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  "body": "\n## Definition\n\nA *type theory* is a formal system with these parts.\n\n1. A collection of *types*, built from base types by type constructors.\n2. A collection of *terms*, built from variables and constants by term constructors.\n3. *Judgments*, in particular the typing judgment $\\Gamma \\vdash t : A$, which says that in the context $\\Gamma = (x_1 : A_1, \\dots, x_n : A_n)$ the term $t$ has type $A$.\n4. *Inference rules*, which say which judgments can be derived from which, and a notion of computation or equality on terms.\n\nEvery well-formed term has a type, and a term can only be used where its type allows.\n\n## Remarks\n\nExample: Church's simply typed lambda calculus is a type theory.\nIts types are built from base types by the arrow $A \\to B$.\nIts rules include: if $\\Gamma, x : A \\vdash t : B$, then $\\Gamma \\vdash \\lambda x.\\, t : A \\to B$; and if $\\Gamma \\vdash f : A \\to B$ and $\\Gamma \\vdash a : A$, then $\\Gamma \\vdash f\\, a : B$.\nNon-example: the untyped lambda calculus is not a type theory, because its terms carry no types.\nIn it, the self-application $\\lambda x.\\, x\\, x$ is a legal term, while in the simply typed calculus it has no type.\nIn *dependent* type theories, types may depend on terms, as in the type $\\mathrm{Vec}(n)$ of lists of length $n$.\nUnlike a [[set|0BSJ7RQ9BW]], a term has its type as part of its form, and \"$t$ has type $A$\" is a judgment, not a proposition that can be true or false.\nThis is how type theory avoids [[Russell's paradox|0HESMMSYEG]].\n",
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