Singular Cardinals Hypothesis

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Regard a cardinal $\kappa$ as its initial ordinal. Its cofinality $\operatorname{cf}(\kappa)$ is the least cardinality of a subset $A\subseteq\kappa$ unbounded in $\kappa$: for every $\alpha<\kappa$ there is $\beta\in A$ with $\alpha\le\beta$. An infinite cardinal is singular when $\operatorname{cf}(\kappa)<\kappa$. It is a strong limit when $2^\lambda<\kappa$ for every cardinal $\lambda<\kappa$, where $2^\lambda=|\mathcal P(\lambda)|$. Write $\kappa^+$ for the least cardinal greater than $\kappa$. The underlying axioms are ZFC.

Statement recorded in the supplied source.

The singular cardinals hypothesis (SCH) asserts \[ \forall\kappa\quad [\kappa\text{ is an infinite singular strong limit cardinal}] \ \Longrightarrow\ 2^\kappa=\kappa^+. \] Is this universal cardinal-arithmetic assertion valid? [S2] The generalized continuum hypothesis, which asserts $2^\lambda=\lambda^+$ for every infinite cardinal $\lambda$, implies SCH. There are also relative-consistency constructions of failures of SCH from suitable large-cardinal hypotheses; those hypotheses are stronger than merely the consistency of ZFC. Source S2 discusses both the assertion and the consistency strength of its failure. [S2]

Short English statement

Does the power set of every singular strong limit cardinal have the smallest cardinality it could have, namely the next larger cardinal? Its behavior is sensitive to additional set-theoretic assumptions.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 41 (SET-002). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 William J. Mitchell, On the Singular Cardinal Hypothesis, arXiv:math/9204202 (1992), Section 1, opening definition and Theorem 1.1. https://arxiv.org/pdf/math/9204202.