Re: Goedel - interesting problem?

From: Acme Diagnostics (LFinezapthis_at_partpostmark.net)
Date: 06/05/04 

Date: 4 Jun 2004 22:52:16 -0500

Aatu Koskensilta <aatu.koskensilta@xortec.fi> wrote:
>Acme Diagnostics wrote:

Thanks for an interesting message.

>> But in either of
>> those two cases, it is then left vague that those statement/sentences
>> might be true both within and without the set of axioms,
>
>I'm afraid you're deepening my confusion here. What on earth do you mean
>by a statement being true "within" or "without" some set of axioms? This
>is not a notion of truth I'm familiar with, and certainly not the one
>employed in Gödel's theorems in any way.

I believe that's true, also that I understand your legitimate
objection. Let's throw those terms into the garbage then. I am a
terrible explainer of Goedel's theorom compared to someone who
obviously understands the technical detail much better than I (for lack
of interest only). But my issue here is not with the technical detail,
it is the *effect* of the theorem as a whole. It is like the difference
between understanding why simultaneous equations can be solved by
subtracting them, and the fact that a future point on a graph can be
predicted with a certain probability, with the notion of probability
applying directly.

To understand that probability, and how that might apply
to making money in the stock market, one simply does not need to
know anything at all about why subtracting simultaneous equations
works. Not in the slightest. The student might not even encounter
the equations, having same calculated by a commercial program.

Similarly, Goedel's theorem has significance outside of its proof. I
can't call it a tool like simultaneous equations because a proof isn't
a handy tool like a formula, but its *effect* does relate to larger
contexts in a similar way. What we want the student to understand is
that no level of description is complete, and that one instance of that
is something called Goedel's Theorem, but not necessarily *why* that is
so. Another instance is the statistical prediction example I gave
(sometimes doesn't work in application due to new factors in that next
level of description).

(still building up to an explanation of "set of axioms.")

The concept that is important about Goedel here, then, is that
logicians have made a discovery which is proven and is quite profound.
That in math (remember the student doesn't know what "math" means the
way a mathematical logician knows, but has only used it as a tool),
which used to be a closed system that proved itself, is now known to be
not so, and everyone accepts this because, for instance, you can google
it right now and get an amazing number of hits to say that more-or-less.
That my description doesn't match your more precise description doesn't
matter all that much because *nobody* with your knowledge of the
theorem will need the explanation we are discusssing, and those who
do need it won't understand the way you do anyway. Further, this
doesn't just apply to math, but to logic and, in fact, any level of
description that has formal rules or laws that are self-proving, i.e.
"as rich as the axioms of..." because in mathematical logic these are
usually called "axioms." Since the concept of "sets" is all the rage
(in general education) and has been for quite some time, the term "set
of axioms" is widely explanatory of the level of description being
described to members of any of a wide variety of fields.

Last, regarding the reason for repeating "set of axioms": 1) It could
be argued it is where the theorem fits into the larger picture in the
most profound sense, 2) which sense is expected to often be central to
discussions for which the explanation you are criticizing is intended,
3) the true fact that there is a level of description which is complex
and well defined "...at least as rich as the axioms of arithmetic," and
4) that some things proven true witihin that level can suddenly be
found to be false when applied to, taken into, considered within, etc.,
etc.
a higher level of description.

That is the reason for repeating the
first term "set of axioms" and empasizing explicitly that these proofs
("true statements") can only be considered true up to the boundaries of
"set of axioms," and it is important to enforce the concept that "set
of axioms" can be replaced wtih "X" or *any* substitute you like, in
both uses - any level of description works from QM to logic to math to
statistics to moving any theory into the real-world to moving from the
observable universe to all existence, and the seemingly useless
repetition of "set of axioms" now becomes quite important in that
regard, if only in analogy. Just as important, though you probably
don't know it, all or most of those examples were included in the
dialog engagement I just recently had with the one and only person to
whom I presented the explanation, that dialog just being recently in
this group and available to any who wish to criticize my
characterization of the explanation. To repeat, the operative concepts
are: use, reliability, intended reader, context, context, and context.

Well there's another probably botched attempt to explain the principles
of editing, context, probability, and reliability in argumentation in
terms of an explanation of Goedel's theorem, probably a hopeless goal
in any event. But if you want another try, maybe I'll try some other
tack. I do need practice editing the principles just itemized into some
understandable form useful in a newsgroup post.

>> Also, though proven with arithmatic in the first instance, I believe
>> the theorem is generally agreed to extend beyond arithmetic. But
>> feeling unqualified, I'll let someone else address that.
>
>Gödel's theorems certainly apply to theories which do not count as
>"theories of arithmetic". However, the unprovable statement produced by
>the proof of Gödel's theorem is a statement about natural numbers.

Thanks. I will take your word for it. That was approximately my
understanding.
>
>>>or "true sentence about
>>>natural numbers formulated in the language of the formal theory"?
>>
>> It's many more than 28 characters, "natural numbers" is a technical
>> term, and "language of the formal theory" is also too technical and not
>> as easily read by intended readers as the original.
>
>Complaining that an explanation of what Gödel's theorem is

You really like changing my "effect of Goedel's theorem" to "what
Goedel's theorem is" don't you? Why is that?

>uses
>"technical" terms like natural numbers or formal theory is like
>complaining about the statement of Fermat's last theorem using the
>notion of exponentiation. The fact is that Gödel's theorem is a
>mathematical theorem and in order to understand it at all you must
>understand the basic concepts it uses.

Or that modems contain printed circuits. However, I once explained
to the CEO of a business, who knew nothing at all about computers,
much less modems, entirely by analogy of how air conditioners work
(his product that he understood), and he was thus able to make an
important decision about the computer system, i.e. what modems,
who to buy from, who to believe about them (among two competing
vendors), how to best use them in a particular context, etc. I am
reporting this experience to demonstrate why I am convinced that
one does not need to know anything at all about the
technical proof of Goedel's theorem to use it profitably in other
contexts. One does, however, need to judge the reliability of 1,000 web
pages all saying the same thing about it more-or-less, not to mention
textbooks. Obviously that is much more reliable than the reliability
said business owner could invest in my analogy, yet it worked, it was
reliable, it had a probability to satisfy an appropriate test, and
that's all that matters in some contexts, such as in business.
>
>> But I'm claiming it is as close to the long-winded complete explanation
>> in all detail as I could find in an hour of googling with that
>> character count by a factor of two at least, assuming the intended
>> reader is not a logician, but from any of a large variety of fields.
>
>The description would be acceptable, provided you did not try to squeeze
>any specific implications from it,

"Implications" in the narrow sense - agree fully. "Implications" in
the larger reasoning sense - "Any" is too universal. I agree that there
will be any number of implications that cannot be squeezed, and "any"
and "all" implications about the proof itself cannot be squeezed. The
explanation we are discussing obviously says nothing about the proof,
why I have been careful to always say "effect of theorem" rather than
"theorem" or "proof of theorem" in every use of the term since my
first post in this thread. Though in one case I used "about the
theorem" carelessly as has already been noted and repaired.

>if the strange notion about truth
>"within" or "without" an axiom system was dropped

Those were my words, not the author's, I am happy to drop them.

>and it was made clear
>that "has statements which are true" does not mean that the statements
>are elements of the set of axioms.

I don't know what "elements of the set of axioms" means because I
do not recognize it as a standalone term, and there is not enough
context for me to inference it. I'll take your word for it. But there
is something within the level of description we are discussing (to
avoid the term you don't like) that is true, but which can be false in
a higher context, whatever you want to call it.

Call it "X" if you like. The explanation called it "statements."
You've suggested "sentence." I think to recall "axioms" and maybe
propositions. Here pops up the word "elements." I am unfazed by such
switches in terminology because I have a perfect solution. We'll just
call it "X" (or whatever) from now on, with X = whatever is true at one
level of description which can be false at another. This "X=" business
works every time, but I have to tell you that it is usually silly, not
required, and only the result of uncooperative dialog, i.e. someone
purposely trying to confuse, to weasel out of what was said or implied.
Not that you are doing that, and I assume that now "elements" has some
precise distinction that is important in Goedel's proof.

>I still insist that a mention about
>it applying to formal theories with mechanically specified rules of
>inference be made, so that people don't incorrectly conclude that it
>applies to, say, the theory of evolution or what not.

Is that your problem with "set of axioms?" Ok, perhaps a light dawns.
If one thought of evolution as a set of axioms (perhaps one could, I
don't know) and the facts about evolution that we know as "statements
that are true," then it does seem that that situation could fit the
explanation we are discussing. These facts aren't proven with any
formal system such as math or logic, but empirically. There intuitively
seems like there is a lot more wrong with this misapplication because I
can't imagine anyone applying that explanation to a science such as
evolution, but of course intuition doesn't mean anything. I'll have to
give it some more thought. Thanks for the example.

>>>For example:
>>>
>>> Gödel's 1st incompleteness theorem says that for any consistent formal
>>> theory T containing elementary arithmetic we can[1] construct a true
>>> sentence about natural numbers which is not provable in T.
>>
>> That would be confusing to many for whom the explanation is intended.
>> Most confusing would be "consistent formal theory T". Why would anyone
>> talk about an inconsistent theory?
>
>Because they didn't know it was an inconsistent theory?

Sorry, I was impersonating an imagined reader of the article who did
not understand the term, not myself. I appreciate your explanation here
to me, but the point is that, once such explanation is needed by a
reader, the explanation piece we are discussing would have failed the
"masterpiece of explanatory text" test. <g>

>Several examples
>of this are known from history: Frege, Church, Quine, Rosser, &c. were
>all interested in theories which later turned out to be inconsistent.

Your substitute words are unnecessary in context, and don't they only
lead one into the proof, where we both agree one can't go from the
explanation? The words in the term only serve to slow some readers down
and imply that the explanation is something not intended.

>This is entirely irrelevant to Gödel's 1st incompleteness theorem, which
>is an implication of form: If T is a consistent theory containing
>elementary arithmetic, then T is incomplete. Since inconsistent theories
>are trivially complete, you must exclude them in the statement of the
>theorem lest it become false.

Again, such an explanation defeats the purpose of the text, and that
was exactly my point of impersonating the imagined reader.

> > What's an informal theory? What's a
>> "theory T?" Doesn't T go in an equation? How can it stand for words?
>
>A theory T is a mathematical object. It doesn't stand for words.

Again, I was only impersonating an imagined reader.

Many people simply could not understand "mathematical object." It seems
now that you are losing our own context. We are
discussing an explanation of the effect of Goedel's theorem intended
for members of a wide variety of fields, and you know very well that
some of those will not understand "T is a mathematical object" and it
will only slow down many others unnecessarily.

I suggest that you have the logic/math ability and are quite expert
in all aspects of the Goedel proof, but lack even introductory
knowledge of editing English judging by what anecdotal
evidence I have, and thus understandably wish to keep casting my
"effect of the theorem" into technical mathematical logic terminology
relating to the proof itself, but which is the antithesis of
"masterpiece of explanatory text" (to any member of any of a wide
variety of fields).

Larry

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