Re: Fitch's paradox and OWA
Date: Thu, 31 Dec 2009 18:12:19 -0800 (PST)
Message-ID: <8797bda2-6276-40e8-82cd-e2820ee9df51_at_j24g2000yqa.googlegroups.com>
On Dec 31, 5:31 pm, Nam Nguyen <namducngu..._at_shaw.ca> wrote:
> Marshall wrote:
> > 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,
>
> It might be straightforward to you and you might call it "Cheney induction"
> instead of "structural induction" but it's irrelevant and the question is
> the same: how could you demonstrate that your definition would meet the FOL
> standard definition of model of a formal system? Did you already make that
> presentation in the thread and I simply missed it?
What sort of thing would you accept as an answer? What difficulties do you foresee?
If you are convinced it is impossible and that nothing will satisfy you, I'd rather not waste my time. On the other hand if you have a specific idea as to what a correct answer would look like, I might be able to satisfy you.
Marshall Received on Fri Jan 01 2010 - 03:12:19 CET