Hyperreals and Their Applications

NSA can be introduced in multiple ways. Instead of choosing one option, these notes include three introductions. Section 1 is best-suited for those who are familiar with logic, or who want to get a flavor of model theory. Section 2 focuses on some common ingredients of various axiomatic approaches to NSA, including the star-map and the Transfer principle. Section 3 explains the ultrapower construction of the hyperreals; it also includes an explanation of the notion of a free ultrafilter. 1 Existence proofs of non-standard models 1.1 Non-standard models of arithmetic The second-order axioms for arithmetic are categoric: all models are isomorphic to the intended model ⟨N, 0,+1⟩. Dedekind was the first to prove this [Dedekind, 1888b]; his ‘rules’ for arithmetic were turned into axioms a year later by Peano, giving rise to what we now call “Peano Arithmetic” (PA) [Peano, 1889]. The first-order axioms for arithmetic are non-categoric: there exist nonstandard models ⟨∗N, ∗0, ∗+∗1⟩ that are not isomorphic to ⟨N, 0,+1⟩. Skolem proved this based on the Compactness property of first-order logic (FOL) [Skolem, 1934]. With the Löwenheim-Skolem theorem, it can be proven that there exist models of any cardinality. ∗N contains finite numbers as well as infinite numbers. We now call ∗N a set of hypernatural numbers. For a discussion of the order-type of countable non-standard models of arithmetic, see e.g. [Boolos et al., 2007, Chapter 25, p. 302–318] and [McGee, 2002]. More advanced topics can be found in this book: [Kossak and Schmerl, 2006]. 1.2 Non-standard models of real closed fields The second-order axioms for the ordered field of real numbers are categoric: all models are isomorphic to the intended model ⟨R,+,×,≤⟩. A “real closed field” (RCF) is a field which has the same first-order properties as R. Robinson realized that Skolem’s existence proof of nonstandard models of arithmetic could be applied RCFs too. He thereby founded the field of non-standard analysis (NSA) [Robinson, 1966]. The axioms for RCFs (always in FOL) are non-categoric: there exist non-standard models ⟨∗R, ∗+, ∗×, ∗ ≤⟩ that are not isomorphic to ⟨R,+,×,≤⟩. With the Löwenheim-Skolem theorem it can be proven that there exist models of any cardinality; in particular, there are countable models (cf. Skolem ‘paradox’ [Skolem, 1922]).