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Edit 2024: I now think that this answer is wrong, specifically that the result in the last paragraph does not answer the original question. I am reluctant to delete this answer as it would also delete …

Some references to literature: In "Reflection and Partition Properties of Admissible Ordinals" (Annals of Math. Logic vol. 22, iss. 3, 1982), Kranakis defines a $\Sigma_n$-admissible ordinal to be an …

A proof now appears as lemma 1.21 of the poster's bachelor's thesis "The real numbers in inner models of set theory" (2022, arXiv), however I have a doubt about the proof which is expressed in a comme …

Edit Jul 25: These results may be strengthenable by using theorem 7.8 of chapter V of Admissible Sets and Structures instead of lemma 1, I may edit this post in the future with any resulting improveme …

This is in the same vein as Monroe Eskew's comments. For $n\geq 1$, given that $\alpha<\beta$ and $L_\alpha\prec_{\Sigma_n}L_\beta$, it is not always possible to produce even a $\Sigma_1$ end-extensio …

(Turning some comments into an answer) The definition of $L(x,\alpha+1)$ was wrong, instead it should have been $$L(x,\alpha+1)\leftrightarrow\bigvee_{n\in\omega}\bigvee_{\varphi}\exists p_1,\ldots,p_ …

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 …

Let lowercase Greek letters denote ordinals. Recall from Richter and Aczel's "Inductive definitions and reflecting properties of admissible ordinals", for a set of formulae $\Gamma$ and a class of ord …

As Asaf mentioned, this is not true. It's indeed true that when $\alpha$ is a gap ordinal $L_\alpha\vDash\textrm{V=HC}$, but when considering some ordinals above gap ordinals we get points where it fa …