Re: Fitch's paradox and OWA
From: Nam Nguyen <namducnguyen_at_shaw.ca>
Date: Thu, 31 Dec 2009 17:03:24 -0700
Message-ID: <jZa%m.277$Mv3.101_at_newsfe05.iad>
>
> I have no disagreement with the point about finiteness, but I
> don't see how that point leads to saying that a theory is
> the same thing as a definition. That is rather tantamount to
> saying that theories are all there are, and that's just not
> true. There are things such as computational models,
> for examples. It seems entirely appropriate to me to
> use a computational model as the definition of something,
> which is why I gave a computational model of the naturals
> as a definition.
Date: Thu, 31 Dec 2009 17:03:24 -0700
Message-ID: <jZa%m.277$Mv3.101_at_newsfe05.iad>
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. Received on Fri Jan 01 2010 - 01:03:24 CET