Full Idea
Downward Löwenheim-Skolem: a finite or denumerable set of first-order formulas that is satisfied by a model whose domain is infinite is satisfied in a model whose domain is the natural numbers
Gist of Idea
Downward Löwenheim-Skolem: if there's an infinite model, there is a countable model
Source
Stewart Shapiro (Higher-Order Logic [2001], 2.1)
Book Reference
'Blackwell Guide to Philosophical Logic', ed/tr. Goble,Lou [Blackwell 2001], p.34