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Single Idea 10265

[filed under theme 6. Mathematics / C. Sources of Mathematics / 6. Logicism / b. Type theory ]

Full Idea

Chihara's system is a version of type theory. Translate thus: replace variables of sets of type n with level n variables over open sentences, replace membership/predication with satisfaction, and high quantifiers with constructability quantifiers.

Clarification

For 'constructibility quantifiers' see Idea 10264

Gist of Idea

Chihara's system is a variant of type theory, from which he can translate sentences

Source

report of Charles Chihara (Constructibility and Mathematical Existence [1990]) by Stewart Shapiro - Philosophy of Mathematics 7.4

Book Ref

Shapiro,Stewart: 'Philosophy of Mathematics:structure and ontology' [OUP 1997], p.231

The 22 ideas from Charles Chihara

8758 We could talk of open sentences, instead of sets [Chihara, by Shapiro]
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]
10264 Introduce a constructibility quantifiers (Cx)Φ - 'it is possible to construct an x such that Φ' [Chihara, by Shapiro]
18151 Could we replace sets by the open sentences that define them? [Chihara, by Bostock]
10192 We can replace existence of sets with possibility of constructing token sentences [Chihara, by MacBride]
9547 Mathematical entities are causally inert, so the causal theory of reference won't work for them [Chihara]
9549 The set theorist cannot tell us what 'membership' is [Chihara]
9551 What is special about Bill Clinton's unit set, in comparison with all the others? [Chihara]
9550 We only know relational facts about the empty set, but nothing intrinsic [Chihara]
9552 Sentences are consistent if they can all be true; for Frege it is that no contradiction can be deduced [Chihara]
9553 Analytic geometry gave space a mathematical structure, which could then have axioms [Chihara]
9559 If a successful theory confirms mathematics, presumably a failed theory disconfirms it? [Chihara]
9561 The mathematics of relations is entirely covered by ordered pairs [Chihara]
9562 In simple type theory there is a hierarchy of null sets [Chihara]
9563 A pack of wolves doesn't cease when one member dies [Chihara]
9566 No scientific explanation would collapse if mathematical objects were shown not to exist [Chihara]
9568 I prefer the open sentences of a Constructibility Theory, to Platonist ideas of 'equivalence classes' [Chihara]
9571 ZFU refers to the physical world, when it talks of 'urelements' [Chihara]
9572 Realists about sets say there exists a null set in the real world, with no members [Chihara]
9573 The null set is a structural position which has no other position in membership relation [Chihara]
9574 'Gunk' is an individual possessing no parts that are atoms [Chihara]