Re: Fitch's paradox and OWA
From: Nam Nguyen <namducnguyen_at_shaw.ca>
Date: Thu, 31 Dec 2009 18:31:33 -0700
Message-ID: <Yfc%m.288$Mv3.161_at_newsfe05.iad>
>
> Showing that the axioms of PA are true in my definition is
> straightforward, using only structural induction,
Date: Thu, 31 Dec 2009 18:31:33 -0700
Message-ID: <Yfc%m.288$Mv3.161_at_newsfe05.iad>
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,
> which in the
> case of my two-constructor definition is simply case
> analysis of the two cases.
>
> Try it; it's fun!
I'm sure there are a lot of fun things in life but here the interesting thing would be demonstrating your definitions meet the FOL definition of models of formal systems. You haven't tried it; so what you've claimed here isn't interesting!
>
>
> Marshall
Received on Fri Jan 01 2010 - 02:31:33 CET