# Difference between revisions of "Talk:Heights of models"

## 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)