User blog:P進大好きbot/Full References of Arguments on Ordinal Notations with Large Cardinals

I would like to list up references of the proof of statements on which the well-definedness of ordinal notations with large cardinals under \(\txetrm{ZFC}\). If you know precise references which I have not mention, please inform me.

I think that listing up full references helps us to prove the well-definedness of given ordinal notations and the correctness of analyses of them. For example, if a table of analysis contains an ordinal whose well-definedness actually depends on axioms indepedent of \(\textrm{ZFC}\), the analysis will be turned to be incorrect.

There might be googologists working on fixed stronger axioms than \(\textrn{ZFC}\), e.g. \(\textrm{ZFC} + \textrm{Inaccessible}\). It is great that we have googology under each choice of reasnable axioms.

On the other hand, if you do not fix axioms, i.e. assume any additional axioms compatible with your analysis, then this blog is not worth reading. Your analysis is always correct under the strongest axiom \(0 = 1\) or the weaker axiom directly stating that your analysis is correct.

Anyway, I would like to know which axioms analysts here are using in their analysis. To begin with, I asked three analysist here. Please inform me of what axioms you use in your analysis, if you are another analysist here.
 * rpakr: I am waiting for the answer.
 * KurohaKafka: I am waiting for the answer.
 * Alemagno12: I am waiting for the answer.

= Introduction =

Before reading this blog post, it is better to understand common mistakes on use of large cardinals in googology under \(\textrm{ZFC}\), which I wrote in a previous blog post. Also, the arguments in the replies of my another blog post might be worth reading.

Since it is well-known that the topic of large cardinals is absolutely difficult to understand, googologists would always be careful about such arguments. Nevertheless, why such mistakes occurs? Here is one of the answers.

There are several sophisticated articles and blog posts on ordinal notations with large cardinals and analyses of them, but the blog posters do not necessarily quote full references of the original paper.

Then many googologists refer to large cardinals without knowing original statements or original proofs. Without full references, googologists might comfound blog poster's personal descriptions and original arguments. This is the reason.

Also, omitting the full references is awfully problematic by several aspects: One of the worst problem is that googologists regard wrong assertions by blog posters as statements verified by the authors of the original papers. Although it is impossible to thoroughly annihilate all such troubles, it is much better to know the full reference of each statement.
 * It ignores the authority of the original paper.
 * It hides the responsibility of personal descriptions.
 * It prevents us from confirming that a given argument has already been reviewed or not.

I emphasise that I might write incorrect arguments here. Please point out 'explicitly' if there are some errors.

= References =


 * 1) [WM] M. Rathjen, Ordinal notations based on a weakly Mahlo cardinal, Archive for Mathematical Logic, Volume 29, Issue 4, pp. 249--263, 1990.
 * 2) [WI] M. Rathjen, Proof-theoretic analysis of KPM, Archive for Mathematical Logic, Volume 30,Issue 5--6, pp. 377--403, 1991.
 * 3) [KPM] M. Rathjen, Provable wellorderings of KPM, (in preparation).
 * 4) [PT] K. Schutte, Proof Theory, Part of the Grundlehren der mathematischen Wissenschaften book series, Volume 225, Springer-Verlag, 1977.

= Weakly Mahlo Cardinal =

This topic is mainly written in [WM].

Convention

 * Following the convention in [WM], \(M\) denotes the smallest weakly Mahlo cardinal.
 * I abbreviate "the existence of a weakly Mahlo cardinal" by \(\exists M\).

Definition of the Ordinal Notation System (T(M),<)
Safe!

It is defined just under \(\textrm{ZFC} + \exists M\) in Section 6 in [WM], but it will be translated into a system under \(\textfm[ZFC}\) in Section 7 in [WM].

Primitive Recursiveness of (T(M),<)
Safe!

It is verified under \(\textrm{ZFC}\) in Section 7 by applying Lemma 2.2 (vi), Proposition 2.4, Corollary 3.14, Lemma 5.5 (v) and (vi), Proposition 5.10, Lemma 5.13, and Lemma 5.14, Lemma 7.2, and Lemma 7.5 in [WM].

Well-foundedness of (T(M),<)
Dangerous?

It is verified just under \(\textrm{ZFC} + \exists M\) in Section 7 in [WM]. Rathjen wrote that it could be proved under \(\textrm{ZFC}\) by replacing \(M\) by its recursve analogue, and would be actually proved in a forthcoming paper. But I do not know in which paper Rathjen actually proved it.

Also, if the well-foundedness is not proved, then the well-definedness of the ordinal notation with weakly inaccessible cardinals is unknown because the proof of the well-definedness of the correspondence from ordinal terms to ordinals heavily uses the transfinite induction, which obvously depends on the well-foundedness.

I note that several googologists state that it is well-defined, and use the ordinal notation with weakly inaccessibles in their analyses. Therefore this just means that I lack knowledge.

Please tell me a reference of the proof.

= Weakly Inaccessible Cardinal =

This topic is mainly written in [WI]. I note that the function \(\psi\) appear in [WI] is a variant of (and actually different from) the original one defined in [WM].

Convention

 * Following the convention in [WM], \(\chi\) denotes the \(2\)-variable enumeration function of limits of higher inaccessible cardinals.
 * I denote by \(M\) the smallest weakly Mahlo cardinal, and abbreviate "the existence of a weakly Mahlo cardinal" by \(\exists M\).
 * I abbreviare \(\chi_0(0)\) to \(I\).

I note that \(M\) denotes a constant term symbol in [WI], and hence my convention here is differeent from the original one.

Definition of the Ordinal Notation System (T(M),<)
Safe!

It is defined under \(\textrm{ZFC}\) in Section 2 in [WI]. Unlike the ordinal notation system \((\textrm{T}(M),<)\) with a weakly Mahlo cardinal in [WM], the symbol \(M\) is directly used in the system as a constant term symbol but not as the smallest weakly Mahlo cardinal.

Primitive Recursiveness of (T(M),<)
Safe!

It is verified under \(\textrm{ZFC}\) in Theorem 2.8 (i) in [WI] by the constructive means in Theorem 14.2 in [KPM].

Well-foundedness of (T(M),<)
Dangerous?

It is verified just under \(\textrm{ZFC}\) plus the well-foundedness of the ordinal notation system with \(M\) in Theorem 2.8 (ii) in [WI]. As I mentioned above, I could not find a proof of the well-foundedness of the ordinal notation system with \(M\) under \(\textrm{ZFC}\).

If it is just proved under \(\textrm{ZFC} + \exists M\), then the well-definedness of the ordinal notation with weakly inaccessible cardinals is 'NOT' verified.

I note that several googologists state that it is well-defined, and use the ordinal notation with weakly inaccessibles in their analyses. Therefore this just means that I lack knowledge.

Please tell me a reference of the proof.

Definition of the OCF ψ_I
Safe!

It is originally defined as a function symbol in \(\textrm{ZFC}\) in the Section 2 in [WI]. Therefore in order to regard it as anOCF, we need to transfer the ordinal notation system to a set of ordinals. Then we need \(\textrm{ZFC} + \exists M\) by the last subsection.

On the other hand, \(\psi_I(0)\) coincides with the omega fixed point under \(\textrm{ZFC} + \exists M\). Therefore \(\psi_I(0)\) makes sense under \(\textrm[ZFC}\) as a syntax sugar of the omega fixed point.

Similarly, \(\psi_I\) can be regarded as the enumertion function of the function \(\textrm{ON} \to \textrm{ON}, \ \alpha \mapsto \Omega_{\alpha}\), and hence makes sense under \(\textrm{ZFC}\).

Relation to KPM
Dangerous?

Here, I consider the equality \(\textrm{PTO}(\textrm{KPM}) = \psi_I(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0))\) under \(\textrm{ZFC} + \exists M\). If it is true, then \(\psi_I(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0))\) can be regarded as a syntax sugar of \(\textrm{PTO}(\textrm{KPM})\) under \(\textrm{ZFC}\).

The inequaility \(\textrm{PTO}(\textrm{KPM}) \leq \psi_I(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0))\) is verified just under \(\textrm{ZFC} + \exists M\) in Theorem 7.14 (iii) in [WI].

Moreover, the opposite inequality \(\textrm{PTO}(\textrm{KPM}) \geq \psi_I(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0))\) is just stated in Theorem 7.15 (iii) in [WI]. Rathjen wrote that it follows from his preprint [KPM], but it seems NOT published or even open.

Anyway, can we regard \(\psi_I(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0))\) as a syntax sugar of \(\textrm{PTO}(\textrm{KPM})\) under \(\textrm{ZFC}\)? Nope. If there is no proof of the equality under \(\textrm{ZFC} + \exists M\), then the collapsing rule of \(\psi_I(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0))\) under \(\textrm{ZFC} + \exists M\) is not applicable to \(\textrm{PTO}(\textrm{KPM})\). Therefore the analysis based on the property that \(\textrm{PTO}(\textrm{KPM})\) admits such a collapsing rule is incorrect.

I note that several googologists state that the equality \(\textrm{PTO}(\textrm{KPM}) = \psi_I(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0))\) is true, and use this result in their analyses. Therefore this just means that I lack knowledge.

Please tell me a reference of the proof.

= Weakly Compact Cardinal =

WIP.

= Huge Cardinal =

Tell me references.

= Stage Cardinal =

Give me definitions.

= Rank-in-Rank Cardinal =

I do not know why there is an artictle of a rank-in-rank cardinal in googology wiki. Has anyone used it in googology?