
Re: Torkel Franzen argues
Posted:
May 8, 2013 9:28 AM


Nam Nguyen wrote: > > On 05/05/2013 8:45 AM, Frederick Williams wrote: > > 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 20130425, FredJeffries <fredjeffries@gmail.com> 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. > > That does _not_ mean there be only two, actually. > > > >>> > >>>> 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. > > The circularity rests with the argument on the _actual and objective_ > state of consistency of PA, _not_ on the _wishful and subjective_ > "acceptance" of anything.
Mathematicians (like the rest of humanity) are forever accepting things. It is no big deal.
 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

