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  "body": "\n## Statement\n\nLet $L$ be a signature, let $\\mathcal{U}$ be an [[ultrafilter|J64Y3JC0RE]] on a nonempty set $I$, and let $(M_i)_{i \\in I}$ be $L$-structures with nonempty domains.\nLet $M = \\prod_{i \\in I} M_i / \\mathcal{U}$ be their [[ultraproduct|AX9Q86GB8M]].\nLet $\\varphi(x_1, \\dots, x_n)$ be a first-order [[$L$-formula|HYJ10AF6RT]], and let $a_1, \\dots, a_n \\in \\prod_{i \\in I} M_i$.\nThen\n$$M \\models \\varphi([a_1], \\dots, [a_n]) \\iff \\set{i \\in I : M_i \\models \\varphi(a_1(i), \\dots, a_n(i))} \\in \\mathcal{U}.$$\n\n## Remarks\n\nThe proof is an induction on the formula.\nAtomic formulas hold by the definition of the ultraproduct, and conjunction uses that $\\mathcal{U}$ is closed under finite intersections.\nNegation uses that $\\mathcal{U}$ contains exactly one of each set and its complement, which is where the ultrafilter property enters.\nFor $\\exists y \\, \\psi$, the axiom of choice picks a witness $b(i)$ in each factor where one exists.\n\nIn particular, a sentence holds in an ultraproduct exactly when the set of factors where it holds lies in $\\mathcal{U}$.\nFor an ultrapower $M^I / \\mathcal{U}$, the diagonal map $m \\mapsto [(m, m, \\dots)]$ is an elementary embedding of $M$.\nFor example, every ultrapower of $\\R$ is a real closed ordered field, since these properties are first-order.\nThe theorem also gives a proof of the compactness theorem.\nIf every finite subset of a theory $T$ has a model, an ultraproduct of these models over a suitable ultrafilter is a model of $T$.\n\nThe ultrafilter is necessary: for a filter that is not an ultrafilter, the corresponding reduced product of fields can fail to be a field.\n\n## Explanations\n\n### L1 human\n\nAn ultraproduct glues many structures into one, and an ultrafilter decides which sets of indices count as \"most\".\nŁoś's theorem says that a first-order statement is true in the glued structure exactly when it is true in \"most\" of the pieces.\nSo the glued structure keeps every first-order property that most pieces share.\n\n### L2 human\n\nŁoś's theorem is the reason ultraproducts are useful in model theory.\nIt gives the compactness theorem without any proof system, and it shows that a class of structures closed under ultraproducts and elementary equivalence is axiomatisable.\nWith a non-principal ultrafilter on $\\N$, the ultrapower of $\\R$ gives the hyperreal numbers, and the theorem becomes the transfer principle of nonstandard analysis.\nUltraproducts of the finite fields $\\F_p$, indexed by the primes $p$, over a non-principal ultrafilter are fields of characteristic $0$.\nThis links statements about large primes with statements in characteristic $0$.\n",
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