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What is the consistency strength of existence of a nontrivial elementary embedding $j:L_δ→L_δ$ for some ordinal $δ$? The consistency strength is strictly between totally ineffable and $ω$-Erdős cardi …

Question: What is the consistency strength of existence of a $(κ^{++})^L$-Mahlo cardinal $κ$? I am particularly interested in how the strength compares to weakly compact cardinals (and other levels …

What is the consistency strength of Basic Theory of Elementary Embeddings (BTEE) from The spectrum of elementrary embeddings j : V → V by Paul Corazza? BTEE uses the language of $(V,∈,j)$ and asserts …

Is it consistent that there is a measurable cardinal $κ$, a $κ$-complete normal nonprincipal ultrafilter $U$ on $κ$, and $S∈U$ such that for every $T⊂S$ with $T∈U$ and $T$ ordinal definable from $S$ a …

If consistent, is existence of a proper class of extendible cardinals provably equivalent to a $Σ^V_5$ statement? Recall that in ZFC, a cardinal $κ$ is extendible iff for every $λ>κ$ there is an elem …

Is it consistent that for all ordinals $α$ and $λ$ and infinite regular cardinals $κ$, games on $V_λ$ with game length $κα$ and $\mathrm{OD}(\mathrm{On}^κ)$ payoff that depends only on the set of all …

Has an inner model theory been developed on the basis of indiscernibles rather than measures? Is there a reasonable formalization at the level of overlapping extenders? Fine-structural models beyond …

Are Reinhardt and Berkeley cardinals (and even a stationary class of club Berkeley cardinals) consistent with $V=\mathrm{HOD}(\mathrm{Ord}^ω)$ ? It seems natural to expect no, but I do not see a proof …

To better understand the transition from large cardinal axioms consistent with the constructible universe $L$ to large cardinal axioms transcending $L$, I am looking for natural equiconsistencies betw …

According to Lightface mice with finitely many Woodin cardinals from optimal determinacy hypotheses by Yizheng Zhu, theorem 1.1, over $\mathbf{Σ^1_{2n+1}}$ determinacy, $Δ^1_{2n+2}$ determinacy is equ …

Are there known statements in $V_{ω+ω}$ independent through forcing after $\mathrm{Col}(ω,<κ_1)*\mathrm{Col}(κ_1,<κ_2)*\mathrm{Col}(κ_2,<κ_3)*...$ where $κ_1<κ_2<κ_3<...$ are supercompact? If no, what …

Question: Is the following theory conservative over ZFC? And if not, what is its strength? Language: $∈$, $j$ (unary function symbol) Axioms: 1. ZFC (without separation and replacement for formulas us …

Under the diamond principle $◊$ and large cardinal axioms, which of the two pointclasses $Π^2_2$ or $Σ^2_2$ is expected to have the scale property? Because conditional $Σ^2_2$ absoluteness under $◊$ …

Is it consistent that the set of ordinal definable real numbers is countable, but for every $y∈\mathrm{OD}∩ℝ$, every true $Σ_2^{V,y}$ statement holds in $\mathrm{HOD}$? Note that $Σ_2^V$ is the best …

In inner model theory, what is the intuition behind the expectation that under appropriate conditions, we should have a single preferred branch to continue an iteration at a limit stage? At the level …