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  "body": "\n## Definition\n\nThe functions $\\varphi_\\gamma$, for ordinals $\\gamma$, map [[ordinals|0RDJK7EQYY]] to ordinals.\nThey are defined by [[transfinite recursion|0ZHQMQ2C1N]] on $\\gamma$:\n\n1. $\\varphi_0(\\alpha) = \\omega^\\alpha$, with ordinal exponentiation from [[ordinal arithmetic|2C7MBG93AZ]].\n2. $\\varphi_{\\gamma + 1}$ lists in increasing order the fixed points of $\\varphi_\\gamma$, the ordinals $\\alpha$ with $\\varphi_\\gamma(\\alpha) = \\alpha$.\n3. For a nonzero limit ordinal $\\gamma$, $\\varphi_\\gamma$ lists in increasing order the common fixed points of all $\\varphi_\\delta$ with $\\delta \u003c \\gamma$.\n\nThe two-variable function $(\\gamma, \\alpha) \\mapsto \\varphi_\\gamma(\\alpha)$, also written $\\varphi(\\gamma, \\alpha)$, is the Veblen function.\n\n## Remarks\n\nEach $\\varphi_\\gamma$ is a [[normal function|PXTF18X433]], and this makes clauses 2 and 3 meaningful, since the fixed points of normal functions form a closed unbounded class.\nExample: $\\varphi_1(\\alpha) = \\varepsilon_\\alpha$, the $\\alpha$-th epsilon number, so $\\varphi_1(0) = \\varepsilon_0$ is the least $\\alpha$ with $\\omega^\\alpha = \\alpha$.\nCheck: $\\varphi_1$ lists the fixed points of $\\varphi_0$, and these are by definition the epsilon numbers.\nNon-example: the function $\\alpha \\mapsto \\omega^\\alpha + 1$ cannot start the hierarchy, since it is not normal and has no fixed points.\nThe least ordinal $\\alpha$ with $\\varphi_\\alpha(0) = \\alpha$ is the Feferman–Schütte ordinal $\\Gamma_0$.\nMore generally, one may start from any normal function $f$ in place of $\\alpha \\mapsto \\omega^\\alpha$.\n",
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