O tto R ajala . An Inner Model from Stationary Logic and the Next Admissible Set . University of Helsinki. 2025. Supervised by Juliette Kennedy and Jouko Väänänen. MSC: 03E45, 03E47, 03E50. Keywords: Inner models, extended constructibility, stationar

This thesis investigates the inner model C ( aa + ) ${C({\texttt{aa}}^+)} $ upper C left parenthesis monospace aa Superscript plus Baseline right parenthesis obtained from the aa + ${\texttt{aa}}^+ $ monospace aa Superscript plus -quantifier, which is a variant of the aa ${\texttt{aa}} $ monospace aa -quantifier of stationary logic. The intuitive meaning of the aa + ${\texttt{aa}}^+ $ monospace aa Superscript plus -quantifier is that there is a closed unbounded set of countable subsets of the model such that a given property holds in the next admissible set built from the model and any one of the subsets in the closed unbounded set. The research presented in the thesis belongs to the framework of extended constructibility, which is a research program that studies inner models obtained by replacing first-order logic by a stronger logic in the definition of Gödel’s constructible hierarchy. The most extensively studied model within the program has been the model C ( aa ) ${C({\texttt{aa}})} $ upper C left parenthesis monospace aa right parenthesis , obtained from stationary logic, which forms the immediate background for the research developed here. After an introductory chapter, we give the definition of the aa + ${\texttt{aa}}^+ $ monospace aa Superscript plus -quantifier and discuss some of the basic properties of the logic L (