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  "body": "\n## Statement\n\nLet $b \\ge 2$ be an integer.\nThe *hereditary base-$b$ notation* of $n \\ge 1$ writes $n$ as a sum $b^{e_1} c_1 + \\cdots + b^{e_r} c_r$ with $e_1 \u003e \\cdots \u003e e_r$ and $0 \u003c c_i \u003c b$.\nThen it writes each exponent $e_i$ in hereditary base-$b$ notation again.\nLet $B_b(n)$ be the number obtained by replacing every $b$ in this notation by $b + 1$, with $B_b(0) = 0$.\nThe *Goodstein sequence* of $m \\in \\N$ is $G_1 = m$ and $G_{k+1} = B_{k+1}(G_k) - 1$ while $G_k \u003e 0$.\n\n1. For every $m \\in \\N$ the Goodstein sequence of $m$ reaches $0$.\n2. Statement 1, written as a sentence of first-order arithmetic, is not provable in [[Peano arithmetic|0QFT7Z76N5]] $\\mathsf{PA}$.\n\n## Remarks\n\nProof idea for 1: replace each base in the hereditary notation of $G_k$ by the [[ordinal|0YSCSX5YZC]] $\\omega$.\nThis gives an ordinal $o_k \u003c \\varepsilon_0$, and the change of base does not change it while subtracting $1$ makes it strictly smaller.\nA strictly decreasing sequence of ordinals is finite, so the sequence reaches $0$.\nGoodstein proved part 1 in 1944; Kirby and Paris proved part 2 in 1982, by showing that $\\mathsf{PA}$ plus statement 1 proves the consistency of $\\mathsf{PA}$.\n\nExample: for $m = 3$, the sequence is $3 = 2 + 1$, then $3 + 1 - 1 = 3$, then in base $3$, $3 = 3^1$ becomes $4 - 1 = 3$, then $2$, $1$, $0$.\nFor $m = 4$ the sequence also reaches $0$, but only after more than $10^{100{,}000{,}000}$ steps.\n\n## Explanations\n\n### L1 human\n\nStart with a number and write it in base $2$, using only $2$s even in the exponents.\nChange all $2$s to $3$s and subtract $1$; then repeat with bases $3$ and $4$, and so on.\nThe numbers first grow enormously, yet the theorem says they always come back to $0$.\nThe proof uses infinite ordinals, and in fact the theorem cannot be proved with the usual axioms of arithmetic alone.\n\n### L2 human\n\nThe theorem shows how ordinals below $\\varepsilon_0$ measure the strength of $\\mathsf{PA}$: transfinite induction up to [[epsilon nought|KT50SMH3EC]] proves it, while $\\mathsf{PA}$ does not.\nThis ties it to [[Gentzen's consistency proof|QPV0HNP22S]] and to [[ordinal analysis|C5E5TG8PGP]].\nLike the [[Paris–Harrington theorem|509ZWA0R26]], it is a natural true statement independent of $\\mathsf{PA}$.\n",
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