Talk: Stable ordinal

Redirect

Isn't it better to create a separated article than to redirect the long article on KP?

p-adic 01:54, June 20, 2020 (UTC)

I will copy it over, but I might have to fix citations ~~ —Preceding unsigned comment added by C7X (talk • contribs)

Thank you.

p-adic 02:53, June 20, 2020 (UTC)

Model

I guess that the one who originally added the desciption of an elementary substructure is hyp cos. Please fix the error by yourself, because I do not know a definition applicable to any class.

p-adic 02:41, June 20, 2020 (UTC)

IDK which axioms are used by the base theory in this definition, but Jan-Carl Stegert defined the theory "Stability" as "KPi augmented by the axiom \(\forall\alpha\exists\kappa\ge\alpha(L_\kappa\prec_{\Sigma_1}L_{\kappa+\alpha})\)" from this paper (page 104) C7X (talk) 02:54, June 20, 2020 (UTC)

Usually, we define an elementary substructure in terms of a set model, while the original description uses a class model, which heavily depends on the choice of axioms even if we consider definable classes. If we restrict a class to a definable class so that we can deal with in in a usual set theory, the notion of a class model is not equivalent to a set model. If we do not restrict in that way, then there is no agreed-upon definition of a model. Therefore I think that your reference is irrelevant. (Or do you think that I should check it understanding this issue of a class model?)

p-adic 02:59, June 20, 2020 (UTC)

Sorry; I wasn't sure if KPi was relevant to this or not, but I thought that it was related because it's how the theory Stability was defined. Also, is the class model problem fixed for stable \(\alpha\) by using \(L_\alpha\prec_{\Sigma_1}L_{\omega_1}\) instead of \(L_\alpha\prec_{\Sigma_1}L\) (an equivalent condition from zoo2.25 ), or is the problem from "A class M" from the definition of elementary substructures?C7X (talk) 03:07, June 20, 2020 (UTC)

The issue is that there does not seem to be an agreed-upon definition of an elementary embedding applicable to any class such as M in the article. If you restrict classes to definable classes such as L, the notion is not equivalent to the usual one.

p-adic 06:14, June 20, 2020 (UTC)

What is the problem in the definition of \(M\prec_{\Sigma_n}N\)? And how is it related to the base theory? My first contact from this notion is from Rathjen's paper, which still seems to have the problem you said. {hyp/^,cos} (talk) 03:56, June 20, 2020 (UTC)

You mean, Rathjen extended the notion of elementary embedding in that paper in the way which you referred to? Please tell me the exact page, because it will costs much time to read all of it.

p-adic 06:12, June 20, 2020 (UTC)

Definition 5.1 (page 18) and 5.4 (page 19) are where elementary substructures apply to proper classes. {hyp/^,cos} (talk) 00:14, June 21, 2020 (UTC)

But Rathjen only used the notion of an elementary embedding for definable classes. Why do you think that the definition can be extended to arbitrary proper classes?

p-adic 01:43, June 21, 2020 (UTC)

Oh. I was not aware of this. Why can elementary embedding apply to sets and definable classes, but not to other classes? {hyp/^,cos} (talk) 04:10, June 21, 2020 (UTC)

It is simply because the satisfaction relation "|=" is formalised in a standard way only for tuples of sets and formulae, and for tuples of definable classes and formulae with fixed bound of the quantifier rank. There is no agreed-upon way to define "|=" for tuples of (not necessarily definable) classes and formulae (with fixed bound of the quantifier rank). It means that the notion of a Σ_1-elementary substructure is defined only for sets and definable classes in a standard convention (as far as I know). (I do not state that it is impossible to define it. For example, I have explicitly defined it in my system using quite strong base theory. If Rathjen had already defined it, then I would like to know it.)

p-adic 06:22, June 21, 2020 (UTC)

Since you just seem to misunderstand the required condition of the truth predicate, I replace the wrong definition by a correct one. Also, since many googologists use intuitively misunderstood truth predicates, please be more certain not to write an ill-defined truth predicate.

p-adic 02:58, June 22, 2020 (UTC)

This might be a bit off-topic. I have seen some definitions of large cardinals using elementary embeddings \(j:V\rightarrow M\) where M has some special property. They apply elementary embeddings on arbitrary class M, which seems to have the problem p-adic pointed out. {hyp/^,cos} (talk) 09:50, June 24, 2020 (UTC)

Maybe you are talking about rank-into-rank. Right, the "naive definition" does not directly make sense, because even the truth predicate on V is known to be unformalisable (Tarski's theorem). For example, Kanamori (The Higher infinite) refers to such an issue at the beginning, and it says "the issue is discussed as it arises." It does not mean that the satisfaction at any proper class is formalisable in ZFC (as it conflicts Tarski's theorem).

p-adic 11:59, June 24, 2020 (UTC)

Madore's Definition

Is Madore's definition the property "\(L_\alpha\preceq_1L_{\alpha+1}\)" from 2.7 in A Zoo of Ordinals? C7X (talk) 07:26, August 14, 2020 (UTC)

Right. (Is there any other definition in the zoo? If so, please tell me.) According to hyp cos's comment above, stable ordinals in his context and in Cantor's attic are defined by using the relection property for Σ_1 formalised by using Levy's truth predicate, and hence it is completely different from this definition. (I checked this article, because I could not understand how you misunderstood what you wrote in your blog post. So I guess that your misconseption is based on the definition in hyp cos's context and in Cantor's attic, while you were unintensionally applying the results on the definition in Madore's zoo. Is it correct?)

p-adic 07:40, August 14, 2020 (UTC)

Was Madore's definition of \(\preceq_1\) different than Cantor's Attic's definition using a reflection property? C7X (talk) 03:09, August 16, 2020 (UTC)

I do not know the precise definition in Cantor's Attic, because it is inaccessible now, At least, the article (perhaps based on Cantor's attic) is different from Madore's one. Both use distinct reflection properties if I correctly understand. (You can literally compare the article and Madore's zoo by yourself. If there is a point which you do not understand, please explicitly tell me the point. Or you can directly ask hyp cos, who wrote the definition and who knew those notions well.)

p-adic 05:08, August 16, 2020 (UTC)

I must not have understood Madore's definition, is it different than elementary substructures for \(\Sigma_1\)-formulae? C7X (talk) 16:26, August 24, 2020 (UTC)

You have asked it in your second question, and I have already answered it. Why do you repeat the same question?

p-adic 23:06, August 24, 2020 (UTC)

According to C7X, Madore simply made a mistake on mixing two distinct hierarchies. Therefore I remove the description on Madore's zoo. If someone thinks that we should keep Madore's original definition without removing the mixing, please continue to discuss it here.

p-adic 10:52, August 30, 2020 (UTC)

IDK if Madore made a mistake about the two hierarchies, but \(\preceq_1\) doesn't contain the symbol "Σ", so maybe \(\preceq_n\) is defined using the Levy hierarchy without denoting a hierarchy C7X (talk) 15:11, August 30, 2020 (UTC)

Edit: I saw your new edit

Madore actually refers to the statement that it is the smallest Π^1_0-reflecting ordinal. If Π denoted the analytic hierarchy for arithmetic as Madore clarified, then the statement would conflict your assumption that Madore used the Σ_1-elementary substructure, didn't it? That is why I thought that Madore used the hierarchy for arithmetic or made a mistake on mixing two hierarchies. Am I missing a point?

p-adic 15:18, August 30, 2020 (UTC)

You're right (I had forgotten about the statement about the smallest \(\Pi_0^1\)-reflecting ordinal) C7X (talk) 15:59, August 30, 2020 (UTC)

I see. Thank you.

p-adic 22:39, August 30, 2020 (UTC)

References

Is it possible to put a reference inside of a footnote? C7X (talk) 01:57, August 31, 2020 (UTC)

It is possible, but is a bit tiresome. (It requires a complicated syntax on a nested reference.)

p-adic 02:11, August 31, 2020 (UTC)

Proof

Is this proof correct?

Let S denote the set of stable ordinals \(<\omega_1\).

Lemma: If \(\beta\) is stable, then \(\beta\) is admissible.

According to [RichterAczel p.46], if A and B are transitive sets, and if \(A\prec_{\Sigma_1}B\) and \(A\subseteq C\subseteq B\), then \(A\prec_{\Sigma_1}C\). Because \(L_\beta\subseteq L\), then \(L_\beta\prec_{\Sigma_1}L\) (i.e. \(\beta\) is totally stable) implies \(L_\beta\prec_{\Sigma_1}L_{\beta+1}\) (i.e. \(\beta\) is (+1)-stable). Also according to [RichterAczel p.46], if \(\gamma\) is (+1)-stable, then \(\gamma\) is admissible, therefore \(\beta\) is admissible.

Theorem: For an ordinal \(\alpha\), if \(\textrm{sup}(S\cap\alpha)=\alpha\) then \(\alpha\) is recursively inaccessible (i.e. all limits of stable ordinals are recursively inaccessible). Each member of \(S\cap\alpha\) is stable, so by the lemma each member is admissible. Also \(\alpha\) is admissible by the lemma, so \(\alpha\) is admissible and a limit of admissibles (i.e. recursively inaccessible) C7X (talk) 17:09, August 31, 2020 (UTC)

Your proof has several errors:

1. The "p.46"s are typos of "p.48".
2. The original statement does not assume that A and B are transitive sets. Please tell me the intention of your assumption (instead of deleting it, so that I can help you to understand the argument).
3. Since L is not a tansitive set and hence your statement of A, B, and C is not applicable to this setting.
4. What does "totally stable" mean? (It might not be an error, but I would like to know your intention.)
5. The theorem is wrong, because it has an obvious counterexample, In particular, the proof is wrong. Please try to search the error in order to understand the argument in a precisely manner. (Please answer the error so that I can check the correctness.)

p-adic 23:26, August 31, 2020 (UTC)

1. Should I use the page numbers on the PDF file or written in the paper?
2. I must have made a mistake (I saw that A and B must be transitive sets from the definition of \(\prec_{\Sigma_1}\) written on (p.15 using the numbers in the PDF, or p.17 using the PDF numbers))
3. You're right; in this case I can replace the definition of "\(\alpha\) is stable" with \(L_\alpha\prec_{\Sigma_1}L_{\omega_1}\) (from A Zoo of Ordinals 2.25) and use "\(L_\beta\subseteq L_{\omega_1}\)" in the proof
4. By "totally stable" I meant "stable"
5. I will guess the counterexample is the limit of the first \(\omega\) stable ordinals, but IDK C7X (talk) 23:36, August 31, 2020 (UTC)

Edit: Is there a counterexample to the statement that S's enumerating function is continuous?

1. Usually we refer to the written page number. But if you clariy that you are using the number on the PDF file, it is ok.
2. You are right. (Sorry, there are many formulation of <_{Σ_1^0}.) But then you have oother problems: C is not necessarily transitive, and hence your statemente does not make sense if you strictly follow the formulation in the article. Also, it is still inapplicable to L, which is not a transitive set, and hence your argument does not make sense if you strictly follow the formulation in the article.
3. It is a good solution, although I do not know the proof of the unsourced fact in Madore's zoo. (Is it obvious for you? If so, I appreciate if you tell me a brief proof.)
4. Then ok. Is it a standard convention?
5. The counterexample is more elementary one.You can define very small α which satisfies sup(S∩α) = α, after noting that sup X denotes the least ordinal which is greater than or equal to any element of X. Namely, you need an additional assumption, and you actualy implicitly used it in your proof.

> Edit

I do not think so. It can be verified by the fact that the limit of a tower of elementary embeddings is again an elementary embedding. Could you tell me the reason why you think the description in its source, which you quoted, might not be true for you? (I might make a mistake, and hence I would like to know the reason of the doubt.)

p-adic 01:30, September 1, 2020 (UTC)

Concerning the replacement of L by L_{ω_1}, Madore does not state that it is equivalent in any case, i.e. he stated that it is equivalent if we consider the smallest one. Is it really equivalent? Please tell me the idea of the proof. Also, why are the ordinals in the main article assumed to be countable? There is not such an assumption in the sources, right? If so, the equivalent seems not to hold for uncountable ordinals. Please tell me a source which requires the countability in the definition of the stabilities.

p-adic 11:37, September 1, 2020 (UTC)

I'm not sure if L can always be replaced by \(L_{\omega_1}\) for countable stable ordinals, so I won't continue this proof.

I thought that when there was a counterexample, that it implied that the remark was false.

I have a question relating to limit points of classes: For any class \(A\subseteq\textrm{Ord}\) and some ordinal \(\alpha\in\textrm{Ord}\), using "\(\textrm{sup}(A\cap\alpha)=\alpha\)" for the definition of "\(\alpha\) is a limit point of \(A\)" doesn't work because \(\textrm{sup}(A\cap 0)=0\) regardless of whether 0 is a limit point of A or not, then should I use "\(\alpha>0\land\textrm{sup}(A\cap\alpha)=\alpha\)" for the definition of "\(\alpha\) is a limit point of \(A\)"?C7X (talk) 22:57, September 1, 2020 (UTC)

Right. Great. (Although there is a convention that 0 is allowed to be a limit, but the convention is inconsistent with your proof. Therefore you should use the convention that 0 is not a limit.) The errors due to the convention are...

1. α is not necessarily recursively inaccessible, while your proof states so. (Please check the definition of the recursive inaccessibility, and check that 0 does not satisfies it.)
2. α is not necessarily admissible, while your proof states so. (Please check the definition of the admissibility, and check that 0 does not satisfies it.)
3. α is not necessarily stable, while your proof states so. (Please check the definition of the continuity, and check that it does not ensure that α is stable.)

p-adic 23:42, September 1, 2020 (UTC)

One of Arai's results

I don't understand how the last remark follows from this formula, on page 14 of this paper:

Why is it that because of this formula \(\varphi\), we're able to iterate \(\Sigma_1\)-stability's proof theory to analyze \(\Sigma_{k+1}\)-stability? C7X (talk) 18:36, 8 January 2021 (UTC)

The formula allows us to reduce the universe L_κ with (Lim)_k to a smaller universe L_α with st_k(α), which perhaps implies (Lim)_{k-1} from the context, as long as we deal with a single Σ_{k+1}-formula φ. By st_k(α) (in the universe L_κ), L_κ|=θ(y,x) (for parameters y and x in L_α) is equivalent toimplies L_α|=θ(y,x). Therefore several arguments on (L_κ,φ) seem to be reduced to those on (L_α,θ). Iterating the process, those are reduced to (L_β,ψ), where st_1(β) is true (in the preceding universe) and a Σ_1-formula ψ. (I do not know explicit arguments to which this reduction process is applicble, and hence my guess might not be correct.)

p-adic 00:18, 9 January 2021 (UTC)

I'm probably misunderstanding while writing this message, but it seems like there are two possibilities relating to \(\theta(y,x)\):

1. \(\theta(y,x)\) is \(\Sigma_{k+1}\), in which case the \(\Sigma_k\)-elementary-substructure doesn't apply to \(\theta\)
2. \(\theta(y,x)\) is \(\Pi_k\), in which case the \(\Sigma_k\)-elementary-substructure only applies to \(\theta\) iff \(\theta\) is \(\Delta_k\)

C7X (talk) 06:32, 9 January 2021 (UTC)

The y and x are not variables, but verctors of variables, I guess. For example, when y = (y_1,y_2), ∃y is just a shorthand of ∃y_1∃y_2. Therefore we can ignore the first case, and I guessed that it is the second case. Then since L_α⊂L_κ and st_k(α) in L_κ, L_κ|=θ(y,x) (for parameters y and x in L_α) implies L_α|=θ(y,x). (Sorry, I wrote "is equivalent to" instead of "implies". It was a typo.)

p-adic 06:40, 9 January 2021 (UTC)

OK C7X (talk) 06:54, 9 January 2021 (UTC)

Also, is the "st_k(α) in L_κ" necessary for that deduction? For example, would "since L_α⊂L_κ, then L_κ|=θ(y,x) (for parameters y and x in L_α) implies L_α|=θ(y,x)" also be the case?

Yes for the first question, and no for the second question. Let us denote θ(y,x) by ∀zψ(z,y,x). For any parameters z, y, and x in L_α, L_α|=ψ(z,y,x) is equivalent to L_κ|=ψ(z,y,x) because ψ is Σ_k and st_k(α) is true in L_κ. Therefore for any parameters y and x in L_α, L_κ|=θ(y,x) implies L_α|=θ(y,x). Since we used st_k(α) here, a naive deduction cannot remove the assumption.

p-adic 07:26, 9 January 2021 (UTC)

\(\Pi_1^1\)-reflection and stability

In order to better understand Richter & Aczel's result "\(\alpha\) is \(\alpha^+\)-stable iff it's \(\Pi_1^1\)-reflecting", I'm trying to consider smaller cases first. For example, "\(\alpha\) is \(\Pi_1^1\)-reflecting" can be shown to imply "\(\alpha\) is (+1)-stable" in this way:

• Given an arbitrary \(\Pi_1^1\)-formula \(\phi(b)\) with parameters from \(L_\alpha\) such that \(L_\alpha\vDash\phi(b)\), we denote here the first-order part of \(\phi\) (i.e. the part with no 2nd-order quantifiers and no atomic subformulae containing 2nd-order variables) by \(\phi^-\). By \(\Pi_1^1\)-reflection, there exists a \(\beta\in\alpha\) such that \(b\in L_\beta\) and \(L_\beta\vDash\phi(b)\), and in fact \(L_\alpha\vDash\phi^-(b)\) and \(L_\beta\vDash\phi^-(b)\). So we have \(L_\alpha\vDash\ulcorner\exists\beta((\phi^-)^{L_\beta})\urcorner\) iff \(L_\beta\vDash\phi^-(b)\) iff \(L_{\alpha+1}\vDash\ulcorner\exists\beta((\phi^-)^{L_\beta})\urcorner\), and since all \(\Sigma_1\) assertions can be made of the form \(\exists x(x\vDash\chi)\) for \(\chi\) of any complexity, then \(L_\alpha\prec_{\Sigma_1}L_{\alpha+1}\).
• I'm not sure about the deduction that "all \(\Sigma_1\) assertions can be made of the form \(\exists x(x\vDash\chi)\) for \(\chi\) of any complexity", but it seems to follow from "... \(\exists \alpha\ V_\alpha\models\varphi(x)\)... all \(\Sigma_2(x)\) assertions can be made in that form" from the Cantor's Attic source, replacing \(V_\alpha\) with an existentially quantified set to possibly reduce complexity.

How would this play out for stronger cases, such as (+2)-stable α, (*2)-stable α, or (ε_α+1)-stable α? C7X (talk) 22:52, 11 February 2021 (UTC)