Real closed fields and models of Peano arithmetic

Abstract

Shepherdson [14] showed that for a discrete ordered ring I, I is a model of I Open iff I is an integer part of a real closed ordered field. In this paper, we consider integer parts satisfying PA. We show that if a real closed ordered field R has an integer part I that is a nonstandard model of PA (or even IΣ4), then R must be recursively saturated. In particular, the real closure of I, RC (I), is recursively saturated. We also show that if R is a countable recursively saturated real closed ordered field, then there is an integer part I such that R = RC(I) and I is a nonstandard model of PA.

Real closed exponential fields

“…(b) In s · t, the coefficient of g is the sum of the products a g ′ b g ′′ , where g = g ′ · g ′′ . 4. k((G)) is ordered anti-lexicographically by setting s > 0 if ag > 0 where g =: max(Supp(s)).…”

“…Recursive saturation has already come up in connection with integer parts. In [4], it was shown that a countable real closed field has an integer part satisfying Peano arithmetic if and only if the real closed field is Archimedean or recursively saturated.…”

A construction of real closed fields

“…In § , we study real closed fields with an extra structure. In , they studied real closed fields R and showed that R is recursively saturated if and only if R has an integer part satisfying . Here, we introduce the notion that a function on R is well‐approximated and prove the following.…”

Uncountable Real Closed Fields With Pa Integer Parts

“…For every x ∈ R(M) we can find unique m ∈ M and r ∈ R(M) such that x = 2 m r and 1 ≤ r < 2. Note that • if 1 ≤ r, s < 2, then v(2 m r) = v(2 n s) if and only if m ≡ n mod Z • v(2 m r · 2 n s) = v(2 m+n rs); • v(2 m r) > 0 if and only if m < Z.4 Real Closures of ω 1 -like Models D'Aquino, Knight and Starchenko conclude in[3] that if M and N are countable models of PA with the same standard system, then their real closures are isomorphic. In contrast we prove that this fails badly for ω 1 -like models.…”

“…One interesting consequence of [3] is that two countable nonstandard models of PA have isomorphic real closures if and only if they have the same standard systems. Can this be generalized to ω 1 -like models?…”