Singular cardinals hypothesis

Remarks

An infinite cardinal that is not singular is a regular cardinal. SCH is equivalent to the statement that $2^\kappa = \kappa^+$ for every singular strong limit cardinal $\kappa$, where strong limit means $2^\lambda < \kappa$ for all $\lambda < \kappa$. The generalized form of the continuum hypothesis, $2^\lambda = \lambda^+$ for every infinite $\lambda$, implies SCH. So SCH is consistent with ZFC whenever ZFC is consistent. SCH is not provable in ZFC if suitable large cardinals are consistent: Magidor built models where it fails at $\aleph_\omega$ from large cardinal hypotheses. Its failure needs large cardinals: by Jensen's covering lemma, if $0^\sharp$ does not exist, then SCH holds. Gitik and Mitchell showed that the failure of SCH is equiconsistent with a measurable cardinal $\kappa$ of Mitchell order $\kappa^{++}$. Solovay proved that SCH holds at all singular cardinals above a strongly compact cardinal. Silver proved that if $\kappa$ is singular of uncountable cofinality and $2^\lambda = \lambda^+$ for all infinite $\lambda < \kappa$, then $2^\kappa = \kappa^+$.

Going deeper

For a student

In ZFC, cardinal exponentiation $\kappa^\lambda$ is governed by the continuum function $\lambda \mapsto 2^\lambda$ on regular cardinals together with the values $\kappa^{\operatorname{cf}(\kappa)}$ at singular cardinals. Under SCH, the second part is determined by the first, so all of cardinal arithmetic is computed from the values of $2^\lambda$ at regular $\lambda$. The study of how SCH can fail led to Shelah's pcf theory, which gives bounds in ZFC alone. A famous such bound is that if $\aleph_\omega$ is a strong limit, then $2^{\aleph_\omega} < \aleph_{\omega_4}$.