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  "body": "\n## Statement\n\nLet ${}^*\\R$ be the field of [[hyperreal numbers|421QMTFG9T]] built as an ultrapower of the [[real numbers|08J2M1JYAG]].\nFor each $k$-ary function $f$ on $\\R$, let ${}^*f$ be its extension to ${}^*\\R$, defined termwise on sequences.\nExtend relations on $\\R$ in the same way.\nLet $\\varphi$ be a sentence of [[first-order logic|086QKPF0A6]] built from real constants, functions and relations on $\\R$.\nLet ${}^*\\varphi$ be the sentence obtained by replacing each function $f$ by ${}^*f$ and each relation $R$ by ${}^*R$.\nThen $\\varphi$ is true in $\\R$ if and only if ${}^*\\varphi$ is true in ${}^*\\R$.\n\n## Remarks\n\nThe principle is a special case of [[Łoś's theorem|AGYBSTD7BG]].\nThat theorem says that a first-order formula holds at a tuple of classes exactly when it holds termwise on a set of indices in the ultrafilter.\n\nExample: the sentence $\\forall x \\, (x \u003e 0 \\to \\exists y \\, (y \\cdot y = x))$ is true in $\\R$.\nBy transfer, every positive hyperreal has a square root in ${}^*\\R$.\n\nThe restriction to first-order sentences, where quantifiers range over elements and not over arbitrary subsets, is necessary.\nEvery nonempty subset of $\\R$ that is bounded above has a supremum, but in ${}^*\\R$ the set of infinitesimals is bounded above by $1$ and has no supremum.\nIndeed, a supremum $s$ would be positive; if $s$ were infinitesimal then $2s$ would be a larger infinitesimal, and otherwise $s/2$ would be a smaller upper bound.\n\n## Explanations\n\n### L1 human\n\nThe hyperreals contain new numbers, infinitely small and infinitely large ones.\nThe transfer principle says that despite this, the hyperreals obey exactly the same elementary rules as the reals.\nTake any statement that uses only \"for all numbers\", \"there is a number\" and the usual functions and relations.\nIt is true for the reals exactly when it is true for the hyperreals.\n\n### L2 human\n\nTransfer is what makes nonstandard proofs valid.\nOne proves a statement in ${}^*\\R$ using infinitesimals, checks that it is the star version of a first-order statement, and transfers it back to $\\R$.\nFor instance, a real function $f$ is continuous at a real $a$ exactly when ${}^*f(x) - f(a)$ is infinitesimal for every $x$ with $x - a$ infinitesimal.\nTransfer is the key step in proving this equivalence.\nStatements about all subsets of $\\R$ transfer only to statements about internal subsets of ${}^*\\R$, such as ${}^*A$ for $A \\subseteq \\R$.\n",
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