Re: Fitch's paradox and OWA

On Dec 31, 4:03 pm, Nam Nguyen <namducngu..._at_shaw.ca> wrote:
> Marshall wrote:
> > On Dec 31, 1:08 pm, Barb Knox <Barb..._at_LivingHistory.co.uk> wrote:
> >>  Marshall <marshall.spi..._at_gmail.com> wrote:
> >>> On Dec 30, 8:16 pm, Barb Knox <s..._at_sig.below> wrote:
> >>>> Marshall <marshall.spi..._at_gmail.com> wrote:
> >>>> By the nature of the construction of predicate logic, every arithmetic
> >>>> formula must be either true or false in the standard model of the
> >>>> natural numbers.
> >>>> But, we have no satisfactory way to fully characterise that standard
> >>>> model! We all think we know what the natural numbers are, but Goedel
> >>>> showed that there is no first-order way to define them, and I don't know
> >>>> of *any* purely formal (i.e., syntactic) way to do do.
> >>> I was more under the impression that Goedel showed there
> >>> was no complete finite theory of them, rather than no
> >>> way to define them. Are you saying those are equivalent?
> >> Yes, in this context.  Since we are finite beings we need to use finite
> >> systems.
>
> > I have no disagreement with the point about finiteness, but I
> > don't see how that point leads to saying that a theory is
> > the same thing as a definition. That is rather tantamount to
> > saying that theories are all there are, and that's just not
> > true. There are things such as computational models,
> > for examples. It seems entirely appropriate to me to
> > use a computational model as the definition of something,
> > which is why I gave a computational model of the naturals
> > as a definition.
>
> You seemed to have confused between the FOL definition of models of formal
> systems in general and constructing a _specific_ model _candidate_. In defining
> the naturals, say, from computational model ... or whatever, you're just
> defining what the naturals be. It's still your onerous to prove/demonstrate
> this definition of the naturals would meet the definition of a model for,
> say Q, PA, .... So far, have you or any human beings successfully demonstrated
> so, without being circular? Of course not.

Showing that the axioms of PA are true in my definition is straightforward, using only structural induction, which in the case of my two-constructor definition is simply case analysis of the two cases.

Try it; it's fun!

Marshall Received on Fri Jan 01 2010 - 01:15:38 CET