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

[theory that maths is a hierarchy of set types]

15 ideas
Frege's logic has a hierarchy of object, property, property-of-property etc. [Smith,P on Frege]
The ramified theory of types used propositional functions, and covered bound variables [George/Velleman on Russell/Whitehead]
The Russell/Whitehead type theory was limited, and was not really logic [Friend on Russell/Whitehead]
In 'x is a u', x and u must be of different types, so 'x is an x' is generally meaningless [Magidor on Russell]
Type theory seems an extreme reaction, since self-exemplification is often innocuous [Swoyer on Russell]
Russell's improvements blocked mathematics as well as paradoxes, and needed further axioms [Musgrave on Russell]
The 'simple theory of types' distinguishes levels among properties [Grayling on Ramsey]
Simple type theory has 'levels', but ramified type theory has 'orders' [Bostock]
Chihara's system is a variant of type theory, from which he can translate sentences [Shapiro on Chihara]
We can replace type theory with open sentences and a constructibility quantifier [Shapiro on Chihara]
In the unramified theory of types, the types are objects, then sets of objects, sets of sets etc. [George/Velleman]
Type theory has only finitely many items at each level, which is a problem for mathematics [George/Velleman]
The theory of types seems to rule out harmless sets as well as paradoxical ones. [George/Velleman]
Type theory prohibits (oddly) a set containing an individual and a set of individuals [George/Velleman]
Set theory was liberated early from types, and recently truth-theories are exploring type-free [Halbach]