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 selfreference? (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 selfreference, 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: selfreference. (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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