41 problems

Let $x$ be an infinite set, let $\omega$ be the first infinite ordinal, and let $\mathcal P(\omega)$ denote the power set of $\omega$. Cantor's continuum hypothesis. There is no ca…

Shelah's Strong Hypothesis. For every singular cardinal $\sigma$,

Shelah's PCF conjecture.

Let $\aleph_\delta$ be the first fix point, meaning the first ordinal index satisfying $\aleph_\delta=\delta$, and hence having cofinality $\aleph_0$. Shelah's first-fix-point pseu…

Let $\theta$ be any uncountable cardinal. Shelah's consistency conjecture for pseudopowers. It is consistent that, for some cardinal $\lambda$, both of the following hold: (A)…

A singular strong limit cardinal is a singular cardinal $\gk$ such that $2^{\gk'}$ is less than $\gk$ for every cardinal $\gk'<\gk$. The Singular Cardinal Hypothesis. If $\gk$ is a…

Let $\theta$ be a regular cardinal with $\theta \geq \omega_2$. The Mapping Reflection Principle, $\mathrm{MRP}$, is a forcing axiom concerning open stationary set mappings on coun…

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 $A$ be a set, let $\theta\neq 0$ be a limit cardinal number, and let $\operatorname{Card}$ denote the class of cardinal numbers. A binary relation on $A$ is a subset…

Let $A$ be a set, let $\theta\neq 0$ be a limit cardinal number, and let $\operatorname{Card}$ denote the class of cardinal numbers. A binary relation on $A$ is a subset…

Let $A$ be a set, let $\theta$ be an infinite cardinal number, and write $2^{2^{\theta}}$ for the cardinal obtained by iterating the power-set operation twice. A binary relation on…

The discussion concerns a countably iterative covering construction in which subsets are assigned to subintervals of a real interval, with $N$ possibly nonclosed and with singular…

Let $T$ be a countable theory and let $f_T(kappa,lambda)$ be the associated two-cardinal counting function for infinite cardinals $kappaleqlambda$. Adler's counting-spectrum conjec…

Let $alekappa$, $alelambda$, and $alemu$ be cardinals. A tree is special when it has the relevant specialization property, and ${\sf RC}(\kappa,\lambda,\mu)$ denotes the correspond…

Let $\omega_\alpha$ denote the cardinal at ordinal index $\alpha$, and let $\mathcal{P}(\omega_\alpha)$ be its power set. Generalized Continuum Hypothesis. For each ordinal number…

Sargsyan's conjecture. Conclusion (2) of that conjecture should hold: the model $N$ has

Solovay-sequence conjecture. Then $N\vDash\mathsf{ZF}$, $\Theta^N=\theta_\alpha$, $N$ satisfies that every cardinal $\eta\geq\Theta$ has regular successor $\eta^+$, and, for every…

Cardinal-regularity conjecture. Then

Let $\aleph_n$ denote the $n$th infinite cardinal, and let the power set of a set of cardinality $\aleph_n$ have cardinality $2^{\aleph_n}$. Generalized continuum hypothesis. The p…

In set theory without assuming the Axiom of Choice, consider the various notions of well-foundedness for cardinals discussed in the paper. Well-foundedness-to-choice conjecture. At…

For a singular cardinal $\theta$, let $\rm pp(\theta)$ denote its pseudopower, as defined by the supremum of true cofinalities of reduced products of regular cardinals below…

A cardinal is called a revision cardinal when it has the revision property developed above. Revision Hypothesis. Any limit cardinal is a revision cardinal. This hypothesis is prese…

Shelah's chain-length conjecture. For every $\mu\geq\theta$, there are no chains in

Shelah's conjecture. If for some $\lambda\geq|\tau_1|$, $\operatorname{Df}(\lambda)>\operatorname{Ded}(\lambda)$, then for every $\lambda$,…

Let $\kappa$ be a singular cardinal, and let ${}^{<\kappa}\square\,2^\kappa$ denote the corresponding middle box product space. The paracompactness conjecture for singular cardinal…