Re: Godel proved maths inconsistent not incompleteness theorem

On Fri, 18 Apr 2008 04:22:01 -0700 (PDT), Charlie-Boo
<shymathguy@xxxxxxxxx> said:
On Apr 17, 5:37 pm, Chris Menzel <cmen...@xxxxxxxxxxxxxxxxxxxx>
wrote:
On Thu, 17 Apr 2008 13:50:04 -0700 (PDT), Charlie-Boo
<shymath...@xxxxxxxxx> said:

...
One CBL formal proof of the unsolvability of the Halting Problem
begins:

1. HALT(I,J)*  Assumption: Assume that the Halting Problem is solvable
i.e. the Halting Set is recursive.
2. P(I)* => P(I)*  NOT Rule: The complement of a recursive set is
recursive.

You mean P(I)* => ~P(I)*, right?

Yes. Very good.

3. ~HALT(I,J)*  2,1: The Nonhalting Set is recursive
etc

Your proof is a joke. Your "proofs" are nothing but trivial
propositional logic at best.

Which step of the following proof is invalid and why?

1. Assume the HP is solvable i.e. the halting set is recursive.
2. So he nonhalting set is recursive.
3. So he nonhalting set is r.e.
4. The set of programs that halt no is r.e.
5. So the set of programs that halt no or don't halt is r.e.
6. Since no program halts yes and no, (5) is the set of programs that
don't halt yes.
7. The set of programs that don't halt yes is not r.e. by direct
diagonalization.
8. Contradiction.
qed

Well, in one sense, nothing at all. The problem is, you think that that
proof, in itself, formalized in CBL, constitutes a complete,
self-contained, systematic proof of the unsolvability of the halting
problem. It does not. *Properly presented*, it would simply be the
final step in a lengthy process in which all of the component concepts
are defined in terms of axiomatized, or easily axiomatizable,
primitives. That's what I've referred to as the underlying
"infrastructure" such proofs require. And it's what you find laid out,
in painstaking detail, in the early chapters of Peter Smith's Gödel
book, B&J's Computability and Logic, Torkel Franzen's Inexhaustibility,
and countless others, and your proof above is in fact *meaningless*
without it. Yet you have the temerity to disparage the proofs of such
results as the incompleteness theorem and the unsolvability of the
halting problem in these works for their length and tout the terseness
of your CBL proofs as evidence of their superiority. *Of course* their
proofs are longer and more complex -- unlike you, they do not leave the
fundamental concepts undefined; unlike you, they are actually doing
mathematics. By simply citing Gödel 1931 and Turing 1937 instead of
actually developing the relevant concepts from those classic texts in
its own terms, CBL has, as Russell would have put it, all the advantages
of theft over honest toil. (This, of course, ignores CBL's many
elementary flaws and confusions.)

.

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