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An ordinal $\alpha$ is called nonprojectible if and only if $\forall \beta (\beta \in X \cap \alpha \implies \min\{\gamma \in X: \beta \in \gamma\} \in \alpha)$, where $\delta \in X \iff L_\delta \text{ is a } \Sigma_1 \text{-elementary substructure of } L_\alpha$. Many theorems regarding these types of ordinals have been proven, for example, that $\alpha$ being nonprojectible is equivalent to $L_\alpha$ being a model of $KP + \Sigma_1-Sep$. Now, I have recently been very interested in nonprojectible ordinals and their properties, especially due to their importance to a potential ordinal analysis of $\Pi^1_2 - CA_0$, and I ran across a statement (I can't remember where) that said that $\alpha$ being nonprojectible is equivalent to the set of ordinals $\{\eta \in \alpha: L_\eta \preceq_1 L_\alpha\}$ being unbounded in $\alpha$. It had no proof, but I believe it's true. Is it true, or is the website spreading misinformation?

Also, would replacing "$\Sigma_1 \text{-elementary substructure}$" with "$\Sigma_2 \text{-elementary substructure}$" make it so that $L_\alpha$ is a model of $KP + \Sigma_2-Sep$ rather than $KP + \Sigma_1-Sep$? This is just my thought, so it might be false.

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  • $\begingroup$ This definition of nonprojectibility may be vacuous, for example if $\alpha=5$ then $X\cap\alpha=\varnothing$. $\endgroup$
    – C7X
    Sep 8, 2023 at 1:02

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It's true that an ordinal $\alpha$ is nonprojectible iff the $\alpha$-stable ordinals are unbounded in $\alpha$. A standard reference for this could be Barwise's Admissible Sets and Structures, although I'll have to check later if it's present. A less common reference I'm sure it's in is p.21 of this page of Jensen's notes on admissible sets, namely the theorem "$L_\alpha[u]$ is nonprojectible iff there is a normal function $\langle\alpha_\nu\mid\nu<\lambda\rangle$ ($\textrm{Lim}(\lambda)$) such that $\alpha=\textrm{sup}_\nu\alpha_\nu$ and $L_{\alpha_\nu}[u]\prec_{\Sigma_1}L_\alpha[u]$ for $\nu<\lambda$."

For the general case of models of $\Sigma_n$-separation, it's a theorem in Marek's "Some comments on the paper by Artigue, Isambert, Perrin, and Zalc" that "$J_\alpha\vDash\Sigma_n\textrm{-separation scheme}$ iff $J_\alpha$ possesses a cofinal tower of transitive $\Sigma_n$-elementary subsystems", so your second thought is also correct. ("Cofinal tower" here means all ordinals in $J_\alpha$ are contained within some level of the tower. We may use the Jensen hierarchy here since stability for the Jensen and constructible hierarchies are equivalent.)

@davidlowryduda Is quoting the results OK?

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  • $\begingroup$ The content of this answer is a very good answer. It's relevant and answers both the primary question and the more theoretic followup. But I worry about a time when links die if websites change/rot. I suggest that you edit to add the names and locations in the cited papers and book. $\endgroup$
    – davidlowryduda
    Mar 1, 2022 at 15:09
  • $\begingroup$ I agree with @davidlowryduda. By the way, C7X, the doc for our collaboration is here: docs.google.com/document/d/… $\endgroup$
    – Binary198
    Mar 1, 2022 at 16:03

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