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  "body": "\n## Definition\n\nThe *Elementary Theory of the Category of Sets* (ETCS) is the first-order theory of a [[category|A2DE66937D]] $\\mathcal{S}$ with the following axioms.\n\n1. $\\mathcal{S}$ is an elementary [[topos|6837RS7N82]].\n2. $\\mathcal{S}$ is nondegenerate: its [[terminal object|EK85VSSRMR]] $1$ is not an initial object.\n3. $\\mathcal{S}$ is well-pointed: for every two distinct morphisms $f, g \\colon X \\to Y$ there is a morphism $x \\colon 1 \\to X$ with $f \\circ x \\ne g \\circ x$.\n4. $\\mathcal{S}$ has a natural numbers object $(N, z, s)$, as described below.\n5. Every [[epimorphism|RVXDSY04KG]] $e \\colon X \\to Y$ in $\\mathcal{S}$ has a section, a morphism $m \\colon Y \\to X$ with $e \\circ m = \\id_Y$.\n\nA natural numbers object is an object $N$ with morphisms $z \\colon 1 \\to N$ and $s \\colon N \\to N$ with this property.\nFor every object $X$ with morphisms $a \\colon 1 \\to X$ and $h \\colon X \\to X$ there is a unique $u \\colon N \\to X$ with $u \\circ z = a$ and $u \\circ s = h \\circ u$.\n\n## Remarks\n\nExample: the category $\\mathbf{Set}$ of sets and functions, built in a model of ZFC, satisfies all five axioms.\nIts natural numbers object is $\\N$ with $0$ and $n \\mapsto n + 1$, and the sections come from the axiom of choice.\nNon-example: the category of sets with an action of the group $\\Z/2$ is a nondegenerate topos with a natural numbers object, but it is not well-pointed.\nIts terminal object is a one-point set, and a morphism $1 \\to X$ is a fixed point of $X$.\nOn $X = \\Z/2$ with the action by translation there are no fixed points, so the identity and the swap of $X$ are distinct but agree on all morphisms $1 \\to X$.\nIn ETCS an element of $X$ is a morphism $1 \\to X$, and the subsets of $X$ are classified by maps into the [[subobject classifier|NKP5TPQ79X]], which turns out to be $1 + 1$.\nETCS has no axiom schema of replacement.\n",
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