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  "body": "\n## Summary\n\nThe Veblen functions $\\varphi_\\gamma$ are a hierarchy of [[normal functions|ERTAM0SFVB]] on the [[ordinals|0YSCSX5YZC]], each one listing the fixed points of the ones before.\nThey start from $\\varphi_0(\\alpha) = \\omega^\\alpha$, and they give names to large countable ordinals such as the epsilon numbers and the Feferman–Schütte ordinal $\\Gamma_0$.\nThe [[precise definition|6QFEVWKE25]] is by transfinite recursion on the index $\\gamma$.\nThey are a basic tool of ordinal notations and of ordinal analysis in proof theory.\n\n## Explanations\n\n### L1 human\n\nStart with the function $\\alpha \\mapsto \\omega^\\alpha$.\nIts fixed points, the ordinals with $\\omega^\\alpha = \\alpha$, are the epsilon numbers, and listing them gives a new function $\\varphi_1$.\nThe fixed points of $\\varphi_1$ give $\\varphi_2$, and so on, also through infinite stages.\nEach new function grows so fast that its values outrun everything the earlier functions can name.\nFor instance, $\\varphi_1(0) = \\varepsilon_0$, the first ordinal with $\\omega^{\\varepsilon_0} = \\varepsilon_0$.\n",
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