Re: Goedel - interesting problem?

From: George Greene (greeneg_at_greeneg-cs.cs.unc.edu)
Date: 06/17/04 

Date: 17 Jun 2004 14:57:44 -0400

"Acme Diagnostics" <LFinezapthis@partpostmark.net> writes:
 : Why is it wrong for Theorems to pop into my head when I hear
 : "statements that are true in a set of axioms"

Because that's just not the definition.
In math, things have definitions.

 : when there are over 100 sites
 : that explain Goedel's true statements as true theorems that cannot be
 : proved?

It doesn't matter; they're all lying.
They are all written by people who don't know the definition of
the word "theorem". They are also written by people who weren't
speaking in THIS room. This is OUR room. You have to use OUR
definitions in HERE. THEY didn't, because they weren't in here.

 : Here's about 10:
 :
 : 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.

That is just a lie.
It almost means the right thing but it is just ignorant of the
actual correct definition of "theorem".

 : 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).

This one is almost correct; here, there are two DIFFERENT classes of theorems:
theormes proved by the "theory axiomatisable in the type system of PM" and
"theorems of arithmetic". Theorems of the former REALLY ARE theorems.
"Theorems" of the latter actually aren't, but people reserve the right to
speak loosely, especially when they're expert enough not to get confused
(only the newbies get confused).

 : Godel's Incompleteness Theorem (actually there is more than one, but this will not
 : be ... for producing proofs could be found, all the true theorems of mathematics ...

Everywhere you see "true theorem", you should just red-flag it.
There is no such thing. Theoremhood and truth are as different
as lungs and kidneys. The fact that you almost never see one without
the other might cause you to confuse the two, but you must resist.

 : Godel's first incompleteness theorem showed that no computer
 : program could automatically prove certain true theorems i.

Again, the "true theorems" mistake.
Theoremhood comes from axioms.
Truth comes from interpretations.

 : So Gödel's incompleteness result is that if you assume that only true theorems
 : are provable, then this ``This statement is unprovable FAS !''. ...

Almost right, eventually. It rightly says "this statement is unprovable...";
the point being that they SHOULD have been saying "statement" INSTEAD of
"theorem" ALL ALONG.

 : turned the mathematical world with the publication of his incompleteness theorem. ... proved
 : that, for most sets of axioms, there are true theorems that cannot be proved ...

That's actually just a straight-up contradiction.

 : > say that, they say "set of axioms" like 3,000 times. And they usually
 : >> say "statements" not "theorems."

 : >Yes, CRITICALLY.

Yes, CRITICALLY.
In THESE treatments, it is being said RIGHT.

 : > The point is that, for any set of axioms you please,
 : >there are true statements about the numbers that are NOT theorems of
 : >those axioms.
 :
 : But while they "usually" call them statements, they "often" seem to call
 : them "theorems."

Indeed.
THIS ROOM is one of the few places on earth where
they will get CALLED on that error.
Most popular treatments just overlook this as nit-
picking. But it isn't really. What it REALLY is is a classic
example of just how ignorant you can be about a subject and
still get to publish a book on it.

 : A few more excerpts to show what I mean:
 :
 : Godel Incompleteness Theorem. Any formal system powerful enough to express arithmetic
 : must have true theorems that can not be proven within the formal system.
                  ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
That is a contradiction.

 : Please excuse my pasting just the google listings, as I think they illustrate my
 : point well enough. Like I said, there must be 100 more according to google counts.

Well, 40 million people voted for Reagan and 50 million voted for Bush.
Sheer multitudinousness doesn't mean they're not All Just Wrong.

 : Of course I'm not trying to prove anything here about the Theorem, or intending
 : to imply that anything you say technically incorrect.
 :
 : I'm only wondering why is it "*really* wrong" for me to intuitively assume
 : "Theorems" when reading in context of Goedel "statements that are true in
 : a set of axioms"?

I would be glad to tell you but I would probably get interrupted.

 : In spite of my promise in my first reply, I think it will be hard for me to forget
 : "theorems" without more explanation about what the "true statements" are.

You need to start in the other direction.
First, you need to learn what a theorem is.
All these dozens of authors didn't know.

 : More than that I want to know how anybody knows they are "true," and
 : which kind of "true" is implied.

Indeed. In classical first-order logic there is a special kind
of truth that is not as important in other places. 

-- 
 --- The history of our nation has demonstrated that separate is seldom, if ever, equal.
 --- (Feb.3,2004) Supreme Judicial Court of Massachusetts (4-3), adv.Sen.#2175

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