Contributed Talks LOGICLESS NONSTANDARD ANALYSIS: AN AXIOM SYSTEM

We give an axiomatic framework for getting full elementary extensions such as ultrapowers. From five axioms, all properties of a nonstandard extension are derived in a rather algebraic manner, without the use of any logical notions such as formulas or satisfaction. For example, when applied to the real number system, it provides a complete framework for working with hyperreals. This has possible pedagogical and expository applications as presented in, e.g., [2, 3], but we avoid use of special logical axioms such as the transfer axiom of [2, 3]. Terminology. An n-ary partial function f on a set X is a function whose domain is a subset of Xn and whose range is a subset of X (here n ∈ ω). For n> 0 and 1 ≤ k ≤ n, let PX,nk be the k-th n-ary projection function on X, i.e. the total function PX,nk: X n → X satisfying PX,nk (x1,..., xn) = xk. For a ∈ X, CX,na: Xn → X is the n-ary constant function taking the value a. We let f, g, h, etc, denote partial functions. The Axioms. Let A ⊂ B be non-empty sets and suppose that for each partia