Pat, (01)
Lainaus Pat Hayes <phayes@xxxxxxx>: (02)
> At 11:28 PM +0200 1/30/08, Avril Styrman wrote:
> >
> >We do not need Gödel numbering to understand that 1+1=2
> >cannot be proved. It is so deeply tied with out cognitive
> >capabilities, that without understanding that 1+1=2, we could
> >not understand anything. If we try to prove that 1+1=2, we
> >have to use the same cognitive capabilities in the proof,
> >that we used when we understood that 1+1=2.
>
> Nonsense.
>
> > This is the idea
> >of Gödel numbering: the things that are to be proved have to
> >be used in their own proof.
>
> Apparently you know very little about formal arithmetic or Goedel's
> theorem.
>
> a. 1+1=2 is provable in any formal arithmetic. (03)
So, prove it Pat! Prove it here. After that I'll prove that you
used the very ability to distinguish between I and II. (04)
> b. "the things that are to be proved have to be
> used in their own proof" is not the idea of
> Goedel numbering (05)
What else is the idea in the end, than to prove that
proving X requires self-reference? (06)
This is copied from Wikipedia: (07)
Gödel specifically used this scheme at two levels: first,
to encode sequences of symbols representing formulas, and
second, to encode sequences of formulas representing proofs.
This allowed him to show a correspondence between
statements about natural numbers and statements about the
provability of theorems about natural numbers, the key
observation of the proof. (08)
If the key idea is not self-reference, then what is it? (09)
Suppose that I'm totally wrong. This does not change the fact
that proving 1+1=2 requires understanding the difference of
I and II. It does not change anything if Gödel took a longer
road and included conventions used by modern mathematicians.
It is still the same old story: self-reference. (010)
If you 'prove' something that is as obvious as can be like
1+1=2, you only prove that you yourself feel more comfortable
after having used some conventions. Tell me, do you need to
prove that 1+1=2? Why do you? After you have proved it, do you
feel more certain about 1+1=2? (011)
Avril (012)
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