[FOM] comments from Lou van den Dries

[Carl G. Jockusch]

Lou van den Dries sent me the following response
to "Question on Decidability of Exponentiation and Bounded Sine"
(Dmytro Taranovsky).  Although FOM does not normally accept
postings from non-members, I think that in this case FOMers
might really want to see this authoritative reply.

Carl Jockusch


>From vddries at charisma.math.uiuc.edu  Thu Dec 18 14:16:38 2003
X-Authentication-Warning: u78.math.uiuc.edu: vddries owned process doing -bs
Date: Thu, 18 Dec 2003 14:16:36 -0600 (CST)
From: Lou van den Dries <vddries at math.uiuc.edu>
To: "Carl G. Jockusch" <jockusch at math.uiuc.edu>
Subject: Re: question on FOM

Carl,

Let me give my thoughts on this. If you like you can forward it to
FOM of which I am not a member.

First, the message claims that the theory of the real exponential field is
decidable "provided no surprising identity holds." That's new to me,
unless what is meant is "if Schanuel's Conjecture holds for
real numbers" (which to me sounds like a much stronger assumption.)

I haven't checked this, but it seems very plausible to me
that the analogue  for the real exponential field and sine restricted to
[-1,1] holds: its theory should be decidable if the full Schanuel
Conjecture holds (i.e. for complex, not just real numbers). My guess is
that the Wilkie-Macintyre proof of "decidability for the real exponential
field modulo the real Schanuel Conjecture" simply goes through for this
expansion, if you graft them onto the proof of model-completeness for this
expansion in my paper with Chris Miller, Israel J. of Math. 85 (1994),
19-56, and use the full Schanuel Conjecture.
-Lou-