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  "body": "\n## Statement\n\nWork in [[ZFC|0HE5WDQ9FJ]].\nFor an infinite [[cardinal|0EBACQFAZW]] $\\kappa$, let $\\operatorname{cf}(\\kappa)$ be its [[cofinality|BAWHE1W6MH]], and call $\\kappa$ singular when $\\operatorname{cf}(\\kappa) \u003c \\kappa$.\nLet $\\kappa^+$ be the least cardinal greater than $\\kappa$.\nThe *singular cardinals hypothesis* (SCH) is the assertion: for every singular cardinal $\\kappa$ with $2^{\\operatorname{cf}(\\kappa)} \u003c \\kappa$, one has $\\kappa^{\\operatorname{cf}(\\kappa)} = \\kappa^+$.\n\n## Remarks\n\nAn infinite cardinal that is not singular is a [[regular cardinal|13ZV2F8JAR]].\nSCH is equivalent to the statement that $2^\\kappa = \\kappa^+$ for every singular strong limit cardinal $\\kappa$, where strong limit means $2^\\lambda \u003c \\kappa$ for all $\\lambda \u003c \\kappa$.\nThe generalized form of the [[continuum hypothesis|0MN2MER9WN]], $2^\\lambda = \\lambda^+$ for every infinite $\\lambda$, implies SCH.\nSo SCH is consistent with ZFC whenever ZFC is consistent.\nSCH is not provable in ZFC if suitable large cardinals are consistent: Magidor built models where it fails at $\\aleph_\\omega$ from large cardinal hypotheses.\nIts failure needs large cardinals: by Jensen's covering lemma, if $0^\\sharp$ does not exist, then SCH holds.\nGitik and Mitchell showed that the failure of SCH is equiconsistent with a measurable cardinal $\\kappa$ of Mitchell order $\\kappa^{++}$.\nSolovay proved that SCH holds at all singular cardinals above a strongly compact cardinal.\nSilver proved that if $\\kappa$ is singular of uncountable cofinality and $2^\\lambda = \\lambda^+$ for all infinite $\\lambda \u003c \\kappa$, then $2^\\kappa = \\kappa^+$.\n\n## Explanations\n\n### L1 human\n\nHere $2^\\kappa$ is the number of subsets of a set of size $\\kappa$.\nAnd $\\kappa^\\lambda$ is the number of functions from a set of size $\\lambda$ to a set of size $\\kappa$.\nFor regular cardinals, ZFC says very little about the size of $2^\\kappa$: almost any reasonable value is consistent.\nSingular cardinals, such as $\\aleph_\\omega$, behave differently, and ZFC proves real restrictions on their powers.\nThe singular cardinals hypothesis says that these powers take the smallest values that the restrictions allow.\nIt is a weak form of the generalized continuum hypothesis that concerns only singular cardinals.\n\n### L2 human\n\nIn 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.\nUnder SCH, the second part is determined by the first, so all of cardinal arithmetic is computed from the values of $2^\\lambda$ at regular $\\lambda$.\nThe study of how SCH can fail led to Shelah's pcf theory, which gives bounds in ZFC alone.\nA famous such bound is that if $\\aleph_\\omega$ is a strong limit, then $2^{\\aleph_\\omega} \u003c \\aleph_{\\omega_4}$.\n",
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