Nam Nguyen wrote: > > On 04/05/2013 10:07 AM, Frederick Williams wrote: > > Nam Nguyen wrote: > >> > >> On 26/04/2013 11:09 AM, Nam Nguyen wrote: > > > >>> On 2013-04-25, FredJeffries <firstname.lastname@example.org> wrote: > >>>> > >>>> 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, > > > > How does one reach an infinite regression? > > By claiming that the state of consistency of PA can be > proved _IN_ a _different formal system_ .
Your notion of infinite is very modest if does not go beyond two.
> > > >> and at that point > >> we still have to confront with the issue of mathematical relativity. > > > > It is not the case that either we go round in a circle or we regress > > forever. > > That's not a refute. Of course. > > (It's just an unsubstantiated claim).
And yet an obviously true one. Suppose the question of the consistency of PA is raised, a party to the discussion may say 'I accept that PA is consistent and I feel no need to prove it.' No circle, no regression.
-- When a true genius appears in the world, you may know him by this sign, that the dunces are all in confederacy against him. Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting