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Singular cardinal

{title2=$\gimel(\kappa)=\max\{\kappa^+,2^{\operatorname{cf}(\kappa)}\}$}

= SCH
{c}
{synonym}

The singular cardinals hypothesis asserts that every infinite <singular cardinal> $\kappa$ satisfies
$$
2^{\operatorname{cf}(\kappa)}<\kappa\ \Longrightarrow\ \gimel(\kappa)=\kappa^+.
$$
Equivalently, $\gimel(\kappa)=\max\{\kappa^+,2^{\operatorname{cf}(\kappa)}\}$ for every infinite <singular cardinal>. Here $\gimel$ is the <gimel function>. The <Generalized continuum hypothesis> implies this principle, but the principle only constrains the indicated singular-cardinal exponentiation.

Singular cardinals hypothesis

Gimel function

Gimel hypothesis

Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 19 / 3 / i / a / Solution

Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 121 / 1 / a / Solution

codex/sch

Singular cardinals hypothesis ( $ℷ (κ) = max {κ^{+}, 2^{cf (κ)}}$ )

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Singular cardinals hypothesis (ℷ(κ)=max{κ+,2cf(κ)})