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User talk:MarkGall (view source)
Revision as of 03:07, September 24, 2009
, 03:07, September 24, 2009→Incompleteness Theorem: new section
:Sure, sounds good to me. \pi_1(S^1) is about the right level. Thanks for fixing this! --[[User:MarkGall|MarkGall]] 14:19, 22 September 2009 (EDT)
:Sure, sounds good to me. \pi_1(S^1) is about the right level. Thanks for fixing this! --[[User:MarkGall|MarkGall]] 14:19, 22 September 2009 (EDT)
== Incompleteness Theorem ==
Mark:
I'm planning to work on the issues of logic, completeness, undecidability, and the Gödel incompleteness theorem next. Basically, everywhere I look, I see things to do, and I happened to look at the [[Undecidable]] page and the [[Axiom]] page. The latter is fairly straightforward -- the characterization of the axioms of geometry being "challenged" by Non-Euclidean geometry needs to be clarified.
But the "undecidable" stuff needs a lot more care. I'm pretty sure I understand the issues, but I'm not 100% sure, so I want to run it by you. It seems to me that essentially all of the material about "Famous undecidable statements" is wrong. These may be statements that are at the edge of what ZF and ZFC mean, but they don't relate to the Gödel incompleteness theorem. And "undecidablility" is about the Gödel incompleteness theorem, ''not'' about the axioms of ZF.
My understanding is that the Gödel incompleteness theorem says, roughly
:*Any logical system rich enough to express integer arithmetic is "incomplete" in the sense that it is not true that, for every well-formed formula F, either F or not F is a theorem (has a proof in that logic).
The logical systems being analyzed by Gödel are simple logics about integers, with nowhere near the sophistication of ZF or ZFC. The non-provability of AC in plain ZF is not really what the Gödel incompleteness theorem is about. You simply add AC, get ZFC, note that the result is still consistent (that is, you now have a trivial proof of AC because it is an axiom, but you do not have a proof of not-AC) and move on.
Hence, it seems to me that all the things in the article related to AC (Banach-Tarski, Continuum hypothesis, large cardinals, non-measurable sets) belong in a different article, since they distract from Gödel's contribution. Only the halting problem (a result from the closely related field of theoretical computer science) is directly relevant to Gödel. And that's what "decidability" should be about.
Of course, fact that neither AC nor not-AC has a proof in ZF is an example of incompletness, but it's not what Gödel was talking about. That is, plain ZF is incomplete, but incomplete logics are a dime a dozen. Gödel showed that ''no'' logic rich enough for integers could be complete.
Does this make sense?
[[User:PatrickD|PatrickD]] 23:07, 23 September 2009 (EDT)