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Edit 2024: This post was based on an incorrect premise, as can be seen by my conversation with Farmer S in the comments. However the mistake I made and the conversation in comments may be instructive (for example it is a mistake I have heard others make), so I have undeleted it to leave it up.

The original content of the question:

An ordinal $\alpha$ is said to be $\Pi_3$-reflecting if for any $\Pi_3$ formula $\phi$ with parameters from $L_\alpha$, if $L_\alpha\vDash\phi(\vec a)$ then there is an ordinal $\beta<\alpha$ where $L_\beta\vDash\phi(\vec a)$. Call an ordinal $\alpha$ "admissible for $X$", where $X$ is a class of functions $L_\alpha\to L_\alpha$, if there is no function in $X$ whose domain is $\alpha$-finite but whose range is not $\alpha$-finite: this is a generalization of the usual idea of admissibility to any given class of functions instead of the usual class of $\alpha$-recursive functions. The following is a proof that if $\alpha$ is $\Pi_3$-reflecting, $\alpha$ is admissible for the $\Sigma_2$-Skolem functions of $L_\alpha$. Originally this came from a "fake proof" I was going to ask on Math.SE, in which I thought I had proved the false claim that $\Pi_3$-reflecting $\alpha$ was admissible for all $\Sigma_2(L_\alpha)$-functions.

Let $\phi(\vec a)$ be a $\Pi_3$-formula such that $L_\alpha\vDash\phi(\vec a)$. Noting that $\phi(\vec a)\equiv \forall x\exists y\psi(x,y,\vec a)$ for some $\Sigma_2$ formula $\psi$, putting $\phi$ in Skolem normal form results in $\phi(\vec a)\equiv\forall x\psi(x,h(x),\vec a)$ for some $\Sigma_2$-Skolem function $h:L_\alpha\to L_\alpha$, $h$ chosen w.r.t. $\psi$. By Jensen's uniformization theorem (stated e.g. on p.37 of Sy Friedman's Ph.D. thesis "Recursion on Inadmissible Ordinals"), there is such $h$ that is $\Sigma_2$-definable on $L_\alpha$, so we don't need to worry about existence of such an $h$. By $\Pi_3$-reflection of $\alpha$, there is some $\beta<\alpha$ (with $\vec a\in L_\beta$) such that $L_\beta\vDash\phi(\vec a)\equiv\forall x\psi(x,h(x),\vec a)$. We obtain $\forall(x\in L_\beta)\psi(x,h(x),\vec a)$, and that $h(x)\in L_\beta$. We have shown $h:L_\beta\to L_\beta$, since $h$ was an arbitrary $\Sigma_2$-Skolem function of $L_\alpha$, we have shown that $\Pi_3$-reflecting $\alpha$ has the desired admissibility property.

In the case that $\alpha$ is admissible for all $\Sigma_2(L_\alpha)$ functions, we have the much stronger property that $L_\alpha\vDash\textrm{KP}+\Sigma_2\textrm{-collection}$. I have a few questions: if we choose a class of $L_\alpha\to L_\alpha$ functions "intermediate" between the class of $\Sigma_2$-Skolem functions and all the $\Sigma_2(L_\alpha)$ functions, do we get more rarefied properties of ordinals? (E.g., if an ordinal is $\Pi_{17}$-reflecting, is there a way to express this via admissibility for a class of functions? Or for $\alpha$ nonprojectible?) If so, has this been studied before?

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  • $\begingroup$ You said "The following is a proof that if $\alpha$ is $\Pi_3$-reflecting, $\alpha$ is admissible for the $\Sigma_2$-Skolem functions of $L_\alpha$". Can you specify exactly what you mean by "the $\Sigma_2$-Skolem functions of $L_\alpha$" here? $\endgroup$
    – Farmer S
    Commented Oct 4, 2022 at 19:45
  • $\begingroup$ What if $V=L$, $\alpha=\aleph_\omega$, and $f:L_\alpha\to L_\alpha$ is the function where $f(n)=\aleph_n$ for $n\in\omega$, and $f(x)=\emptyset$ for $x\notin\omega$. Then $f$ is $\Sigma_2$-definable over $L_\alpha$ (from no parameters), and is unbounded in $L_\alpha$. But $L_\alpha$ is also $\Pi_3$-reflecting (in fact much more reflecting), just by downward Lowenheim Skolem and condensation. Does your proof somehow not apply to this function? $\endgroup$
    – Farmer S
    Commented Oct 4, 2022 at 19:45
  • $\begingroup$ And in your proof, why should $L_\beta$ agree with $L_\alpha$ about outputs of $h(x)$? (Isn't "$h$" actually shorthand for a definition interpreted over $L_\alpha$ or $L_\beta$, respectively?) $\endgroup$
    – Farmer S
    Commented Oct 4, 2022 at 19:46
  • $\begingroup$ @FarmerS Apologies for not responding sooner, you are correct about $h$ being shorthand and this changes my question drastically! When writing this I thought a canonical $\Sigma_n$-Skolem function was an $h:\omega\times L_\alpha\to L_\alpha$ that's $\Sigma_n(L_\alpha)$ where for a $\Sigma_n$ formula $\phi$ with a parameter $x\in L_\alpha$, if $L_\alpha\vDash\exists y\phi(x,y)$ then $L_\alpha\vDash\phi(x,h_{\ulcorner\phi\urcorner}(x))$. But looking more closely at Friedman's thesis, instead of $\ulcorner\phi\urcorner$, that parameter doesn't necessarily code $\phi$. (contd) $\endgroup$
    – C7X
    Commented Nov 12, 2022 at 1:06
  • $\begingroup$ You were right about $h(x)$ being shorthand for a definition, so instead of $h(x)\in L_\beta$ the correct deduction should have been $L_\beta\vDash h\textrm{ is defined at }x$. Then given some witness $L_\beta$ of $L_{\aleph_\omega}$'s $\Pi_3$-reflection, $L_\beta$ would contain all the witnesses $y$ of $L_\beta\vDash\forall x\exists y(y=f(x)\land\psi(x,y,\vec a))$ - these witnesses $y$ are $L_\beta$-cardinals but not necessarily $V$-cardinals. $\endgroup$
    – C7X
    Commented Nov 12, 2022 at 1:06

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