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Single Idea 10775

[filed under theme 4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX ]

Full Idea

The main objection to the axiom of choice was that it had to be given by some law or definition, but since sets are arbitrary this seems irrelevant. Formalists consider it meaningless, but set-theorists consider it as true, and practically obvious.

Gist of Idea

The axiom of choice now seems acceptable and obvious (if it is meaningful)

Source

Leslie H. Tharp (Which Logic is the Right Logic? [1975], §3)

Book Ref

'Philosophy of Logic: an anthology', ed/tr. Jacquette,Dale [Blackwell 2002], p.40

The 16 ideas from Leslie H. Tharp

10762 In sentential logic there is a simple proof that all truth functions can be reduced to 'not' and 'and' [Tharp]
10763 Completeness and compactness together give axiomatizability [Tharp]
10770 If completeness fails there is no algorithm to list the valid formulas [Tharp]
10771 Compactness is important for major theories which have infinitely many axioms [Tharp]
10772 Compactness blocks infinite expansion, and admits non-standard models [Tharp]
10764 A complete logic has an effective enumeration of the valid formulas [Tharp]
10768 Effective enumeration might be proved but not specified, so it won't guarantee knowledge [Tharp]
10765 Soundness would seem to be an essential requirement of a proof procedure [Tharp]
10773 The Löwenheim-Skolem property is a limitation (e.g. can't say there are uncountably many reals) [Tharp]
10766 Logic is either for demonstration, or for characterizing structures [Tharp]
10767 Elementary logic is complete, but cannot capture mathematics [Tharp]
10769 Second-order logic isn't provable, but will express set-theory and classic problems [Tharp]
10775 The axiom of choice now seems acceptable and obvious (if it is meaningful) [Tharp]
10774 There are at least five unorthodox quantifiers that could be used [Tharp]
10776 The main quantifiers extend 'and' and 'or' to infinite domains [Tharp]
10777 Skolem mistakenly inferred that Cantor's conceptions were illusory [Tharp]