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Single Idea 10773

[filed under theme 5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems ]

Full Idea

The Löwenheim-Skolem property seems to be undesirable, in that it states a limitation concerning the distinctions the logic is capable of making, such as saying there are uncountably many reals ('Skolem's Paradox').

Gist of Idea

The Löwenheim-Skolem property is a limitation (e.g. can't say there are uncountably many reals)

Source

Leslie H. Tharp (Which Logic is the Right Logic? [1975], §2)

Book Ref

'Philosophy of Logic: an anthology', ed/tr. Jacquette,Dale [Blackwell 2002], p.39

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]