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Re: Essential Consistency
Posted:
Mar 21, 2012 3:12 PM
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On Mar 21, 5:55 pm, MoeBlee <modem...@gmail.com> wrote:
> On Mar 21, 12:57 am, Zuhair <zaljo...@gmail.com> wrote:
>
> > On Mar 20, 10:29 pm, MoeBlee <modem...@gmail.com> wrote:
>
> > > On Mar 19, 3:50 pm, Zuhair <zaljo...@gmail.com> wrote:
>
> > > > concrete "standard" natural number n.
>
> > > What is the definition of 'concrete'?
> > Standard.
>
> Then you're saying:
>
> standard "standard" natural number
>
> All you need to say is:
>
> natural number
>
> A proof in a system S is a certain kind of finite sequence of
> formulas. And a set (a sequence is a set) is finite if and only if the
> set is 1-1 with a natural number. Natural number. Period. Nothing
> about nonstandard natural numbers.
>
> And, as I also mentioned, an object is a nonstandard natural only in
> context of some particular ordering of a set that is the universe for
> a model. An object is not just a nonstandard natural number or not a
> nonstandard natural number, but rather, the object is or is not a
> nonstandard natural number PER some given set AND ordering on that
> set.
>
> In this context, there is no gain, except in confusion, in worrying
> about nonstandard natural numbers.
>
> MoeBlee
The idea is that proofs of non standard natural number length are not
real proofs, they are actually not proofs at all, they are only seen
by the theory in question to be proofs, but externally they are
definitely not proofs. I agree it is confusing. I myself need to look
more into that.
Zuhair
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