Hi Jesse:
I think you miss my point. The All contains ALL including Turing machines
that model complete FAS and other inconsistent systems. The All is
inconsistent - that is all that is required.
Godel's theorem is a corollary of Turing's.
As you say a key element of Godel's approach to incompleteness is to assume
consistency of the system in question.
The only way I see to falsify my theory at this location is to show that
all contents of the All are consistent.
Hal
At 11:46 PM 12/6/2004, you wrote:
>Hal Ruhl wrote:
>
>>
>>Hi Jesse:
>>
>>My originating post appeals only to the result of Turing to the effect
>>that there is in general no decision procedure.
>
>There's no single decision procedure for a Turing machine, but if you
>consider more general kinds of "machines", like a "hypercomputer" that can
>check an infinite number of cases in a finite time, then there may be a
>single decision procedure for such a machine to decide if any possible
>statement about arithmetic is true or false. If your "everything" includes
>only computable universes, then such hypercomputers wouldn't exist in any
>universe, but if you believe in an "everything" more like Tegmark's
>collection of all conceivable mathematical structures, then there should
>be universes where it would be possible to construct such a hypercomputer,
>even if they can't be constructed in ours.
>
>By the way, do you understand that Godel's proof is based on the idea
>that, if we have an axiomatic system A, we can always find a statement G
>that we can understand to mean "axiomatic system A will not prove
>statement G to be true"? Surely it is not simply a matter of random choice
>whether G is true or false--we can see that as long as axiomatic system A
>is consistent, it cannot prove G to be false (because that would mean
>axiomatic system A [i]will[/i] prove G to be true), nor can it prove it is
>true (because that would mean it was proving true the statement that it
>would never prove it true). But this means that A will never prove G true,
>which means we know G *is* true, provided A is consistent. I would say
>that we can *know* that the Peano axioms are consistent by consulting our
>"model" of arithmetic, in the same way we can *know* the axiomatic system
>discussed in my post at http://www.escribe.com/science/theory/m4584.html
>is consistent, by realizing those axioms describe the edges and vertices
>of a triangle. Do you disagree that these model-based proofs of
>consistency are valid?
>
>Jesse
>
>
Received on Tue Dec 07 2004 - 00:27:16 PST