The ω in ω-model stands for the set of non-negative integers (or finite ordinals). An ω-model is a model for a fragment of second-order arithmetic whose first-order part is the standard model of Peano arithmetic, but whose second-order part may be non-standard. More precisely, an ω-model is given by a choice "S"⊆2ω of subsets of ω. The first-order variables are interpreted in the usual way as elements of ω, and +, × have their usual meanings, while second-order variables are interpreted as elements of "S". There is a standard ω model where one just takes "S" to consist of all subsets of the integers. However, there are also other ω-models; for example, RCA0 has a minimal ω-model where "S" consists of the recursive subsets of ω.