On Tue, 31 Dec 2002, Sarad AV wrote:
> Does a paradox ever help in understanding any thing?
Yes, it can demonstrate that you aren't asking the right questions within
the correct context.
> We define a paradox on a base of rules we want to
> prove.
No, a paradox is two things we accept that imply two contradictory
answers.
> 2.G�del asks for the program and the circuit design of
> the UTM. The program may be complicated, but it can
> only be finitely long.
Wrong, there is -nothing- that says the program must have finite length
-or- halt.
We -assume- it is so (which relates to the a priori assumption of PM being
complete in order to prove it is undecidable - as opposed to incomplete,
which is not the same thing at all).
> The question is it in a formal system,since we don't
> have paradoexes in a formal system.
Godel has demonstrated that this is untrue, that in fact you -can- have
-undecidable- statements in a formal system. The flaw in our assumption is
that we can reduce everything to a 'T' or a 'F'.
* note that Godel uses 'consistent' where we use 'complete' *
Proposition XI:
If c be a given recursive, consistent class of formulae, then the
propositional formula which states that c is consistent is not c-provable;
in particular, the consistency of P is unprovable in P, it being assumed
that P is consistent (if not, then of course, every statement is
provable).
...further clarification (original italics/bold denoted by -*-)...
It may be noted is also constructive, ie it permits, if a -proof- from c
is produced for w, the effective derivation from c of a contradiction. The
whole proof of Proposition XI can also be carried over word for word to
the axiom-system of set theory M, and to that of classical mathematics A,
and here too it yields the result that there is no consistency proof for M
or of A which could be formalized in M or A respectively, it being assumed
that M and A are consistent. It must be expressly noted that Proposition
XI (and the corresponding results for M and A) represent no contradiction
of the formalistic standpoint of Hilbert. For this standpoint presupposes
only the existance of a consistency proof effected by finite means, and
there might conceivably be finite proofs which -cannot- be stated in P (or
in M and A).
In other words, "There are some proofs that can't be written".
--
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