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 $\Pi _{n}^{m}$- 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 "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 $1\leq n<\omega$, $\sigma _{n}^{m}<\pi _{n}^{m}$ 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)