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  "body": "\n## Description\n\nOrdinal exponentiation is defined by $\\omega^0 = 1$, $\\omega^{\\alpha + 1} = \\omega^\\alpha \\cdot \\omega$, and $\\omega^\\lambda = \\sup_{\\alpha \u003c \\lambda} \\omega^\\alpha$ for a limit ordinal $\\lambda$.\nAn *epsilon number* is an [[ordinal|0RDJK7EQYY]] $\\varepsilon$ with $\\omega^\\varepsilon = \\varepsilon$.\nThe ordinal $\\varepsilon_0$, read \"epsilon nought\", is the least epsilon number.\nIt is the limit of the sequence\n$$\\omega, \\quad \\omega^\\omega, \\quad \\omega^{\\omega^\\omega}, \\quad \\dots,$$\nthat is, $\\varepsilon_0 = \\sup \\set{\\omega_n : n \\in \\N}$ with $\\omega_0 = \\omega$ and $\\omega_{n+1} = \\omega^{\\omega_n}$.\n\nCheck: let $\\varepsilon = \\sup_n \\omega_n$.\nThe map $\\alpha \\mapsto \\omega^\\alpha$ is continuous at limits, so $\\omega^\\varepsilon = \\sup_n \\omega^{\\omega_n} = \\sup_n \\omega_{n+1} = \\varepsilon$.\nIf $\\beta$ is an epsilon number, then $\\omega \\le \\beta$, and induction on $n$ gives $\\omega_n \\le \\beta$, so $\\varepsilon \\le \\beta$.\nSince each $\\omega_n$ is a [[countable|01TQB88H4C]] ordinal, $\\varepsilon_0$ is a countable union of countable sets, hence countable.\n\n## Remarks\n\nEvery ordinal $\\alpha$ with $0 \u003c \\alpha \u003c \\varepsilon_0$ has a Cantor normal form $\\omega^{\\beta_1} c_1 + \\dots + \\omega^{\\beta_k} c_k$ with $\\alpha \u003e \\beta_1 \u003e \\dots \u003e \\beta_k$ and positive integers $c_i$.\nSince each exponent is smaller than $\\alpha$, the exponents can be written in the same form again, and the process ends.\nThis gives a finite notation for each ordinal below $\\varepsilon_0$.\nThe ordinal $\\varepsilon_0$ is the proof-theoretic ordinal of Peano arithmetic.\n\n## Explanations\n\n### L1 human\n\nStart with $\\omega$, the first infinite ordinal, and keep raising $\\omega$ to the power of what you have.\nThe tower $\\omega^{\\omega^{\\omega^{\\cdots}}}$ never ends, and $\\varepsilon_0$ is the first ordinal past all of its stages.\nIt is the first ordinal that does not change when you raise $\\omega$ to it.\nEvery smaller ordinal can be written as a finite expression built from $\\omega$, sums and powers.\n",
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