[FOM] Explaining incompleteness
Roger Bishop Jones
rbj01 at rbjones.com
Sun Oct 23 12:27:55 EDT 2005
On Friday 21 October 2005 05:20, A.P. Hazen wrote:
> Roger Bishop Jones, in the post that strated this string,
> mentions two candidate "explanations" for the incompleteness
> of arithmetic: Gödel's explanation in terms of the
> undefinability of truth (thanks, Jeff Ketland, for the textual
> smoking gun!) and one turning on the fact that arithmetic is
> undecidable and that proofs have to be finite, effectively
> recognizable, "certificates" of theoremhood. It is perhaps
> evidence that the "truth" explanation is "right" ("righter?")
> that it gives you incompleteness in two related situations
> where certificates of "theoremhood" are NOT finite and
> effectively recognizable. One is Jeroslow's notion of an
> "experimental logic" ... Another is
> the case of Second-Order PA, formulated with an Omega rule.
Perhaps you can explain to me:
(1) how a falsehood can explain anything
(or (2) where you find the fallacy in my "disproof")
Relaxing somewhat and allowing that just some instance of Godel's
generalisation be taken as the explanation (which works for PM
and first order arithmetic), could you explain how, by any true
instances of Godel's premise the two incompleteness results you
mention can be said to be explained.
Also I might be interested in your take on PM and first order
arithmetic as well.
My reason for asking this is that if I had been asked myself to
explain how Epimenides' paradox explains the incompleteness of
arithmetic I would have answered that it shows that arithmetic
truth is not arithmetically definable and hence not recursively
enumerable and hence not recursively axiomatisable.
However, this gives me no clue why a deductive system which is
not recursive should be incomplete, and of course, there is one
which is not.
Is there another route from Epimenides to incompleteness?
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