[FOM] What does Peano arithmetic have to offer?

[Vaughan Pratt]

(I have to board a plane to the east coast in a few hours which is why 
I'm replying to Tim's easier message first---will address Harvey's 
longer list of points in due course.)

On 5/3/2010 7:29 PM, Timothy Y. Chow wrote:
> Harvey Friedman wrote:
>> 6. I am sure that we agree that there are incomparably more consumers
>> of elementary numerical mathematics than algebraic number theory. So I
>> don't quite see why you are focused on algebraic number theory.
>
> I wonder if this is a case of "moving the goalposts."
< [...]
>
> Algebraic number theory is currently considered to be real mathematics.
> Once f.o.m. is shown to intrude into algebraic number theory, that status
> will undoubtedly be questioned.

No disagreement there.  What I'm questioning is whether *specific* 
results in one first order domain obtain in another.  I'm more familiar 
with relation algebra than algebraic number theory which is why I had 
specific examples only for that one.

The particular example I cited was the well known first order 
ineffability of transitive closure, which applies to first order 
theories for which the relations are in the language.  A first order 
theory where the relations are instead in the domain can capture 
transitive closure precisely.  I was quite surprised by this when I 
first noticed it in the context of dynamic logic and related logics of 
action.  I would expect algebraic number theory to have analogous examples.

Regarding the latter, Martin's point about ideals of rings brings in 
duality: ideals dualize rings and act like numbers (Kummer christened 
them that because he imagined them as "ideal numbers" which attracted 
Dedekind's interest).  Once you have ideals as well as rings you have 
both sides of the duality and now you're in the position of a physicist 
who has both position and momentum.  Likewise a filter or an ideal in a 
Boolean algebra acts like an element of a set---the ultrafilters of a 
finite Boolean algebra are in bijection with the set of its atoms. 
Boolean algebra can be thought of as one form of algebraic set theory 
(Boole's sense of the term as distinct from Steve Awodey's).

Vaughan