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  "body": "\n## Description\n\nThe *surreal numbers* are built in stages, called days.\nA surreal number is written $x = \\{ L \\mid R \\}$, where $L$ and $R$ are sets of surreal numbers built on earlier days.\nThe condition is that no element of $L$ is greater than or equal to any element of $R$.\nThe order is defined at the same time.\nThe rule is: $x \\le y$ holds when no left option of $x$ is $\\ge y$ and no right option of $y$ is $\\le x$.\nTwo forms $x$ and $y$ name the same number when $x \\le y$ and $y \\le x$.\n\nOn day $0$, only $0 = \\{ \\mid \\}$ exists, with both sets empty.\nOn day $1$, one gets $1 = \\{ 0 \\mid \\}$ and $-1 = \\{ \\mid 0 \\}$.\nOn day $2$, one gets $2 = \\{ 1 \\mid \\}$, $\\tfrac12 = \\{ 0 \\mid 1 \\}$, and their negatives.\nAfter infinitely many days one gets $\\omega = \\{ 0, 1, 2, \\dots \\mid \\}$, which is larger than every natural number.\nOne also gets $\\varepsilon = \\{ 0 \\mid 1, \\tfrac12, \\tfrac14, \\dots \\}$, which is positive but smaller than every positive real number.\n\nAddition and multiplication are defined by recursion on the options.\nWith them the surreal numbers satisfy all the axioms of an [[ordered field|0P3VCBDEE2]].\nThe one caveat is size: they form a proper class, not a set, so strictly they are an ordered field only in this class sense.\nThis field contains a copy of the real numbers and a copy of every [[ordinal number|0YSCSX5YZC]].\nIt is not [[Archimedean|0XH79S06J3]], since $\\omega$ exceeds every natural number, and $\\omega \\cdot \\varepsilon = 1$.\nIt is a [[real closed field|DDY3Q3PJ6W]].\n\n## Remarks\n\nEach real number $r$ is the surreal number $\\{ L \\mid R \\}$ with $L$ the rationals below $r$ with power-of-two denominators and $R$ those above $r$; this mirrors a [[Dedekind cut|0G3RRM41X9]].\nLike the [[hyperreal numbers|421QMTFG9T]], the surreals contain infinite and infinitesimal elements.\nThe surreals were introduced by John Conway, and the name comes from Donald Knuth's book *Surreal Numbers* (1974).\n\n## Explanations\n\n### L1 human\n\nSurreal numbers start from nothing and grow by one rule: a new number sits between a set of numbers on its left and a set on its right.\nFrom the empty pair one gets $0$, then $1$ and $-1$, then $2$, $\\tfrac12$ and so on.\nEvery real number appears, and so do infinite numbers like $\\omega$ and tiny numbers like $1/\\omega$.\nYou can still add, subtract, multiply and divide them, and compare any two of them.\nSo they are one huge number line that holds both the reals and the ordinals.\n",
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