User blog:P進大好きbot/Guideline on How to Use Large Cardinals for Ordinal Notations

This is a simle guideline on how to use large cardinal axioms to construct an ordinal notation in \(\textrm{ZFC}\) set theory.


 * 1) Choose any large cardinal(s) \(K\).
 * 2) Define an ordinal collapsing function(s) \(\Psi\) under \(\textrm{ZFC} + K\).
 * 3) Define an ordinal notation, i.e. a countable set \(C\) equipped with a map \(o\) to \(\textrm{ON}\), by translating constants and functions into constant term symbols and function symbols) under \(\textrm{ZFC} + K\).
 * 4) Compute the binary relation \(\alpha <_C \beta \stackrel{\tetrm{def}}{\Leftrightarrow} o(\alpha) \in o(\beta)\) under \(\textrm{ZFC} + K\).
 * 5) Analyse the ordinal type of segments of \((C,<_C)\) by comparing other known countable ordinals, e.g. \(\textrm{PTO}(A)\) for some an arithmetic or a set theory \(A\), under \(\textrm{ZFC} + K\).
 * 6) Prove that \(<_C\) admits a (recursive) definition under \(\textrm{ZFC}\).
 * 7) Prove that \(<_C\) is well-founded under \(\textrm{ZFC}\).
 * 8) Prove that the analysis on \((C,<_C)\) can be reduced to \(\textrm{ZFC}\).

None would forget to check Step 1, 2, 3, 4, and 5, but I sometimes saw googologists to forget to check Step 6, 7, and 8. I explain more about it.

= Definition under ZFC =

If there is no interpretation of \(<_C\) into a definition under \(\textrm{ZFC}\), then \((C,<_C)\) is useless for googologists to construct a well-defined large number under \(\textrm{ZFC}\).

Moreover, if there is no recursive way to compute a large number under \(\textrm{ZFC}\), then the resulting large number becomes uncomputable. For example, one might need recursive definition of \(<_C\), the subset of standard form, and the system of the fundamental sequence, and so on.

The definition of \((C,<_C)\) under \(\textrm{ZFC} + K\) just ensures that the existence of such a \((C,<_C)\) is consitent with \(\textrm{ZFC}). It does not imply that \((C,<_C)\) is also well-defined under \(\textrm{ZFC}\).

= Well-foundedness under ZFC =

The binary relation \(<_C\) is well-founded by the definition under \(\textrm{ZFC} + K\), because \((\textrm{ON},\in)\) is well-founded.

However, the well-foundedness of \(<_C\) under \(\textrm{ZFC} + K\) just ensures that the well-foundedness of \(<_C\) is consistent with \(\textrm{ZFC}\). It does not imply that the ordinal type of \((C,<_C)\) is well-defined under \(\textrm{ZFC}\). It would be no longer called as an ordinal notation, but just a notation.

= Analysis under ZFC =

The analysis of \((C,<_C)\) heavily depends on \(K\). A result like \(\Psi(\varepsilon_K) = \textrm{PTO}(A)\) under \(\textrm{ZFC} + K\) just ensures that the statement that the limit of \((C,<_C)\) is beyond \(\textrm{ZFC} + K\) is consistent with \(\textrm{ZFC}\). It does not imply that \((C,<_C)\) has that great strength under \(\textrm{ZFC}\).

If you want to analyse \((C,<_C)\) under \(\textrm{ZFC}\), then you need to refer to other known results under \(\textrm{ZFC}\), or sincerely verify your desired "fact" by studying the proof-theoretic ordinal analysis, the reflection property, or something like that.

= Footnotes =