159 problems

Woodin's HOD conjecture. … mathrm{ZFC}+text{“there is a supercompact cardinal”} $$ proves the HOD hypothesis.

Let $(\kappa_1,\kappa_0)$ and $(\mu_1,\mu_0)$ be pairs of cardinals. For a structure $\mathfrak A$ on $\kappa_1$ in a countable language, write $\mathfrak B\prec\mathfrak A$ when…

Let $\mathcal{V}$ be the relevant model and let $\delta$, $\mathcal{Q}$, and $\eta$ be as in the preceding results. The cardinal $\delta^{\mathcal{Q},\eta}_{\infty}$ is considered…

Let $V$ be a model of set theory with a large cardinal, for example an extendible cardinal. Woodin's Axiom of Choice conjecture. If $V$ has a large cardinal, e.g., an extendible ca…

Assume that $V$ satisfies $\mathsf{ZF}$ and that $\delta$ is an extendible cardinal. Let $G$ be $V$-generic for a forcing that collapses $V_\delta$ to be countable. Axiom of Choice…

The theories in question are ${\sf PFA}$ and ${\sf ZFC}+\exists$ a supercompact cardinal. Equiconsistency conjecture. These theories are equiconsistent. This is presented as a conj…

Consistency-strength dichotomy conjecture. The consistency strength of this statement is always either ZFC or ZFC together with the existence of an inaccessible cardinal; moreover,…

Villaveces's conjecture. The consistency strength of the failure of the GCH at an unfoldable cardinal is no greater than that of a strongly unfoldable cardinal.

Assume that $\lambda_n$ is the $n$-th inaccessible cardinal, let $\kappa=\sum_{n<\omega}\lambda_n$, and consider a forcing notion $\mathbb P$. Shelah's forcing conjecture for pseud…

Menas's conjecture. Not every strongly compact cardinal has a function with the Menas property. Menas proved that every strongly compact limit of strongly compact cardinals has suc…

Consistency conjecture.

Let $C_2(0)=0$ and $C_2(n+1)=C_2(n)+2^{C_2(n)}$, and define $f(n)=2^{2^{C_2(n)}}$. For an elementary embedding $j$, write $\jsub{m}$ for its $m$-fold iterate and let $\gamma_k$ den…

For each $n$, let $a,b<2^{n-1}$ satisfy $p_n(a)=p_n(b)=2^k$. Let $c$ be the least element such that $\gamma_k$ is in the range of $c$, and let $c\gamma_i=\gamma_k$. Assume that…

For each $n$, let $A_n$ be the cyclic algebra considered in the paper. If $a<2^n-1$, write $p_n(a)=2^k$, and if $c+1$ is the threshold of $a$ in $A_n$, require that…

Let $\Gamma$ be the set of critical points of the embeddings in $\Cal A$, and let $\Sigma$ be the closure of $\Gamma$ under $a"\alpha=\sup\{a\xi:\xi<\alpha\}$ for $a\in\Cal A$. Let…

Let $\kappa$ be measurable. Assume that Full Reflection holds at $\kappa$, that all canonical stationary sets $E^\kappa _\delta$ exist, and that $C[E^\kappa _\delta]$ denotes the f…

Sufficiency conjecture. c93-chain-condition universal Baireness suffices for the argument above, namely for proving that $0^{\#}$ exists.

Consistency-strength claim. The consistency strength of the theory $

Cover-removal conjecture. The added assumption that $V\subseteq\overline{V}$ satisfies the $\kappa$ cover property is unnecessary: the $\delta$ approximation and cover properties a…

Let ${\sf BMM}$ denote the Bounded Martin's Maximum forcing axiom, and let an inner model with a Woodin cardinal mean an inner model containing a Woodin cardinal. BMM inner model c…

For each formula $\phi(x_1,\ldots,x_n)$ and stationary set $S\subseteq\omega_1$, let $Y_{\phi}$ be the set of all $a\in\{0,1\}^{\omega_1}$ such that there is a stationary…

Assume that there exists a proper class of supercompact cardinals. Let $T$ be the set of all $\Sigma^2_2$ sentences $\phi$ such that $V\vDash\phi$. A set of sentences is Ω-recursiv…

Let (1) be the assertion that … and let (2) be the assertion that if $\kappa$ is singular of uncountable cofinality, then either … contains a closed unbounded subset of $\kappa$. T…

Let $\kappa$ and $\lambda$ be cardinals, and let $NS_{\kappa\lambda}$ denote the non-stationary ideal on ${\mathcal P}_{\kappa}\lambda$. An ideal is precipitous if forcing with its…

Consistency-strength conjecture. The theory