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First-order indescribable with $V_{\kappa +n}$

Is this definition conventional? Cf. e.g. Kanamori 2003 p.58. --Fourier-Deligne Transgirl (talk) 15:56, 4 May 2023 (UTC)[reply]

Forcing comparisons of the least $\Pi _{n}^{m}$ - and $\Sigma _{n}^{m}$ -indescribable cardinals

Let $\sigma _{n}^{m}$ denote the least $\Sigma _{n}^{m}$ -indescribable cardinal and $\pi _{n}^{m}$ denote the least $\Pi _{n}^{m}$ -indescribable cardinal. Theorem 7.1 (p.148) in Huaser's thesis "Hauser's thesis" (1989) seems to state that for any function $F$ with domain $\{(m,n)\mid 2\leq m<\omega \land 1\leq n<\omega \}$ and codomain $\{0,1\}$ , there is a model of ZFC+GCH in which, for all $2\leq m<\omega$ and $1\leq n<\omega$ , $\sigma _{n}^{m}<\pi _{n}^{m}$ if $F(m,n)=0$ , and $\sigma _{n}^{m}>\pi _{n}^{m}$ if $F(m,n)=1$ . I don't know enough about forcing to be sure that this is a consistency result, but if anyone can confirm it may be a good thing to add under the Properties section. C7XWiki (talk) 01:47, 8 September 2023 (UTC)[reply]

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 Is this definition conventional? Cf. e.g. Kanamori 2003 p.58. --[[User:Fourier-Deligne Transgirl|Fourier-Deligne Transgirl]] ([[User talk:Fourier-Deligne Transgirl|talk]]) 15:56, 4 May 2023 (UTC)
+== Forcing comparisons of the least <math>\Pi^m_n</math>- and <math>\Sigma^m_n</math>-indescribable cardinals ==
+Let <math>\sigma^m_n</math> denote the least <math>\Sigma^m_n</math>-indescribable cardinal and <math>\pi^m_n</math> denote the least <math>\Pi^m_n</math>-indescribable cardinal. Theorem 7.1 (p.148) in Huaser's thesis "[https://thesis.library.caltech.edu/1949/3/hauser-k_1989.pdf Hauser's thesis]" (1989) seems to state that for any function <math>F</math> with domain <math>\{(m,n)\mid 2\leq m<\omega\land 1\leq n<\omega\}</math> and codomain <math>\{0,1\}</math>, there is a model of ZFC+GCH in which, for all <math>2\leq m<\omega</math> and <math>1\leq n<\omega</math>, <math>\sigma^m_n<\pi^m_n</math> if <math>F(m,n)=0</math>, and <math>\sigma^m_n>\pi^m_n</math> if <math>F(m,n)=1</math>. I don't know enough about forcing to be sure that this is a consistency result, but if anyone can confirm it may be a good thing to add under the Properties section. [[User:C7XWiki|C7XWiki]] ([[User talk:C7XWiki|talk]]) 01:47, 8 September 2023 (UTC)

Revision as of 01:47, 8 September 2023

First-order indescribable with V κ + n {\displaystyle V_{\kappa +n}}

Forcing comparisons of the least Π n m {\displaystyle \Pi _{n}^{m}} - and Σ n m {\displaystyle \Sigma _{n}^{m}} -indescribable cardinals

First-order indescribable with $V_{\kappa +n}$

Forcing comparisons of the least $\Pi _{n}^{m}$ - and $\Sigma _{n}^{m}$ -indescribable cardinals