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  "body": "\n## Summary\n\nA type theory is a formal system in which every term has a type.\nIts rules say which terms can be formed in which types.\nRussell introduced a theory of types to avoid [[Russell's paradox|0HESMMSYEG]], by forbidding a collection to contain objects of its own level.\nChurch's simple type theory of 1940 and Martin-Löf's intuitionistic type theory are the main later systems.\nType theories serve as foundations of mathematics alternative to set theory, and as the basis of programming languages and proof assistants.\n\n## Explanations\n\n### L1 human\n\nIn a type theory, everything comes with a label, its type, such as \"natural number\" or \"function from numbers to numbers\".\nYou may only apply a function to an argument of the right type.\nSo the expression $f(f)$, which feeds a function to itself, is not allowed for a function $f \\colon \\N \\to \\N$.\nThis simple discipline blocks the paradoxes of naive [[set|0BSJ7RQ9BW]] theory.\nIt also catches many errors in computer programs before they run.\n",
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