On 26/04/2013 11:09 AM, Nam Nguyen wrote: > On 26/04/2013 10:37 AM, Herman Rubin wrote: >> On 2013-04-25, FredJeffries <email@example.com> wrote: >>> On Apr 25, 8:25 am, Alan Smaill <sma...@SPAMinf.ed.ac.uk> wrote: >>>> Newberry <newberr...@gmail.com> writes: >>>>> Torkel Franzen argues that all the axioms of ZFC are manifestly true >>>>> the logic apparatus is truth preserving therefore all is good and the >>>>> system is consistent. >> >>>> Really?? >> >>>> Where did he make this claim? >> >> >>> In "The Popular Impact of Gödel's Incompleteness Theorem" >> >>> http://www.ams.org/notices/200604/fea-franzen.pdf >> >>> he says: >> >>> "we can easily, indeed trivially, prove PA consistent using >>> reasoning of a kind that mathematicians otherwise >>> use without qualms in proving theorems of >>> arithmetic. Basically, this easy consistency proof observes >>> that all theorems of PA are derived by valid >>> logical reasoning from basic principles true of the >>> natural numbers, so no contradiction is derivable in PA" >> >> Mathematicians are willing to assume PA is consistent. > > Agree. For the record I've always assumed PA is consistent, > until of course if one day ones present a proof _IN_ PA > of the form (F /\ ~F). > > But assumption is _not_ assertion, wouldn't you agree? > > (That's all I'm asking.) > >> The >> inconsistency of PA would mean that the basic principles >> of the natural numbers are inconsistent. >> >> I recommend that the discussion of the natural numbers from >> the basic principles be taught very early, and addition, etc., >> be derived from them. They LOOK obvious. But if it is >> consistent, we know we cannot prove it. >> >> Now PA has been proved consistent in ZF or NBG, but then that >> brings the consistency of axioms for set theory.
Exactly right. And exactly my point.
Somewhere, somehow, a circularity or an infinite regression of _mathematical knowledge_ will be reached, and at that point we still have to confront with the issue of mathematical relativity.
There's really no escape to it, I'm afraid from what I could gather.
-- ---------------------------------------------------- There is no remainder in the mathematics of infinity.