$\Pi_0^1$-weakly indescribable cardinals are exactly the regulars

Question

Hi,

I'm not sure if I should ask here or over at math.stackexchange.com, but I think here it's a bit more fitting. This question stems from a homework problem:

Definition:

Given some class of formulas $Q$ we call a cardinal $\kappa$ $Q$-weakly indescribable if for every $Q$-sentence $\phi$ and $R\subset\kappa$, $\langle\kappa,\in,R\rangle\models\phi$ implies that there is some $\alpha<\kappa$ such that $\langle\alpha,\in,R\cap\alpha\rangle\models\phi$.

Background:

The exercise asks to show that $\kappa$ is $\Pi_0^1$-weakly indescribable exactly when it is regular. One part (namely that a regular cardinal is $\Pi_0^1$-weakly indescribable) is easy, but I am unsure about the other direction. If we had $R\subset\kappa\times\kappa$ then it would be fairly easy to "code" the singularity of $\kappa$ into $R$, but I don't see how to do this when $R$ is a subset of $\kappa$. Of course, we have that $\kappa$ can be mapped injectively onto $\kappa\times\kappa$ but -at least to the best of my knowledge- that would require some form of inductive definition, which on principle uses functions, objects that do not exist when our universe is $\langle\kappa,\in,R\rangle$.

After giving it a lot of thought, I actually checked a paper by Levy ("The size of the indescribable cardinals") in which he uses binary predicates, and I also tried to find an old paper by Hanf and Scott ("Classifying inaccessible cardinals") but it turns out that the library threw away most of the Notices of AMS volumes when they moved to a smaller building.

So my question is:

Can we somehow define the bijection inside $\langle\kappa,\in,R\rangle$, or on? And if not is there some other way to prove this?

Thanks,

---

Disclosure: Even though this is formally homework, we are allowed to use that $R\subseteq\kappa\times\kappa$. Hence this is more of a personal question that arose from the fact that I got stuck on this for a long time.

Answer 1

First off, note that being weakly $\Pi^1_0$-indescribable is actually the same as being weakly $\Sigma_1^1$-indescribable. Let $\phi(x_0...x_n,S)$ be some formula, say $\exists X(\psi(X,x_0...x_n,S))$, where $\psi(X,x_0...x_n,S)$ is $\Pi^1_0$. Then let $C$ be witness to this, and let $$(\alpha,\beta)=\operatorname{ot}\big\{(\gamma,\zeta)\lt(\alpha,\beta)|\gamma,\zeta\lt\kappa\big\}.$$ Then, $C\times S\subseteq\kappa$, and so, if $$(\kappa,\in,C\times S)\vDash\psi\big(\operatorname{dom}(C\times S),x_0...x_n,\operatorname{ran}(C\times S)\big),$$ then there is some $\alpha\lt\kappa$ such that $$(\alpha,\in, C\times S\cap\alpha)\vDash\psi\big(\operatorname{dom}(C\times S\cap\alpha),x_0...x_n,\operatorname{ran}(C\times S\cap \alpha)\big),$$ and therefore $(\alpha,\in,S\cap\alpha)\vDash\phi(X,x_0...x_n,S)$. You can formalize $\alpha\in \operatorname{dom}(C\times S)$ by $$\exists\beta,X\;\Big(\forall\gamma,\zeta\big(\exists Y(Y=X\leftrightarrow\gamma=\alpha\land\zeta=\beta)\big)\land X\in C\times S\Big).$$

Now, assume to the contrary that $\kappa$ is irregular. Let $C$ be cofinal in $\kappa$, with $|C|\lt\kappa$. Let $\lambda=|C|$. Then $$(\kappa,\in,\lambda,C)\vDash C\text{ is cofinal in }\mathit{Ord}\land\exists x(x=\lambda\land x=|C|).$$ Then, if $$(\alpha,\in,\lambda\cap\alpha,C\cap\alpha)\vDash C\cap\alpha\text{ is cofinal in }\mathit{Ord}\land\exists x(x=\lambda\land x=|C\cap\alpha|),$$ $\alpha\gt\lambda$, and $|C\cap\alpha|\lt\lambda$. But $|C\cap\alpha|=\lambda$. Contradiction.

For completeness, I will mention the secondary case. Let $\kappa$ be regular uncountable. Let $M_0$ be the Skolem hull of $\emptyset$ in $\kappa$. Then $|M_0|=\omega$ is less than $(\kappa,\in,S)$, so that $\alpha_0=\text{sup}M_0$ is less than $\kappa$. Then let $M_{n+1}$ be the Skolem hull of $\alpha_n$ in $(\kappa,\in,S)$, and $\text{sup}M_n=\alpha_{n+1}$. Then let $\alpha=\lim_{n\rightarrow\omega}\alpha_n$.

Edit: It seems that the question is actually asking about how to define the pairing function $(\alpha,\beta)$ in $\kappa$. I am not exactly sure how you could define this.