Re: CHAT: mathematics

John Cowan <cowan@...> wrote:
> Right.  G is the Goedel sentence, the one whose meta-mathematical
> interpretation is "G has no proof".  (Its purely *mathematical*
> interpretation is just a relationship between some rather large
> numbers, but when you map the numbers onto statements in proofs,
> you find that G asserts that it has no proof.)
>
> However, I blundered above by saying "finite proof for G"; it's
> precisely Goedel's Theorem that G has no proof. I meant "finite
> proof for not-G".  This is an open question, and most number theory people
> believe it's false, but there is no proof either way.
    Goedel proved that a consistent theory's G is neither provable nor
refutable in that theory (and therefore, by Goedel's Completeness Theorem,
neither true nor false).  If G is refutable in, say, Peano Arithmetic, then
PA would be inconsistent, which is a possibility, but one which no
mathematician seriously entertains.

> In standard number theory, we assume G is true, so there are no
> nonstandard numbers.  In nonstandard number theory, we assume ~G
> is (i.e. "G has a proof"), but we still cannot construct a proof
> for G that is not infinitely long (and a good thing too, since it's
> false)!
    I don't understand this.  Are you saying that "nonstandard number
theory" proceeds from the assumption that PA is inconsistent?
    Clearly, if PA is consistent, then there exist theories in which PA's
G is false, PA+~G being an obvious example.  However, that theory would
have its own G, which is undecidable in that theory.

> But if there were an *independent* proof of ~G, then nonstandard
> number theory (analysis, topology, etc. etc.) is the only kind
> there is, and nonstandard numbers must exist whether we want them
> or not.
    It what sense is analysis or topology "nonstandard number theory"?
Certainly, neither field is incompatible with Peano Arithmetic.  In fact,
analysis is ultimately based upon PA, since the reals are based on the
rationals which are based on the integers which are based on the naturals.
    For that matter, in what sense are you using "independent" and "exist"?

    An interesting linguistic aspect of all this is the fact that Goedel
proved both a Completeness Theorem and two Incompleteness Theorems.
"Complete" is one of the most overloaded words in mathematics.  Even in
logic it has several different meanings, which is why Goedel could prove
first order logic both complete and incomplete.

    ObConlang:  Several years ago I had a long discussion on this list with
Mark Line about, among other things, the question "Do numbers exist?"  We
weren't disagreeing on the answer to that question, but on whether the
question is even meaningful.  This lead to, or at least influenced, the
section "Alien Concepts" in the "Lexicon" section of my Gladilatian pages,
http://www.connix.com/~dennis/glad/lexicon.htm#alien.

 ===========================================================================

             Dennis Paul Himes    <>    dennis@himes.connix.com
                  http://www.connix.com/~dennis/dennis.htm

Disclaimer: "True, I talk of dreams; which are the children of an idle
brain, begot of nothing but vain fantasy; which is as thin of substance as
the air."                      - Romeo & Juliet, Act I Scene iv Verse 96-99