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Throughout, we work in $\mathsf{ZFC+V=L+}$ "There is a weakly compact cardinal," $\kappa$ is the first weakly compact cardinal and "tree" means "subtree of $2^{<\kappa}$ of height $\kappa$" Despite t …

For a paper I'm writing I need to use (as a blackbox) the following theorem: if there is a proper class of Woodin cardinals and $G$ is set-generic, then $L(\mathbb{R})$ and $L(\mathbb{R})^{V[G]}$ are …

For $\varphi$ a first-order sentence in the language of set theory and $\kappa$ an ordinal, let $G_\varphi^{\kappa}$ be the game of length $\kappa$ in which players $1$ and $2$ alternately play elemen …

Let $\mathsf{ODet}_{\omega_1}(L(\mathbb{R}))$ be the following principle ("determinacy for simple open length-$\omega_1$ games"): If $\kappa$ is any ordinal and $X\subseteq \kappa^{<\omega_1}$ is in …

Say that a good logic is a regular logic $\mathcal{L}$ containing $\mathsf{FOL}$ and having the finite use property and the strong downward Lowenheim-Skolem property together with, for each finite lan …

(The original version of this question was much narrower and less natural; but see the edit history if interested.) Say that a good logic is a regular logic $\mathcal{L}$ containing $\mathsf{FOL}$ an …

For a cardinal $\kappa$ I'll use the phrase "$(\kappa,\kappa)$-game" to mean "two-player, perfect-information, deterministic game on $\kappa$ of length $\kappa$." Say that a cardinal $\kappa$ is LOI ( …

Throughout, I work in $\mathsf{MK}$ in order to be able to conveniently quantify over logics; if one prefers, we can restrict attention to (say) $\Sigma_{17}$-definable logics and work in $\mathsf{ZFC …

Although self-contained, this question is a follow-up to this earlier one. Also, thanks to Fedor Pakhomov for fixing a trivial early version of this question! Work in $\mathsf{ZFC}$ + "There is a meas …

To motivate things, let me start with a special case of the question I'm interested in. Let $\mathsf{In}(x)\equiv$ "$x$ is an inaccessible cardinal." Question 1: Is it consistent with the theory $$\m …

Throughout, we work in the class theory $\mathsf{MK}$ (although I'm open to tweaking this), "logic" means "set-sized logic whose semantics is definable over $V$," and "$\mathcal{J}\approx\mathcal{K}$" …

This starts with a vaguely-recalled result (which may be false!): that if $\mathcal{U}$ is a measure on the least measurable cardinal $\kappa$, then every elementary $j: L[\mathcal{U}]\rightarrow M$ s …

(Previously asked and bountied at MSE:) Let $\Sigma$ be the language consisting of a single binary relation symbol. Second-order logic can "detect" second-order-elementary-equivalence of $\Sigma$-str …

Previously asked and bountied at MSE, with slight difference. To keep things relatively simple I'm presenting a somewhat-butchered version of the IMH; for more details, see S.-D. Friedman, Internal co …

(Throughout, all ultrafilters are nonprincipal.) Given a property $P$ - really, a sentence in some appropriate logic - say that a ultrafilter $\mathcal{U}$ on a cardinal $\kappa$ averages $P$ iff for …