Are the ordinals $\alpha$ so that the stable ordinals $< \alpha$ are unbounded in $\alpha$ equivalent to the nonprojectible ordinals $\alpha$

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.

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

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.)