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  "body": "\n## Summary\n\nSecond-order logic extends [[first-order logic|086QKPF0A6]] with variables for relations and functions on the domain, and with quantifiers over them.\nIn its standard semantics these variables range over all relations and functions on the domain.\nIt can describe structures such as the natural numbers up to isomorphism, but it loses the completeness and compactness theorems of first-order logic.\n\n## Explanations\n\n### L1 human\n\nFirst-order logic can say \"for every number $x$\", but not \"for every set of numbers $X$\".\nSecond-order logic can say both.\nWith it, the induction principle becomes one sentence: every set of numbers that contains $0$ and is closed under successor contains all numbers.\nThis one sentence, with the other Peano axioms, pins down the natural numbers completely.\nThe price is that no proof system can capture all its valid sentences.\n",
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