arXiv subjects
Publications and source records attributed to Obrad Kasum.
We present an $L$-like construction that produces the minimal model of $\mathsf{AD}_\mathbb{R}+$"$Θ$ is regular". In fact, our construction can produce any model of $\mathsf{AD}^++\mathsf{AD}_\mathbb{R}+V=L(P(\mathbb{R}))$ in which there is no hod mouse with a measurable limit of Woodins.
We give a game-theoretic characterization of when a model of an infinitary propositional formula can be added by a proper, semiproper, and stationary-set-preserving poset. In the latter case, we also give a general sufficient condition for the existence of such a poset. We use this condition to give a somewhat different proof of the theorem of Asperó and Schindler, which states that $\mathsf{MM}^{++}$ implies Woodin's axiom $(*)$.
We show that in $L(\mathbb{R})$, assuming large cardinals in $V$, $\mathsf{HOD} {\parallel}\eta^{+\mathsf{HOD}}$ is locally definable from $\mathsf{HOD} {\parallel}\eta$ for all $\mathsf{HOD}$-cardinals $\eta\in [\boldsymbol{\delta}^2_1,\Theta)$. This is a further elaboration of the statement "$\mathsf{HOD}^{L(\mathbb{R})}$ is a core model below $\Theta$" made by John Steel.
By a virtual model, we mean a model of set theory which is elementary in its transitive closure. Virtual models are first used by Neeman \cite{neeman2014forcing} to iterate forcing. That paper is concerned with proper forcing. The method was then adjusted by Veličković to the case of semi-proper forcing and this was drafted in \cite{velickovic2021iteration}. We here straighten the details and further elaborate on Veličković's method. The first section collects facts about virtual model, the second section describes the iteration, and the third one illustrates the method in the case of getting saturation of $\mathsf{NS}_{ω_1}$ (loosely relying on \cite{schindler2016nsomega1}).