# Re: Fitch's paradox and OWA

Date: Thu, 31 Dec 2009 16:15:38 -0800 (PST)

Message-ID: <841a1d32-bc4d-46a7-8e35-6037d54ff809_at_h9g2000yqa.googlegroups.com>

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