Difference between revisions of "Talk:Heights of models"

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: More on this, Marek and Srebrny showed (Theorem 6.14, contains typo) that if <math>L_\alpha\cap\mathcal P\omega=L_\beta\cap\mathcal P\omega</math>, then <math>\alpha</math> must be <math>\beta</math>-stable. So in accordance with Barwise's "<math>\beta</math>-cardinals are <math>\beta</math>-stable" corollary in this article, <math>\alpha</math> is a plausible candidate for <math>(\omega_1)^{L_\beta}</math>. [[User:C7X|C7X]] ([[User talk:C7X|talk]]) 14:19, 7 August 2021 (PDT)
 
: More on this, Marek and Srebrny showed (Theorem 6.14, contains typo) that if <math>L_\alpha\cap\mathcal P\omega=L_\beta\cap\mathcal P\omega</math>, then <math>\alpha</math> must be <math>\beta</math>-stable. So in accordance with Barwise's "<math>\beta</math>-cardinals are <math>\beta</math>-stable" corollary in this article, <math>\alpha</math> is a plausible candidate for <math>(\omega_1)^{L_\beta}</math>. [[User:C7X|C7X]] ([[User talk:C7X|talk]]) 14:19, 7 August 2021 (PDT)
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:: Answer: No. [http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf#page=7 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 $L_\alpha\vDash\textrm{V=HC}$. [[User:C7X|C7X]] ([[User talk:C7X|talk]]) 18:52, 11 September 2021 (PDT)
 
==Lower attic==
 
==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 [http://cantorsattic.info/Category:Lower_attic Lower attic]? [[User:C7X|C7X]] ([[User talk:C7X|talk]]) 18:16, 21 August 2021 (PDT)
 
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 [http://cantorsattic.info/Category:Lower_attic Lower attic]? [[User:C7X|C7X]] ([[User talk:C7X|talk]]) 18:16, 21 August 2021 (PDT)

Latest revision as of 18:52, 11 September 2021

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)

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)