Re: Goedel - interesting problem?

From: Acme Diagnostics (LFinezapthis_at_partpostmark.net)
Date: 06/16/04 

Date: 16 Jun 2004 03:35:14 -0500

Chris Menzel <cmenzel@remove-this.tamu.edu> wrote:
>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).

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. Suppose
our article reader learned about the theory (say, in the distant past)
under those circumstances?
>
>> 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.

Ditto then. But again, I only copied the google description.

>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.

Again, I get that "theorem" is acceptable under certain circumstances.

>> 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.

An incomplete description. Sorry.

>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.

Thanks very much for explaining. I only copied/pasted the google
descriptions to save time, thinking there was enough copy to make
my point. Also to demonstrate the quantity. Quality was not the point
as explained below.

>*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. I emphasize *seems*. I am not
saying you contradicted yourself. Of course there is a distinction to
be made, probably the POV of intended readers v. POV of experts,
and I just can't understand the latter. Let's forget it, not for you to
waste your good time explaining things I won't understand anyway.

>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 think I understand that pretty well. "Truth *outstrips* provability"
- a fine way to say it (thanks) and this was also my understanding. If
the use of "theorem" confuses that distinction then I can well see your
problem with it.
>
>> 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.

Yes that is a good explanation. My main point, which is at issue, is
whether intended readers of the article in question could be reasonably
expected to think "theorem" when:

1. They did not think axioms were things that were "proved" (rightly or
wrongly). I think this an important distinction. Not all or even most
would think this, but I suspect many would.

2. They only distantly recall standard general math courses, perhaps
never using them as a practice in their careers.

If that person rejects "axioms" as the "true statements which are
unprovable" then what is left for them? Quadratic? Cotan? Pythagorean?
Derivative? I mean, there are only so many technical words they are
going to remember, and I am sure "theorem" would be one of them since
they already (think they) know something about axioms, also that it is
most closely "associated" with "axioms" (Why I used "associated").

One thing I am doing which is obviously confusing things: Sometimes I
am impersonating the intended readers and sometimes not, but I am
forgetting to make that distinction. Also I am losing the perspective
to get into the mind of the intended readers because of so much
googling for Goedel.
>
>> 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.

I recall seeing "derived" on some web sites. I won't bother to google
them. I assume the explanation is again, garbage on the web and failure
to make fine distinctions.

>> 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 has always been my understanding too, though lately the suggestion
of empirical truth has been somewhat confusing. "True" is being used in
the basic logic sense of a logical truth condition. It is not an
empirical or "highly probable" truth or philosophical Truth or faith
TRUTH, for instance.

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 believe this is probably not true in one
sense, i.e. I have understood that Goedel was asking a question about
the system as a whole, and of course if so, or in that particular case,
he and his question would be outside the system. But that's not what I
mean. The "truth" of the statements and the provability, formulations,
etc., are all instrinsic.

>That took way too long again. :-)

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

Larry

Relevant Pages

  • Re: Goedel - interesting problem?
    ... >>any given axioms, there are true sentences NOT provable from that set. ... There are two reasons for this. ... for "truth of mathematics". ... there are true statements that are unprovable. ...
    (sci.logic)
  • Re: Godels incompleteness and formal language
    ... > there are true statements that are unprovable. ... That means that, for any such set of axioms, there are ... > Tr is the smallest set X of arithmetic sentences such tha t, ... > the best of my knowledge is again a definitional truth. ...
    (sci.logic)
  • Re: Godels incompleteness and formal language
    ... > there are true statements that are unprovable. ... That means that, for any such set of axioms, there are ... > Tr is the smallest set X of arithmetic sentences such tha t, ... > the best of my knowledge is again a definitional truth. ...
    (sci.math)
  • Re: Goedel - interesting problem?
    ... of the Dolan piece on Goedel and again thank all worthy critics. ... "...any set of axioms at least as rich as the axioms of arithmetic has ... "Arithmetic or math, but not exactly, includes true statements that cannot ...
    (sci.logic)
  • Re: A Manual of Practical Reason
    ... There is no such thing as "the" axioms of arithmetic. ... of axioms "containing a truth". ... > math facts cannot be proved within math, ... math facts" can be proved within math?). ...
    (sci.logic)