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Ideas for 'fragments/reports', 'Must We Believe in Set Theory?' and 'Hermeneutics: a very short introduction'

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4 ideas

4. Formal Logic / F. Set Theory ST / 1. Set Theory
10482 The logic of ZF is classical first-order predicate logic with identity [Boolos]
     Full Idea: The logic of ZF Set Theory is classical first-order predicate logic with identity.
     From: George Boolos (Must We Believe in Set Theory? [1997], p.121)
     A reaction: This logic seems to be unable to deal with very large cardinals, precisely those that are implied by set theory, so there is some sort of major problem hovering here. Boolos is fairly neutral.
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
10492 A few axioms of set theory 'force themselves on us', but most of them don't [Boolos]
     Full Idea: Maybe the axioms of extensionality and the pair set axiom 'force themselves on us' (Gödel's phrase), but I am not convinced about the axioms of infinity, union, power or replacement.
     From: George Boolos (Must We Believe in Set Theory? [1997], p.130)
     A reaction: Boolos is perfectly happy with basic set theory, but rather dubious when very large cardinals come into the picture.
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / d. Naïve logical sets
10485 Naïve sets are inconsistent: there is no set for things that do not belong to themselves [Boolos]
     Full Idea: The naïve view of set theory (that any zero or more things form a set) is natural, but inconsistent: the things that do not belong to themselves are some things that do not form a set.
     From: George Boolos (Must We Believe in Set Theory? [1997], p.127)
     A reaction: As clear a summary of Russell's Paradox as you could ever hope for.
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / e. Iterative sets
10484 The iterative conception says sets are formed at stages; some are 'earlier', and must be formed first [Boolos]
     Full Idea: According to the iterative conception, every set is formed at some stage. There is a relation among stages, 'earlier than', which is transitive. A set is formed at a stage if and only if its members are all formed before that stage.
     From: George Boolos (Must We Believe in Set Theory? [1997], p.126)
     A reaction: He gives examples of the early stages, and says the conception is supposed to 'justify' Zermelo set theory. It is also supposed to make the axioms 'natural', rather than just being selected for convenience. And it is consistent.