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  "body": "\n## Example\n\nFor $n \\ge 0$, the *Laver table* $A_n$ is the set $\\set{1, 2, \\dots, 2^n}$ with the unique binary operation $\\star$ such that:\n\n1. $p \\star 1 = p + 1$ for $p \u003c 2^n$, and $2^n \\star 1 = 1$;\n2. $p \\star (q \\star r) = (p \\star q) \\star (p \\star r)$ for all $p, q, r$.\n\nSo $A_n$ is a [[shelf|PDFC0MRYR8]]: clause 2 is [[left self-distributivity|VEFE5WZMS3]].\nThe table can be computed row by row from the bottom, where row $2^n$ is $2^n \\star q = q$.\nFor $p \u003c 2^n$ one uses $p \\star (q + 1) = (p \\star q) \\star (p + 1)$, where $p \\star q \u003e p$ lies in a row already known.\n\nFor $n = 2$ this gives the table of $A_2$, with $p$ indexing rows and $q$ indexing columns:\n\n| $\\star$ | 1 | 2 | 3 | 4 |\n|---|---|---|---|---|\n| 1 | 2 | 4 | 2 | 4 |\n| 2 | 3 | 4 | 3 | 4 |\n| 3 | 4 | 4 | 4 | 4 |\n| 4 | 1 | 2 | 3 | 4 |\n\nCheck of one instance of clause 2: $1 \\star (2 \\star 1) = 1 \\star 3 = 2$, and $(1 \\star 2) \\star (1 \\star 1) = 4 \\star 2 = 2$.\nAll $64$ instances of clause 2 hold in $A_2$.\nThe shelf $A_2$ is not a rack, since the row of $1$ takes only the values $2$ and $4$.\n\n## Remarks\n\nEach row of $A_n$ is periodic, and its period is a power of $2$.\nLaver proved that, if a rank-into-rank [[large cardinal|X1FG7PSCTY]] exists, then the period of the row of $1$ in $A_n$ tends to infinity as $n$ grows.\nNo proof of this statement in ZFC alone is known.\nThe period of the row of $1$ grows extremely slowly: it is $2$ in $A_2$ and $4$ in $A_3$.\nDougherty showed that the least $n$ for which it reaches $32$, if there is one, is enormous.\n\n## Explanations\n\n### L1 human\n\nA Laver table is a small multiplication table on the numbers $1$ to $2^n$.\nIt is fixed by two rules: multiplying by $1$ on the right adds one, and the operation distributes over itself.\nThese two rules leave no freedom, so each size has exactly one table.\nThe tables look like simple finite puzzles, but a basic fact about how their rows repeat has only been proved using axioms of infinity far beyond ordinary mathematics.\n",
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