^{\psi(\Omega\cdot\omega^2)})[2]=\psi(\Omega^{\psi(\Omega\cdot\omega\cdot2)})\)
\(\psi(\Omega^{\psi(\Omega\cdot\omega\cdot2)})[2]=\psi(\Omega^{\psi(\Omega\cdot(\omega+2))})=\)
\(=\psi(\Omega^{\psi(\Omega\cdot\omega+\Omega+\Omega)})\)
Next step:
According rules 1 and 8
\((\Omega\cdot\omega+\Omega+\Omega)[\eta]=\Omega\cdot\omega+\Omega+\Omega[\eta]=\Omega\cdot\omega+\Omega+\eta\)
and \(\text{cof}(\Omega\cdot\omega+\Omega+\Omega)=\Omega>\omega=\text{cof}(\psi)\)
and, сonsequently, use rule 4
\(\psi(\Omega^{\psi(\Omega\cdot\omega+\Omega+\Omega)})[2]=\psi(\Omega^{\psi(\Omega\cdot\omega+\Omega+\gamma[2])})\)
where \(\gamma[0]=0\)
\( \gamma[1]=\psi(\Omega\cdot\omega+\Omega+0)=\psi(\Omega\cdot\omega+\Omega)\)
\(\gamma[2]=\psi(\Omega\cdot\omega+\Omega+\psi(\Omega\cdot\omega+\Omega))\),
That is why \(\psi(\Omega^{\psi(\Omega\cdot\omega+\Omega+\Omega)})[2]=\)
\(=\psi(\Omega^{\psi(\Omega\cdot\omega+\Omega+\psi(\Omega\cdot\omega+\Omega+\psi(\Omega\cdot\omega+\Omega)))})\)
and so on.
Examples of application of normal form
1) \(\psi_1(\psi_0(\psi_0(1))+\psi_0(1))+\psi_0(\psi_0(1))+\psi_0(1)\)
is valid since
\(\psi_1(\psi_0(\psi_0(1))+\psi_0(1))>\psi_0(\psi_0(1))>\psi_0(1)\)
\(\psi_1(\psi_0(\psi_0(1))+\psi_0(1)), \psi_0(\psi_0(1)), \psi_0(1) \in P\),
and all of arguments also are written in normal form. Note: \(P\) denotes class of additive principal numbers.
2) We can write this expression as
\(\omega^{\Omega_1+\omega^\omega+\omega}+\omega^\omega+\omega\)
this is also valid, since \(\omega=\Omega_0\) and this is
\(\Omega_0^{\Omega_1+\Omega_0^{\Omega_0}+\Omega_0}+\Omega_0^{\Omega_0}+\Omega_0=\)
\(=\Omega_0^{\alpha}\beta+\Omega_0^{\gamma}\beta+\Omega_0^{\delta}\beta\)
where
\(\Omega_0^{\alpha}\beta>\Omega_0^{\gamma}\beta>\Omega_0^{\delta}\beta\)
\(\Omega_0^{\alpha}\beta, \Omega_0^{\gamma}\beta, \Omega_0^{\delta}\beta \in P\)
\(\alpha<\Omega_0^{\alpha}, \gamma<\Omega_0^{\gamma}, \delta<\Omega_0^{\delta}\)
\(\beta=1, \beta<\Omega_0=\omega, \beta \in P\)
and all of exponents \(\alpha, \gamma, \delta\) also are written in normal form.
3) We can write this expression as
\(\Omega_1\cdot(\omega^\omega)+\Omega_1\cdot\omega+\omega^\omega+\omega=\Omega_1\cdot(\omega^\omega)+\Omega_1\cdot\omega+\Omega_1^0\cdot(\omega^\omega)+\Omega_1^0\cdot\omega\)
and this is also valid, since
\(\Omega_1\cdot(\omega^\omega)>\Omega_1\cdot\omega>\omega^\omega>\omega\)
\(\Omega_1\cdot(\omega^\omega),\Omega_1\cdot\omega,\omega^\omega,\omega \in P\)
\(\omega^\omega<\Omega_1,\omega<\Omega_1\)
For example,
\(\omega^\Omega\) is not valid, because the condition \(\beta<\Omega_\nu^\beta\) is not satisfied, but \(\omega^{\Omega+1}\) is valid, since\(\omega^{\Omega+1}=\Omega\omega>\Omega+1\).
\(\Omega_1\cdot(\omega^\omega+\omega)\) is not valid, since \(\gamma=\omega^\omega+\omega\) is not additive principal number (i.e. \(\gamma \notin P\)).
Examples of additive principal numbers are \(\omega^\alpha\) starting with \(\omega^0=1\), since 1 can not be obtained as sum of zeros.
If \(\alpha=\Omega_\nu^\beta\cdot \gamma\) and \(\gamma\) is successor ordinal, then \(\gamma=1\).
That is why the condition \(\Omega_\nu^\beta\cdot\gamma\), where \(\gamma\) is a successor ordinal, was excluded from rule set for FS.
Note: the post was written using materials from Deedlit's posts [1], [2] and Buchholz article [3].