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  "body": "\n## Statement\n\nLet $\\alpha \u003e 0$ be an [[ordinal number|0RDJK7EQYY]], with [[ordinal arithmetic|KAEDCQ8RM6]].\nThen there are an integer $k \\ge 1$, ordinals $\\beta_1 \u003e \\beta_2 \u003e \\cdots \u003e \\beta_k$ and positive integers $c_1, \\ldots, c_k$ such that\n$$\\alpha = \\omega^{\\beta_1} \\cdot c_1 + \\omega^{\\beta_2} \\cdot c_2 + \\cdots + \\omega^{\\beta_k} \\cdot c_k.$$\nThe integer $k$, the ordinals $\\beta_i$ and the integers $c_i$ are unique.\n\n## Remarks\n\nThe proof is by induction on $\\alpha$.\nLet $\\beta_1$ be the largest ordinal with $\\omega^{\\beta_1} \\le \\alpha$.\nDivision with remainder gives $\\alpha = \\omega^{\\beta_1} \\cdot c_1 + \\rho$ with $0 \u003c c_1 \u003c \\omega$ and $\\rho \u003c \\omega^{\\beta_1}$, and the induction hypothesis applies to $\\rho$.\nExample: $\\omega^2 + \\omega \\cdot 3 + 5$ is already in normal form, with exponents $2 \u003e 1 \u003e 0$.\nThe normal form has $\\beta_1 \\le \\alpha$, and equality can occur: for $\\varepsilon_0$, the least ordinal with $\\omega^{\\varepsilon_0} = \\varepsilon_0$, the normal form is $\\omega^{\\varepsilon_0}$.\nFor ordinals below [[epsilon nought|KT50SMH3EC]], the exponents are smaller than the ordinal, and repeating the normal form on them gives a finite notation.\nThe same holds with any base $b \\ge 2$ in place of $\\omega$, with coefficients $0 \u003c c_i \u003c b$.\n\n## Explanations\n\n### L1 human\n\nEvery ordinal can be written as a polynomial in $\\omega$, with decreasing exponents and positive whole coefficients.\nThis is like writing a whole number in base ten, but with base $\\omega$.\nFor instance, $\\omega^2 \\cdot 2 + 7$ means two copies of $\\omega^2$ followed by seven more points.\nThe expression is unique, so it can be used to name and compare ordinals.\n\n### L2 human\n\nThe Cantor normal form makes ordinal arithmetic computable on notations.\nTwo ordinals in normal form are compared by their leading terms, as polynomials are compared by degree.\nIterating the form on exponents gives the standard notation system for ordinals below $\\varepsilon_0$, which appears in Gentzen's consistency proof for Peano arithmetic and in [[Goodstein's theorem|XQMNJF241J]].\nBeyond $\\varepsilon_0$ one needs stronger notations, such as those of the [[Veblen function|JQFAE9S2F4]].\n",
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