Non-standard model

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In model theory, a discipline within mathematical logic, a non-standard model is a model of a theory that is not isomorphic to the intended model (or standard model).^[1]

Existence

If the intended model is infinite and the language is first-order, then the Löwenheim–Skolem theorems guarantee the existence of non-standard models. The non-standard models can be chosen as elementary extensions or elementary substructures of the intended model.

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v t e Mathematical logic
General	Formal language Formation rule Formal system Deductive system Formal proof Formal semantics Well-formed formula Set Element Class Classical logic Axiom Natural deduction Rule of inference Relation Theorem Logical consequence Axiomatic system Type theory Symbol Syntax Theory
Traditional logic	Proposition Inference Argument Validity Cogency Syllogism Square of opposition Venn diagram
Propositional calculus Boolean logic	Boolean functions Propositional calculus Propositional formula Logical connectives Truth tables
Predicate logic	First-order Quantifiers Predicate Second-order Monadic predicate calculus
Naive set theory	Set Empty set Enumeration Extensionality Finite set Infinite set Subset Power set Countable set Uncountable set Recursive set Domain Range Map Function Relation Ordered pair
Set theory	Foundations of mathematics Zermelo–Fraenkel set theory Axiom of choice General set theory Kripke–Platek set theory Von Neumann–Bernays–Gödel set theory Morse–Kelley set theory Tarski–Grothendieck set theory
Model theory	Model Interpretation Non-standard model Finite model theory Truth value Validity
Proof theory	Formal proof Deductive system Formal system Theorem Logical consequence Rule of inference Syntax
Computability theory	Recursion Recursive set Recursively enumerable set Decision problem Church–Turing thesis Computable function Primitive recursive function

↑ Roman Kossak, 2004 Nonstandard Models of Arithmetic and Set Theory American Mathematical Soc.