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  "body": "\n## Example\n\nThe Ackermann function, in the two-argument form of Rózsa Péter, is the function $A \\colon \\N \\times \\N \\to \\N$ given by\n$$A(0, n) = n + 1, \\qquad A(m + 1, 0) = A(m, 1), \\qquad A(m + 1, n + 1) = A(m, A(m + 1, n)).$$\nIt is a total [[computable function|KZSB5E471K]].\nEach right-hand side calls $A$ at a pair that is smaller in the lexicographic order on $\\N \\times \\N$.\nThis order is a well-order, so the recursion stops on every input and gives an algorithm for $A$.\n\n## Remarks\n\nBy induction on $n$, one gets $A(1, n) = n + 2$, $A(2, n) = 2n + 3$ and $A(3, n) = 2^{n+3} - 3$.\nThen $A(4, 0) = A(3, 1) = 13$ and $A(4, 1) = A(3, 13) = 2^{16} - 3 = 65533$.\nThe function $A$ is not [[primitive recursive|5NVA2JRHPV]], see [[Ackermann function is not primitive recursive|97QF3KHSBY]].\n\n## Explanations\n\n### L1 human\n\nThe Ackermann function is a simple rule that produces enormous numbers very quickly.\nIts first argument chooses the operation: adding, then something like multiplying, then something like taking powers, and so on.\nThe values $A(4, n)$ are already huge: $A(4, n) + 3$ is a tower of $n + 3$ twos, such as $A(4, 1) + 3 = 2^{2^{2^2}}$.\nIt was built to show that some computable functions cannot be defined with bounded loops alone.\n",
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