Talk:Indescribable cardinal

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)


 * The paper "Infinitary Compactness without Strong Inaccessibility" (Boos, 1976) defines it using $$V_{\kappa+n}$$, but in that paper it's denoted $$R(\kappa+n)$$. C7XWiki (talk) 22:31, 11 September 2023 (UTC)

Forcing comparisons of the least $$\Pi^m_n$$- and $$\Sigma^m_n$$-indescribable cardinals
Let $$\sigma^m_n$$ denote the least $$\Sigma^m_n$$-indescribable cardinal and $$\pi^m_n$$ denote the least $$\Pi^m_n$$-indescribable cardinal. Theorem 7.1 (p.148) in Huaser's thesis "Independence Results for Indescribable Cardinals" (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 $$1\leq n<\omega$$, $$\sigma^m_n<\pi^m_n$$ if $$F(m,n)=0$$, and $$\sigma^m_n>\pi^m_n$$ 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)