[FOM] What does Peano arithmetic have to offer?

[Vaughan Pratt]

Finally able to get back to this.

On 5/3/2010 7:09 PM, Harvey Friedman wrote:
 > > With regard to "normal mathematics", PA has enormous expressive power.
 > > Of course the language has very few primitives, so that this
 > > expressive power, when put into action, is rather ugly. But one can
 > > design what are called "strong conservative extensions of PA", where
 > > one adds lots of new primitives without changing the power, so that
 > > the expressive power, when put into action, is rather pretty.

No disagreement there.

 > > If there is a serious part of algebraic number theory for which it is
 > > not obvious how to formulate the statements in PA, and there is no
 > > gratuitous generality involved, then that would make it far more
 > > likely that an incompetent algebraic number theorist like me could
 > > come up with statements of the kind in 
http://www.cs.nyu.edu/pipermail/fom/2010-May/014703.html
 > >    which are in the spirit and culture of algebraic number theory.

No disagreement there either.  What I'm questioning is whether the 
particular limitations of PA are of intrinsic interest number 
theoretically or are specific to the logical framework within which one 
chooses to do number theory.  The disagreement may lie in what we take 
as the scope of number theory.  You may feel that any framework that 
can't quantify over numbers couldn't be a framework for number theory. 
More below on this.

 > > As for literally carrying over statements in ??? to algebraic number
 > > theory, this is possible, but more likely is that they have to be
 > > adapted.

If PA doesn't quantify over rings and ??? doesn't quantify over numbers 
then they may constitute incomparable perspectives on number theory.

 > > "appeal only to logicians and those number theorists still working
 > > inside PA" makes no sense to me. There is such a thing as "general
 > > intellectual interest".
 > >
 > > Would you say:
 > >
 > > Beethoven's Fifth Symphony is the sort of thing that would appeal only
 > > to musicians and those number theorists who have children in 
orchestras?

Good point.  Replace "logicians" by "those logically inclined," which 
hopefully also addresses your point about "general intellectual 
interest."  (I'm assuming Beethoven's 5th appeals only to those 
musically inclined, and that those with no mathematical inclination will 
be unlikely to find Theorems 4-5 intellectually interesting.)  I had a 
harder time with your second analogy (number theorists working in PA vs. 
number theorists with children in orchestras).

 > > What incomparable logical frameworks? Aren't all of the usual
 > > reasonable logical frameworks for doing classical mathematics
 > > comparable?
 > > Specifically, what reasonable logical framework do you have in mind
 > > which will not be sensitive to the special status of the statements 
in http://www.cs.nyu.edu/pipermail/fom/2010-May/014703.html

Well, for example any framework formulated for the purpose of 
formalizing Hilbert's 9th and/or 12th problems axiomatically.  This is 
still number theory, but it needn't provide for quantifying over 
numbers, and I don't see right off how one would translate your Theorems 
1-3 into such a framework, a prerequisite for anything to do with 4-5. 
Presumably this is where Martin's point about ideals comes in, but are 
Theorems 4-5 still meaningful when numbers are exhibited as ideals in an 
algebraic number field?

Vaughan