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### 6. Mathematics / C. Sources of Mathematics / 6. Logicism / b. Type theory

#### [theory that maths is a hierarchy of set types]

23 ideas
 10607 Frege's logic has a hierarchy of object, property, property-of-property etc. [Frege, by Smith,P]
 10093 The ramified theory of types used propositional functions, and covered bound variables [Russell/Whitehead, by George/Velleman]
 8691 The Russell/Whitehead type theory was limited, and was not really logic [Friend on Russell/Whitehead]
 21555 For 'x is a u' to be meaningful, u must be one range of individuals (or 'type') higher than x [Russell]
 18003 In 'x is a u', x and u must be of different types, so 'x is an x' is generally meaningless [Russell, by Magidor]
 23457 Type theory cannot identify features across levels (because such predicates break the rules) [Morris,M on Russell]
 21556 Classes are defined by propositional functions, and functions are typed, with an axiom of reducibility [Russell, by Lackey]
 10418 Type theory seems an extreme reaction, since self-exemplification is often innocuous [Swoyer on Russell]
 10047 Russell's improvements blocked mathematics as well as paradoxes, and needed further axioms [Russell, by Musgrave]
 23478 Type theory means that features shared by different levels cannot be expressed [Morris,M on Russell]
 6409 The 'simple theory of types' distinguishes levels among properties [Ramsey, by Grayling]
 21557 Russell confused use and mention, and reduced classes to properties, not to language [Quine, by Lackey]
 18127 Simple type theory has 'levels', but ramified type theory has 'orders' [Bostock]
 10265 Chihara's system is a variant of type theory, from which he can translate sentences [Chihara, by Shapiro]
 8759 We can replace type theory with open sentences and a constructibility quantifier [Chihara, by Shapiro]
 21703 Types are 'ramified' when there are further differences between the type of quantifier and its range [Linsky,B]
 21714 The ramified theory subdivides each type, according to the range of the variables [Linsky,B]
 21721 Higher types are needed to distinguished intensional phenomena which are coextensive [Linsky,B]
 10092 In the unramified theory of types, the types are objects, then sets of objects, sets of sets etc. [George/Velleman]
 10095 Type theory has only finitely many items at each level, which is a problem for mathematics [George/Velleman]
 10094 The theory of types seems to rule out harmless sets as well as paradoxical ones. [George/Velleman]
 17901 Type theory prohibits (oddly) a set containing an individual and a set of individuals [George/Velleman]
 16308 Set theory was liberated early from types, and recent truth-theories are exploring type-free [Halbach]