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  "body": "\n## Statement\n\nLet $L$ be the [[constructible universe|6TF2JBTPPJ]].\n\n1. The axioms of [[ZF|0HE5WDQ9FJ]] and the [[axiom of constructibility|4QKRPRCSXH]] $V = L$ hold in $L$.\n2. ZF together with $V = L$ proves the [[axiom of choice|0HTGB13FE2]] and the generalized [[continuum hypothesis|0MN2MER9WN]] $2^{\\aleph_\\alpha} = \\aleph_{\\alpha + 1}$ for every ordinal $\\alpha$.\n3. Hence, if ZF is consistent, then ZF together with the axiom of choice and the generalized continuum hypothesis is consistent.\n\n## Remarks\n\nThe construction of $L$ can be carried out inside any model of ZF, and it is absolute, so $L$ computed inside $L$ is $L$ again; this gives part 1.\nThe axiom of choice holds in $L$ because $L$ has a definable well-order.\nA set enters at a least stage and is defined there by a formula with parameters, and these data can be ordered.\nFor the continuum hypothesis, assume $V = L$; the condensation lemma shows that every subset of $\\omega_\\alpha$ lies in $L_{\\omega_{\\alpha + 1}}$, a set of size $\\aleph_{\\alpha + 1}$.\nThe result is a relative consistency proof: by [[Gödel's second incompleteness theorem|X9WYNYE29C]], ZF cannot prove its own consistency, so the hypothesis \"ZF is consistent\" cannot be removed.\nTogether with [[Cohen's theorem|EMGBZ46GCD]], it shows that the continuum hypothesis is independent of ZFC.\n\n## Explanations\n\n### L1 human\n\nThe continuum hypothesis says that there is no size of infinity strictly between the natural numbers and the real numbers.\nGödel found a world of sets, the constructible universe, in which all the usual axioms hold and the continuum hypothesis is true as well.\nSo the usual axioms can never disprove the continuum hypothesis.\nThe same world also shows that the axiom of choice cannot be disproved from the other axioms.\n\n### L2 human\n\nThe theorem is the first half of the independence of the continuum hypothesis; the second half is Cohen's forcing argument.\nThe method is the method of inner models: one finds a definable class in which the axioms hold together with the desired statement.\nInner models alone cannot show that the continuum hypothesis can fail: if $V = L$, then $L$ is the only inner model, and the continuum hypothesis holds in it.\nTo show that the continuum hypothesis can fail, one needs a way to enlarge the universe, and this is what forcing does.\n",
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