Re: Godel's theorem is invalid?

sradhakr says...

>> > > But there is a big difference between
>> > >
>> > > (1) For all n, we can prove P(n).
>> > >
>> > > and
>> > >
>> > > (2) We can prove "for all n, P(n)"

>In fact we can *know* that (1) is true, and hence a meaningful
>assertion, if and only if:
>
> (3) We can prove P(n) for an *arbitrary* natural number n, where
>n is left in symbolic form.

Yes, you're right. The only way to know that (1) is true is if
something like (2) holds. However, that wasn't the point. The
point is that (1) can be *true* without our *knowing* that it
is true.

Imagine that you have some formula Phi(n). For example, n might
be the statement "2*n is a number that can be expressed as the
sum of two prime numbers". You can check to see that Phi(0) is
true, you can check to see that Phi(1) is true. For any n, you
can check whether Phi(n) is true. Yet there may be no known way to
check whether "forall n, Phi(n)" is true.

>Hence there *should* really be no distinction between (1) and (3), a
>distinction which classical first-order logic dubiously tries to
>maintain.

Why do you say that there should be no distinction?

--
Daryl McCullough
Ithaca, NY

.

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