Editing Reverse mathematics (section)

==&omega;-models and &beta;-models==
{{main|Beta-model}}

The &omega; in &omega;-model stands for the set of non-negative integers (or finite ordinals). An &omega;-model is a model for a fragment of second-order arithmetic  whose first-order part is the standard model of Peano arithmetic,<ref name="Simpson2009" /> but whose second-order part may be non-standard. More precisely, an &omega;-model is given by a choice <math>S\subseteq\mathcal P(\omega)</math> of subsets of <math>\omega</math>. The first-order variables are interpreted in the usual way as elements of <math>\omega</math>, and <math>+</math>, <math>\times</math> have their usual meanings, while second-order variables are interpreted as elements of <math>S</math>.  There is a standard &omega;-model where one just takes <math>S</math> to consist of all subsets of the integers. However, there are also other &omega;-models; for example, RCA<sub>0</sub> has a minimal &omega;-model where <math>S</math> consists of the recursive subsets of <math>\omega</math>.

A &beta;-model is an &omega; model that agrees with the standard &omega;-model on truth of <math>\Pi^1_1</math> and <math>\Sigma^1_1</math> sentences (with parameters).

Non-&omega; models are also useful, especially in the proofs of conservation theorems.