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From: Nam Nguyen <namducnguyen@shaw.ca>
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Subject: Re: Fitch's paradox and OWA
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Date: Thu, 31 Dec 2009 17:03:24 -0700
Xref:  news.cambrium.nl sci.logic:158576 comp.databases.theory:38083

Marshall wrote:
> On Dec 31, 1:08 pm, Barb Knox <Barb...@LivingHistory.co.uk> wrote:
>>  Marshall <marshall.spi...@gmail.com> wrote:
>>> On Dec 30, 8:16 pm, Barb Knox <s...@sig.below> wrote:
>>>> Marshall <marshall.spi...@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.