[FOM] What does Peano arithmetic have to offer?
Vaughan Pratt
pratt at cs.stanford.edu
Sat May 1 20:38:39 EDT 2010
On 5/1/2010 11:25 AM, Harvey Friedman wrote:
> The key divide is whether one is interested in
>
> #) *HAVING A SCIENTIFIC THEORY OF MATHEMATICAL PROOF*
>
> I offer up some facts about PA of a certain kind that show PA "in
> action".
>
> [THEOREMs 1-5]
While these very nice theorems strikingly address limitations of
elementary arguments about numbers, do they carry over for example to
algebraic number theory as described at
http://en.wikipedia.org/wiki/Algebraic_number_theory , or they specific
to the proof theory of the elementary theory of numbers?
While I have no problem with the idea that *any* theory of a domain such
as numbers (whether in a finitary or infinitary language) will have
limitations of one kind or another, as a sort of Heisenberg-Goedel
uncertainty principle for mathematics, what evidence is there that the
limitations of a suitably sharply defined formulation of algebraic
number theory will bear any resemblance to those of PA? This was the
point of item 3 in my response to Martin, about abstract algebra being
just as problematic for FOM as category theory.
(And Andrej Bauer and I would both like to see a representative sample
of answers to my question 1 in that response, about whether there's any
important difference beween the free monoid on one generator and the
initial successor-zero algebra.)
Perhaps Andreas Blass has some thoughts on this sort of thing since he's
closer to the elementary-vs-algebraic boundary than many on this list.
Vaughan
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