On 2/28/2014 3:07 AM, Virgil wrote: > In article <firstname.lastname@example.org>, > email@example.com wrote: >> On Thursday, 27 February 2014 22:52:54 UTC+1, fom wrote: >>>> The question was whether Peano defines the natural numbers. He fails. >>> Why do you say that? >> I say that because it is widely assumed that Peano defined the natural >> numbers. > > They were around long before Peano, but he axiomatized them successfully > to the point that we still use his axiomatizaton, at least those of us > outside of WMytheology do. > >> People assume the natural numbers and find that Peano is rigth. But >> the other way dos not work. Assuming the Peano axioms does not yield N. > > It produces something order-isomorphic to |N every time!.
Then WM is right.
Given your statement, the correct foundational reduction is to study order theory and to not make foundational claims for arithmetic.
You may use the Peano axioms without concern for foundations. But, then you are not really using them at all since they are mere representations of your practice.