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  "body": "\n## Definition\n\nA set $X$ is transitive when each element of an element of $X$ is an element of $X$.\nAn ordinal number, in the definition of von Neumann, is a transitive set $\\alpha$ that is [[well-ordered|0F1D6PK9RZ]] by the relation $\\in$, with $\\beta \u003c \\gamma$ when $\\beta \\in \\gamma$.\n\n## Remarks\n\nThe first ordinals are $0 = \\varnothing$, $1 = \\{0\\}$, $2 = \\{0, 1\\}$, and in general $n + 1 = n \\cup \\{n\\}$.\nThe set $\\omega = \\{0, 1, 2, \\dots\\}$ of all finite ordinals is the first infinite ordinal.\nEach well-ordered set is [[order isomorphic|0Y74KN99A5]] to exactly one ordinal, its order type.\nEach ordinal $\\alpha$ is the set of all ordinals smaller than $\\alpha$.\n\n## Explanations\n\n### L1 human\n\nOrdinals count positions: first, second, third, and then beyond all finite positions.\nAfter $0, 1, 2, \\dots$ comes $\\omega$, then $\\omega + 1$, and so on.\nThey let you continue a step-by-step construction past infinity.\n",
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