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. >
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