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]