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  "body": "\n## Definition\n\n*Internal set theory* (IST) is the first-order theory in the language of set theory with one extra unary predicate $\\operatorname{st}$, read \"is standard\".\nA formula is *internal* if it does not contain $\\operatorname{st}$, and *external* otherwise.\nWrite $\\forall^{\\mathrm{st}} x$ for \"for all standard $x$\", $\\exists^{\\mathrm{st}} x$ for \"there is a standard $x$\", and $\\forall^{\\mathrm{st\\,fin}} z$ for \"for all standard [[finite|0BBTH3YSZA]] sets $z$\".\nThe axioms of IST are the following.\n\n1. The axioms of [[ZFC|0HE5WDQ9FJ]], where the schemes of separation and replacement are taken only for internal formulas.\n2. Idealization: for each internal formula $B(x, y)$, possibly with other free variables,\n$$\\forall^{\\mathrm{st\\,fin}} z \\, \\exists x \\, \\forall y \\in z \\; B(x, y) \\;\\leftrightarrow\\; \\exists x \\, \\forall^{\\mathrm{st}} y \\; B(x, y).$$\n3. Standardization: for each formula $C(z)$, internal or external, possibly with other free variables,\n$$\\forall^{\\mathrm{st}} x \\, \\exists^{\\mathrm{st}} y \\, \\forall^{\\mathrm{st}} z \\, \\bigl(z \\in y \\leftrightarrow (z \\in x \\wedge C(z))\\bigr).$$\n4. Transfer: for each internal formula $A(x, t_1, \\dots, t_k)$ with no other free variables,\n$$\\forall^{\\mathrm{st}} t_1 \\cdots \\forall^{\\mathrm{st}} t_k \\, \\bigl(\\forall^{\\mathrm{st}} x \\, A(x, t_1, \\dots, t_k) \\to \\forall x \\, A(x, t_1, \\dots, t_k)\\bigr).$$\n\n## Remarks\n\nExample: there is a nonstandard natural number.\nApply idealization to the internal formula $B(x, y)$: \"$x \\in \\N$, and if $y \\in \\N$ then $y \u003c x$\".\nThe left side holds, since every finite set $z$ has only finitely many natural numbers, and some natural number exceeds them all.\nSo some $\\nu \\in \\N$ is larger than every standard natural number, and $\\nu$ is not standard.\n\nNon-example: the collection of standard natural numbers is not given by separation, because the formula \"$n$ is standard\" is external, so clause 1 does not apply to it.\nIn fact no such set exists: its complement in $\\N$ would have a least element $m \u003e 0$, and then $m - 1$ would be standard, so $m$ would be standard by transfer.\n\nBy transfer, every object uniquely defined by an internal formula without parameters, such as $0$, $\\N$, $\\R$ or $\\pi$, is standard.\n",
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