### Single Idea 13658

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

Full Idea

A language has the Downward Löwenheim-Skolem property if each satisfiable countable set of sentences has a model whose domain is at most countable.

Gist of Idea

Downward Löwenheim-Skolem: each satisfiable countable set always has countable models

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 employ an infinite model to represent a fact about a countable set.

Related Idea

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