Uncountable Real Closed Fields With Pa Integer Parts

Abstract

D'Aquino, Knight and Starchenko classified the countable real closed fields with integer parts that are nonstandard models of Peano Arithmetic. We rule out some possibilities for extending their results to the uncountable and study real closures of ω 1 -like models of PA.

Incomparable ω1‐like models of set theory

“…For models of ZF, this is equivalent to saying that M is uncountable, but every object has only countably many ‐predecessors, that is, is countable; for models of , it is also equivalent to asserting that the ordinals Ord M are ω 1 ‐like as a linear order. The ω 1 ‐like models constitute a gateway from the realm of countable models to the uncountable, sharing and blending many of the features of both kinds of models, and they have been extensively studied both in the case of models of arithmetic and of models of set theory .…”

63 Years of the MacDowell-Specker Theorem

“…Such models are of independent interest not only in the model theory of arithmetic. For examples, see [17,28,43].…”

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

“…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.…”