User blog:Hyp cos/TON, stable ordinals, and my array notation

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 \(b\), 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. Then \(B_m(a,b)\) iff \(T_m(a,b,0)\).
 * \(B_0(a,b)\) if \(a< b\).
 * \(B_{m+1}(a,b)\) if for every subterm \(c\) of \(a\), "\(c<a\)" or "\(c\) is a subterm of \(d\), where \(d\) is a subterm of \(a\) and \(B_m(d,b)\)".
 * 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 \(T_m(e,b,c)\)

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.
 * Both \(a\) and \(b\) are standard
 * If \(b=C(c,d)\), then \(a\le c\)
 * \(B_n(a,C(a,b))\)

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

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.

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 \(\underbrace{C(C(\cdotsC(}_n\Omega_n2,0)\cdots,0),0)\). The notation may extend up to (but not including) \(\omega^\text{CK}_1\), so we can set \(\Omega_1=\omega^\text{CK}_1\).