5.4 Nonstandard Models
Nonstandard models arise when a first order theory admits models that satisfy all its axioms but differ from the intended or standard interpretation, and their existence follows naturally from the compactness theorem and the Lowenheim Skolem theorems. Definition 5.22 (Standard Model) A standard model is a structure that is intended to represent the usual interpretation of a theory. For example, the standard model of arithmetic is: $$ \mathbb{N} = {0,1,2,\dots}...