Re: Goedel - interesting problem?

From: Chris Menzel (cmenzel_at_remove-this.tamu.edu)
Date: 06/16/04 

Date: 16 Jun 2004 15:05:38 GMT

On 16 Jun 2004 03:35:14 -0500, Acme Diagnostics <LFinezapthis@partpostmark.net>
said:
> Chris Menzel <cmenzel@remove-this.tamu.edu> wrote:
> >On 15 Jun 2004 22:57:07 -0500, Acme Diagnostics <LFinezapthis@partpostmark.net>
> >said:
> >
> >A poor statement of Godel's Theorem and a mistaken use of the word
> >"theorem". The author is using "true theorem" simply to mean "truth of
> >arithmetic". It's wrong (though see qualification below).
>
> Ok. I didn't check quality of the sites. I just copyied/pasted the
> first 10 google descriptions that came up. I assume that in 100+ sites
> there will be university sites with qualified authors, etc., but of
> course I can't know that unless I do the work.
> >
> >The second is that there *is* in fact a legitimate way to understand
> >"true theorem that cannot be proved" if things are set up properly:
>
> That's a little confusing. You seem to be saying that "theorem" is
> *really* wrong but that it's right under some circumstances.

It's a *bit* misleading to put it that way. The point is that,
typically, "theorem" is being used in a way that is badly wrong, as in
one of your examples:

> >> Gödel's Incompleteness Theorem states that any formal system powerful
> >> enough to express arithmetic must have true theorems that can not be
> >> proven within the formal system.

Note it says that the *formal system* in question has "theorems that can
not be proven within the formal system". That is really wrong, indeed
flat contradictory. The theorems of a system are exactly its provable
sentences. So the *only* charitable interpretation of this usage is to
take "theorem" simply (and inaccurately) to mean "sentence". By
contrast, consider the other example:

> >> No theory axiomatisable in the type system of PM (i.e., in Russell's
> >> theory of types) which contains Peano arithmetic and is consistent
> >> proves all true theorems of arithmetic (and no false ones).

Here the author speaks of the theorems of *arithmetic* as opposed to the
theorems of the formal system in question. That makes all the
difference. As noted before, "arithmetic" can be understood in such a
way (viz., as the theory whose axioms are all the sentences that are
true of the natural numbers) that "theorems of arithmetic" makes
perfectly good sense. Again, the important point is the contrast
between the truths (or theorems, in the sense just noted) of
*arithmetic* and the theorems of the *axiomatizable formal system* in
question.

> >*That said*, I will agree that your googling does show this informal
> >usage ("theorem ..." = "truth of math") to be pretty pervasive. IF
> >we're careful about it and not too picky, the gist of Godel's theorem
> >can of course still be conveyed.
>
> Using "theorem?" But the gist is all I'm concerned about as stated many
> times. This seems to contradict your '*really* wrong" statement in my
> clear context of reading the article.

No, it was indeed wrong because, as in the first example above, you were
using "theorem" with regard to the systems that are shown to be
incomplete. What I was suggesting is that one *could* overload
"theorem" so that the expression "theorem of arithmetic" could be used
as a synonym simply for "truth of arithmetic". And as just noted above,
there is even a way of understanding "arithmetic" so that this usage is
perfectly correct, though it takes a bit more technical machinery to do
it right -- which is why it is dangerous to use "theorem" in this
broader sense at all in informal discussions. Better to avoid it and
stick with "truth of arithmetic" so that the contrast between truth and
provability is explicit in the terminology.

> I conclude that the incompleteness is instrinsic to the system, or
> theory. There is no information from outside the system *required* to
> make the system incomplete.

I'm not *quite* sure what you mean by "intrinsic", but yes, the question
of whether or not a system is complete does not require looking "beyond"
the system: it is simply the question of whether, for every sentence A
in the language of the system, either A is provable or not-A is
provable. If so, the system is complete; if not, not.

> It was a very worthwhile post for me to read. I appreciate your time.

Glad you found in helpful.

Chris Menzel

Relevant Pages

  • Re: Goedel - interesting problem?
    ... The theorems of a system are exactly its provable ... >theorems of the formal system in question. ... the important point is the contrast ... >provability is explicit in the terminology. ...
    (sci.logic)
  • Re: The Law of the Excluded Middle again (long)
    ... "Inside the scope of quantification, ... to equate (or confuse) truth with provability." ...
    (sci.math)
  • Re: Godel cant tell us what makes a mathematical statement true
    ... are identifying truth with provability. ... oppose such a truth definition I've given. ... But we can formalize the mathematical defintion of 'truth' so that we ... we must therefore examine the methods of the mathematician." ...
    (sci.logic)
  • Re: A little knowledge is a dangerous thing - THE HALTING PROOF
    ... is NOT "Truth" in some huge mystic medieval sense, ... INDEED, the statement encoding "PA is consistent", ... finite number of things, verification is trivial. ... > Provability Thesis: A true arithmetic relation ...
    (sci.logic)
  • Re: Gdel, Turing, and Cantor
    ... If your formal system is meaningless, then no question of truth arises, only ... The language of arithmetic is therefore an interpreted ... If you really want to know the inductive definition of truth for the ...
    (sci.math)