Talk:Heights of models

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ω₁ exists

Let \(\gamma^+\) denote the next admissible after \(\gamma\). Arai has shown that if \(L_{\beta^+}\cap\mathcal P\omega\subseteq L_\beta\), then \(L_{\beta^+}\vDash\textrm{KP}+``\omega_1\textrm{ exists}\!"\). If we replace \(\textrm{KP}\) with some weaker base theory, obtaining a theory such as \(\textrm{Rud. set theory}+``\omega_1\textrm{ exists}\!"\), is it known how this affects model results?

E.g., if \((L_{\beta\times 2}\cap\mathcal P\omega\subseteq L_\beta)\rightarrow L_{\beta\times 2}\vDash\textrm{Rud. set theory}+``\omega_1\textrm{ exists}\!"\)? Or maybe \((L_{\beta+\omega}\cap\mathcal P\omega\subseteq L_\beta)\rightarrow L_{\beta+\omega}\vDash\textrm{Rud. set theory}+``\omega_1\textrm{ exists}\!"\)? C7X (talk) 14:14, 7 August 2021 (PDT)

More on this, Marek and Srebrny showed (Theorem 6.14, contains typo) that if \(L_\alpha\cap\mathcal P\omega=L_\beta\cap\mathcal P\omega\), then \(\alpha\) must be \(\beta\)-stable. So in accordance with Barwise's "\(\beta\)-cardinals are \(\beta\)-stable" corollary in this article, \(\alpha\) is a plausible candidate for \((\omega_1)^{L_\beta}\). C7X (talk) 14:19, 7 August 2021 (PDT)
Answer: No. Here it's written that $(L_{\alpha^+}\setminus L_\alpha)\cap\mathcal P\omega\neq\varnothing$ and $L_\alpha\vDash\textrm{V=HC}$ are equivalent. So if $\textrm{max}\{\beta:(L_\beta\setminus L_\alpha)\cap\mathcal P\omega=\varnothing\}<\alpha^+$ (i.e. $\alpha$ starts a gap that ends strictly before $\alpha^+$), then $L_\alpha\vDash\textrm{V=HC}$. C7X (talk) 18:52, 11 September 2021 (PDT) Original source showed different equivalence

Lower attic

As far as I know, some of these ordinals can't be proven to exist in ZFC, such as $\textrm{min}\{\alpha:L_\alpha\vDash\textrm{ZFC}\}$. So should this page be moved out of Lower attic? C7X (talk) 18:16, 21 August 2021 (PDT)

Possible incorrect claim

> If we have a first-order theory \(T\) that's consistent with \(V=L\), we may take any model \(M\) of \(T\), and due to the consistency with the axiom of constructibility, then \(M\prec L_\beta\) for some large ordinal \(\beta\)

I think this is false, since even though ZF for example is consistent with V=L, there are models $M$ of ZF with $M\vDash(\lnot\textrm{V=L})$, which is a $\Sigma_2$-sentence that no $L_\beta$ satisfies. C7X (talk) 08:24, 27 April 2022 (PDT)

Update: This is false as written in the article. C7X (talk) Updated. C7X (talk) 20:03, 1 July 2022 (PDT)