Full Idea
The Löwenheim-Skolem theorems (which apply to first-order formal theories) show that any theory with an infinite model has a model of every infinite cardinality.
Gist of Idea
Any theory with an infinite model has a model of every infinite cardinality
Source
Stewart Shapiro (Philosophy of Mathematics [1997], 4.8)
Book Reference
Shapiro,Stewart: 'Philosophy of Mathematics:structure and ontology' [OUP 1997], p.132
A Reaction
This aspect of the theorems is the Skolem Paradox. Shapiro argues that in first-order this infinity of models for arithmetic must be accepted, but he defends second-order model theory, where 'standard' models can be selected.