Single Idea 13660

[catalogued under 5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic]

Full Idea

Tharp (1975) suggested that compactness, semantic effectiveness, and the Löwenheim-Skolem properties are consequences of features one would want a logic to have.

Gist of Idea

Maybe compactness, semantic effectiveness, and the Löwenheim-Skolem properties are desirable

Source

Stewart Shapiro (Foundations without Foundationalism [1991], 6.5)

Book Reference

Shapiro,Stewart: 'Foundations without Foundationalism' [OUP 1991], p.159


A Reaction

I like this proposal, though Shapiro is strongly against. We keep extending our logic so that we can prove new things, but why should we assume that we can prove everything? That's just what Gödel suggests that we should give up on.

Related Ideas

Idea 13658 Downward Löwenheim-Skolem: each satisfiable countable set always has countable models [Shapiro]

Idea 13659 Upward Löwenheim-Skolem: each infinite model has infinite models of all sizes [Shapiro]

Idea 13661 A language is 'semantically effective' if its logical truths are recursively enumerable [Shapiro]