Re: Goedel - interesting problem?
From: Chris Menzel (cmenzel_at_remove-this.tamu.edu)
Date: 06/16/04
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Date: 16 Jun 2004 05:47:49 GMT
On 15 Jun 2004 22:57:07 -0500, Acme Diagnostics <LFinezapthis@partpostmark.net>
said:
>
> Chris Menzel <cmenzel@remove-this.tamu.edu> wrote:
> >On 12 Jun 2004 04:37:29 -0500, Acme Diagnostics
> ><LFinezapthis@partpostmark.net> said:
>
> Please excuse replying to the same message twice. I did some googling
> after replying to the first.
>
> >> After sitting here for a while and looking at these two, I realize
> >> that you are not going to accept my assertion that the intended
> >> reader will substitute "arithemetic" or "math" for "set of axioms."
> >
> >I really don't know why they would.
> >
> >> First time I read it, I just intuitively assumed theorems associated
> >> with the set of axioms.
> >
> >But that would have been *really* wrong, as the theorems of a set of
> >axioms are, by definition, the sentences that are *provable* from the
> >axioms, and the whole point of Godel's Theorem is (roughly) that, for
> >any given axioms, there are true sentences NOT provable from that set.
>
> Why is it wrong for Theorems to pop into my head when I hear
> "statements that are true in a set of axioms" when there are over 100 sites
> that explain Goedel's true statements as true theorems that cannot be
> proved?
There are two reasons for this. The first and by far the most pervasive
reason is that there is lots of garbage on the net, and lots of
incorrect uses of mathematical terminology, as in the following:
> 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.
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).
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:
> 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).
I don't know the context here, but depending on how the terminology was
defined, but this one can actually be acceptable. In some
presentations, "arithmetic" is used to mean the theory whose axioms are
exactly all the sentences in the language of arithmetic that are true of
the natural numbers. For obvious reasons, this is often called the
theory of "true arithmetic". On that definition, "theorem of
arithmetic" -- something provable from the axioms of true arithmetic --
makes perfect sense (though "*true* theorem of arithmetic" is
redundant). However -- and there is again no avoiding some technical
terminology -- this theory is *not axiomatizable*, which means there is
no *decidable* set of axioms that has the same theorems as arithmetic,
no set of axioms such that we can tell what's an axiom and what isn't.
> According to Dummett, incompleteness theorems only rule out the
> possibility that ... the specification of the set, recognise that it
> contains only true theorems.
I can't quite grok that, but there is no problem I can see. If your
axioms are true, your theorems are true.
The remaining examples all simply continue mistakenly to treat
"theorem", or "true theorem", or "theorem of mathematics" as a synonym
for "truth of mathematics". (Some of the examples also seem to have
been munged in some way or another in the copy/paste process). But
don't take my word for it; check *any* mathematical logic text or any
other formal presentation of Godel.
*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. But the potential for confusion is
quite powerful -- for the heart (well, one heart) of Godel's Theorem is
that truth *outstrips* provability; any attempt to capture mathematical
(specifically, arithmetical) truth in the *theorems* of some (decidable)
system is bound to fail. If we overload "theorem" in the manner
suggested, it can indicate *both* provability and truth, and confusion
has its foot in the door.
> 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"?
A reasonable question. I hope the above is a reasonably clear answer.
> Are the "true statements" derived in any way?
In some cases, yes, but not by reasoning in the axiom system being shown
to be incomplete. This is another pretty subtle point that can't be
appreciated if we confine ourselves to informal discussions.
> More than that I want to know how anybody knows they are "true,"
Well, in fact, we don't ever need to be able to *tell* which of the
unprovable statements are true (though often we can) to know *that*
there are true statements that are unprovable (in a given system).
Godel showed that any reasonable set of axioms for arithmetic will be
incomplete. That means that, for any such set of axioms, there are
statements A in the language of arithmetic such that both A itself *and*
its negation not-A are unprovable. By basic logic, one or the other has
to be true. So even if we don't know which it is, we know that there
are true statements that are not provable from the axioms.
That took way too long again. :-)
Chris Menzel
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