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  "body": "\n## Statement\n\nLet $M$ be a countable transitive model of ZFC in which the generalized continuum hypothesis holds.\nLet $F$ be a class function, definable in $M$, from the [[regular cardinals|ZCE333FAWC]] of $M$ to the [[cardinals|0QKPNZKNEA]] of $M$.\nAssume that in $M$ the function $F$ satisfies both conditions:\n\n1. If $\\kappa \\le \\lambda$, then $F(\\kappa) \\le F(\\lambda)$.\n2. $\\operatorname{cf}(F(\\kappa)) \u003e \\kappa$ for each $\\kappa$, where $\\operatorname{cf}$ is the [[cofinality|NTE5C260S1]].\n\nThen there is a [[forcing|F2DTWW9HP7]] extension $N$ of $M$ with the same cardinals and the same cofinalities as $M$.\nIn $N$, $2^\\kappa = F(\\kappa)$ for every regular cardinal $\\kappa$.\n\n## Remarks\n\nThe two conditions are exactly the constraints that ZFC proves for $\\kappa \\mapsto 2^\\kappa$ on regular cardinals.\nMonotonicity is clear, and $\\operatorname{cf}(2^\\kappa) \u003e \\kappa$ is König's theorem.\nThe proof uses a class product of Cohen-style forcings, one for each regular $\\kappa$, with supports chosen so that the whole product preserves cardinals.\nExample: one can have $2^{\\aleph_0} = \\aleph_2$ and $2^{\\aleph_1} = \\aleph_2$, or $2^{\\aleph_0} = \\aleph_{\\omega + 1}$.\nBut $2^{\\aleph_0} = \\aleph_\\omega$ is impossible, since $\\operatorname{cf}(\\aleph_\\omega) = \\omega$.\nThe theorem says nothing about singular cardinals, where the situation is far more constrained, as in Silver's theorem and the singular cardinals hypothesis.\n\n## Explanations\n\n### L1 human\n\nFor an infinite cardinal $\\kappa$, the number $2^\\kappa$ counts the subsets of a set of size $\\kappa$.\nThe usual axioms of set theory say very little about it.\nEaston's theorem makes this precise for regular cardinals: apart from two simple rules, any pattern of values is possible.\nFor example, the axioms do not decide whether there are $\\aleph_1$ or $\\aleph_2$ real numbers.\n\n### L2 human\n\nForcing with Cohen reals shows that $2^{\\aleph_0}$ can be any cardinal of uncountable cofinality.\nEaston's theorem extends this to all regular cardinals at once.\nIt shows that monotonicity and König's theorem are the only ZFC constraints on the function $\\kappa \\mapsto 2^\\kappa$ on regular cardinals.\nAt singular cardinals the story changes: the values of $2^\\kappa$ below a singular cardinal can restrict its value at that cardinal.\nThis is the subject of Silver's theorem, of the singular cardinals hypothesis and of Shelah's pcf theory.\n",
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