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Single Idea 13659

[from 'Foundations without Foundationalism' by Stewart Shapiro, in 5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems ]

Full Idea

A language has the Upward Löwenheim-Skolem property if for each set of sentences whose model has an infinite domain, then it has a model at least as big as each infinite cardinal.

Gist of Idea

Upward Löwenheim-Skolem: each infinite model has infinite models of all sizes

Source

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

Book Reference

Shapiro,Stewart: 'Foundations without Foundationalism' [OUP 1991], p.158


A Reaction

This means you can't have a countable model to represent a fact about infinite sets.

Related Idea

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