Merlin Carl: Real Closed Exponential Fields and Models of Peano Arithmetic

Monday, June 25, 2012, 16.30-18.00
Romm 1.007, Mathematical Institute, University of Bonn

Speaker: Merlin Carl (University of Konstanz)

Title: Real Closed Exponential Fields and Models of Peano Arithmetic

Abstract:

A real closed field (RCF) is a first-order structure which is elementary equivalent to the real numbers. An integer part (IP) of a real closed field K is a discretely ordered subring R with smallest element 1 such that, for every x in K, there is a unique k in R such that k≤x<k+1. By a theorem of Shepherdson, integer parts of real closed fields are exactly the models of open induction, which is Peano Arithmetic with induction restricted to quantifier-free formulas. We consider the question when an RCF can have an IP that is a model of full PA. By a recent result of D’Aquino, Knight and Starchenko, this is the case for a countable real closed field K iff K is recursively saturated. We will demonstrate that this characterization fails in the uncountable case. In particular, we will prove that no field of formal power series can admit an IP modeling PA.

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