On the value group of a model of Peano Arithmetic

Abstract

Abstract. We investigate IP A -real closed fields, that is, real closed fields which admit an integer part whose non-negative cone is a model of Peano Arithmetic. We show that the value group of an IP A -real closed field is an exponential group in the residue field, and that the converse fails in general. As an application, we classify (up to isomorphism) value groups of countable recursively saturated exponential real closed fields. We exploit this characterization to construct countable exponential real closed fields which are not IP A -real closed fields.

Mini-Workshop: Surreal Numbers, Surreal Analysis, Hahn Fields and Derivations

“…This is closely related to recursively saturated RCF, see [34]. It was further observed in [72] that if K admits a left exponential, then the value group G of K is an exponential group in K. In [23] we show that if K admits an IPA (i.e K is an IPA real closed field), then K admits a left exponential, therefore the value group of K is an exponential group in K. Remark: There are plenty of DOAG that are not exponential groups in K. For example, take the Hahn group G = H γ∈Γ A γ where the archimedean components A γ are divisible but not all isomorphic and/or Γ is not a dense linear order without endpoints (say, a finite Γ). Alternatively, we could choose all archimedean components to be divisible and all isomorphic, say to C, and Γ to be a dense linear order without endpoints, but choose the residue field so that K not isomorphic to C. A class of not IPA real closed fields: Let k be any real closed subfield of R. Let G = {0} be any DOAG which is not an exponential group in k. Consider the Hahn field k((G)) and its subfield k(G) generated by k and {t g : g ∈ G}.…”

Models of true arithmetic are integer parts of models of real exponentation

“…Our result can be seen as a further variation of one direction of a well-known theorem of Shepherdson [21], according to which each model of open induction-Peano arithmetic with induction restricted to open (ie quantifier-free) formulas-is an IP of a real closed field. The result of Carl, D'Aquino and S. Kuhlmann [3] mentioned above implies that each model of I∆ 0 + EXP is an exponential IP of a real closed left-exponential field. Finally, by Krapp [13,Theorem 7.26] any model of PA is an IP of a real closed exponential field.…”

“…Since by F.-V. Kuhlmann, S. Kuhlmann and Shelah [14] such an exponential on K cannot exist, K does not admit an IP modelling PA. The methods of [18] were refined in Carl, D'Aquino and S. Kuhlmann [3] to the theorem that real closed fields with IPs modelling I∆ 0 + EXP always allow a weak form of exponentiation known as 'left-exponentiation', that is, an isomorphism from an additive group complement of the valuation ring for the We show that models of true arithmetic are always IPs of real closed fields that are very similar to the real numbers with exponentiation in a model theoretic sense: Namely, let us say-following the usual terminology-that a function E : K → K on a real closed field K is an exponential if it defines an isomorphism between (K, +, 0, <) and (K >0 , •, 1, <). The structure (K, +, •, 0, 1, <, E) is then called a real closed exponential field.…”

Order Types of Models of Fragments of Peano Arithmetic

“…In contrast, theories with the full induction scheme (or sufficiently large fragments of the induction scheme) require model theoretic methods to analyse their models. 4 We shall discuss and illustrate this notable difference in Section 6 with some elementary examples.…”