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Author Topic: Gödel's Incompleteness Theorem
Kent
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Linky

It blows my mind! How many of you scientists/mathmaticians have come across this?

quote:
Gödel showed that within a rigidly logical system such as Russell and Whitehead had developed for arithmetic, propositions can be formulated that are undecidable or undemonstrable within the axioms of the system. That is, within the system, there exist certain clear-cut statements that can neither be proved or disproved. Hence one cannot, using the usual methods, be certain that the axioms of arithmetic will not lead to contradictions ... It appears to foredoom hope of mathematical certitude through use of the obvious methods. Perhaps doomed also, as a result, is the ideal of science - to devise a set of axioms from which all phenomena of the external world can be deduced.

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LetterRip
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It is pretty well known, I think it tends to be at least mentioned in 200 level science classes.

LetterRip

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Liberal
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When I was back in college, I got this in Calc. 2, most of my friends got it in that same class or in descriptive stats.
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kenmeer livermaile
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Seems to me that Godel's theorem shows how it is possible for variety to exist and yet mostly conform to laws.

'Assuming a perfect sphere'... [Wink] ...actually, assuming perfect nothingness, Godel's theorem suggests that even nothingness would, in time, deform from itself and... voila! Something!

It takes a deep subtle riddle to shed light on deep subtle riddles like the fact of existence, says me.

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Tom Curtis
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kenmeer, anyone who gets mystical about Godel's Theorem has failed to understand it.
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Tezcatlipoca
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quote:
kenmeer, anyone who gets mystical about Godel's Theorem has failed to understand it.
I wouldn't be so sure about that. After all, kenmeer has discovered the secret of self-generated perpetual motion. [Wink]
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kenmeer livermaile
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Mytical? It's a simple thing. Rules can be designed that ultimately won't adhere to themselves, that deviate from themselves not by intent or design but by... what shall we call it? Mathematical entropy? Digital decay?

Godel's Theorem gets mysty by itself. A tad blurry around the outcome. For reasons that can be proven but that defy even their means of proof.
Godel's Theorem says that its own conclusions can be deviated, yes?

The mystical bit about cosmogony ex nihilo is based on the idea expressed in Godel's theorem that even a stopped clock 'ticks' twice a day...

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Colin JM0397
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Sounds like someone's Waiting for Godel.
"Nothing to be done."

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Wayward Son
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You really don't know Godel's theorem, do you, keemer?
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kenmeer livermaile
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Of course not. I'm not a mathematician. I've read lots about it, though, and the semantic gist is that a system of axioms can be shown to disprove or at least render moot the very axioms by which they disprove or render moot themselves. Starting with absoluteness and adamancy, one can follow said adamant absolutes along the course set forth by their dictates, and in so doing render them less than adamantly absolute. (I assume this process can be reversed so that using ambiguous terms one can render said ambiguous terms more bsolute than their original premises would allow, but that's just extrapolation on my part.)

Care to correct me? I warn you: I strongly believe in the axiom that my most influential science mentor -- Ben Hull -- told me about his years of education in physics, geology, and chemistry:

If I can't explain the fundamental idea of a theory or law to someone of decent intelligence in simple terms that neither paint it over with crude analogy nor bury it in incomprehensible details, then I don't understand what I know -- or at least think I know.

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Omega M.
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quote:
Originally posted by kenmeer livermaile:

Of course not. I'm not a mathematician. I've read lots about it, though, and the semantic gist is that a system of axioms can be shown to disprove or at least render moot the very axioms by which they disprove or render moot themselves. Starting with absoluteness and adamancy, one can follow said adamant absolutes along the course set forth by their dictates, and in so doing render them less than adamantly absolute.

I'm not sure this is quite it. As I understand it (nontechnically), it says that if you take any consistent system of axioms (i.e., one in which you can't prove a contradiction) that uses ordinary (first-order) logic and in which you can prove that basic arithmetic (addition and multiplication of natural numbers) works, you'll be able to write statements in the language of those axioms that you can't prove true or false.

The problematic statements are all along the lines of "This statement can't be proved true." If you can prove that statement true, then you can't prove it true; and if you can prove that statement false (which by noncontradiction would mean that the statement can't be proved true), then the statement can be proved true. Either way it's a contradiction.

So the system of axioms doesn't undermine itself; it just can't settle everything you can ask in the language of the system. The axioms can still be "absolute" as far as they go, I think. Maybe it shows that there's no magic set of rules that explains everything or tells you what to do in any situation, but maybe not, since Godel's theorem is about systems of rules in which you can do arithmetic. Also, most of the ambiguity we face in everyday life is probably due to the fact that terms we use have fuzzy boundaries, not to statements that refer to themselves.

Here's the Wikipedia page I got this from:

http://en.wikipedia.org/wiki/Gödel's_incompleteness_theorems

[ October 18, 2006, 03:40 PM: Message edited by: Omega M. ]

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kenmeer livermaile
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Close enough, eh? Such a theorem by nature can't be completely understood, no? The meaning of Godel's Theorem is, like the result of certain operations on certain axioms, moot.
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pickled shuttlecock
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Omega: Yep that's it, except you forgot the other direction. Here they are:

1) If a powerful enough logical system is noncontradictory, it is possible to write statements in it that are true but unprovable.

2) If a powerful enough logical system can prove every true statement that can be expressed in it, it is necessarily contradictory.

In other words: any powerful enough logical system is incomplete if and only if it is noncontradictory.

You may be right that the problematic statements are along the lines of "This statement can't be proved true" - that the space of problematic statements is probably dense with them. There are a few notable exceptions that mathematicians actually care about. One is the continuum hypothesis, which postulates that there is no cardinality between that of the integers and the reals. It turns out that you can assume it's either true or false, and all of lower mathematics works out just fine either way, but it does have implications to more exotic things like topology and measure theory.

With regards to the original post:

quote:
Perhaps doomed also, as a result, is the ideal of science - to devise a set of axioms from which all phenomena of the external world can be deduced.
It's a good thing they put "perhaps" in there, because science generally infers truth from reality rather than asserting it logically from a set of axioms. If something observed doesn't conform, we may revise the axioms. Mathematics is much less malleable in this respect.

[ October 18, 2006, 04:02 PM: Message edited by: pickled shuttlecock ]

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Colin JM0397
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There is no spoon.
Does that sum it up?

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pickled shuttlecock
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Kenmeer: I think you're misapplying the principle to nonformal systems, and concentrating way too much on direction #2. You're free to do this, of course, and even divine some truth from it, but you'll drive the mathematicians around you insane.

I get excited about #1, personally. I read a study once that seemed to suggest that most true statements expressible in the mathematics derived from ZF set theory (currently-accepted math, in other words) are unprovable, and I almost wet myself.

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Kent
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So, it all comes down to faith?
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kenmeer livermaile
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"1)If a powerful enough logical system is noncontradictory, it is possible to write statements in it that are true but unprovable.

+++

I get excited about #1, personally. I read a study once that seemed to suggest that most true statements expressible in the mathematics derived from ZF set theory (currently-accepted math, in other words) are unprovable, and I almost wet myself."

Waxing mystological again, to be able to write statements that are true but unprovable seems to provide excellent means for extending planks into the void in a non-random fashion.

No, I have no real idea of what that means, that is to say, I don't know how (or am too tired to think I know how) to express this in precise verbal terms.

This:

"In other words: any powerful enough logical system is incomplete if and only if it is noncontradictory"

reads like something that one senses might should be a paradox but can't say why (at least not without great mental effort).

It's inverse:

'any powerful enough logical system is complete if and only if it is contradictory'

however, makes perfect and easy sense to me. A Daoist would recognize the ying-yang symbol within its language. A Shudean, one of the very very few acolytes of THE PROPHET John Smith # 38, would see it as a classical 'contradiction spiral'.

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velcro
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Try readiing Goedel, Escher and Bach. Pretty dense, but it goes through step by step.

ISBN: 0465026567

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pickled shuttlecock
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quote:
Originally posted by Kent:
So, it all comes down to faith?

More or less. Or intimidation - that works, too.

Modern mathematics is built on set theory, which is built on predicate logic, which is built on truth tables. Faith version: "We believe these truth tables represent reality, therefore all that is derived from them is correct." Intimidation version: "Any fool can see these truth tables represent reality, therefore all that is derived from them is correct."

Further, all those "built on" verbs represent a logical derivation (or ten), using a logical system we believe in. Or are intimidated into using, whichever you prefer. [Big Grin] In the end, though, the only justification we have is "it seems to work."

This has nothing to do with Godel's incompleteness theorem so far. But it does show the kind of "truth" we work with when we do mathematical proofs: from the ground up it's actually a rigorously-defined social construct and it doesn't necessarily represent reality (though we strongly believe it does) - it really exists as an entity of its own, almost independent of reality.

With mathematics as such, Godel's incompleteness theorem doesn't really bother me. It does have some interesting tie-ins to undecidable problems (problems that no computational model we know of can ever solve) - but a general search for truth really isn't impacted at all. If our current axioms can't explain a true statement, we'll invent some more.

More interesting to me is the idea that no matter what your ideology, religion, or belief system happens to be, it cannot explain everything that's true unless you're willing to extend your set of axioms or live with an inconsistent belief system. This may be what Kenmeer is getting at - I can't tell, though, because he's too poetical for a simple-minded computer scientist like me. [Big Grin]

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Tom Curtis
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No, it's not built on faith. It is built on pragmaticsm.

There are many formal systems of logic, including formal systems in which not every proposition is either true or false, and formal systems in which propostions can be both true and false. There are even intuitionist systems in which Goedels' theorem can't be proved.

The answer to the question, "Which formal system best describes the world?" is mu. No formal system describes the world. Rather, they are tools for describing the world. As to which is best for describing the world, that depends on what you are doing. For most purposes, classical systems are best.

As I pointed out to my (dialethist) logic professor, even dialethists use classical logic for their metatheory.

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kenmeer livermaile
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"More interesting to me is the idea that no matter what your ideology, religion, or belief system happens to be, it cannot explain everything that's true unless you're willing to extend your set of axioms or live with an inconsistent belief system. This may be what Kenmeer is getting at - I can't tell, though, because he's too poetical for a simple-minded computer scientist like me."

That, and it's flip-flop: that such inevitable inconsistency might well model something like a basic impulse for how ANYTHING ever came to be in the first place: inconsistency. THe loophole of cosmogenesis.

Yeah, I'm poetic alright.

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Kent
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quote:
Gödel showed that provability is a weaker notion than truth, no matter what axiom system is involved ...

How can you figure out if you are sane? ... Once you begin to question your own sanity, you get trapped in an ever-tighter vortex of self-fulfilling prophecies, though the process is by no means inevitable. Everyone knows that the insane interpret the world via their own peculiarly consistent logic; how can you tell if your own logic is "peculiar' or not, given that you have only your own logic to judge itself?

This quote in particular has me stewing. This is likely why we rely on others to tell us our value as a human being.
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kenmeer livermaile
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The concept of 'truth' is itself axiomatic.
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Wayward Son
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How about the concept of "real?"

Does the universe itself follow an axiomatic system of "basic laws" which combine to produce what we see about us? If so, can't we then divide what we see as being real ("true") and fantasy ("false")? And if both of these apply, doesn't then Godel Theorem apply to reality, too?

Does this mean that even reality is inconsistent? [Eek!]

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kenmeer livermaile
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Wayward hs penetrated to the essence of my mystological musings on the implications of Godel's Theorem.
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Kent
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Tomorrow all the grass could be purple rather than green (or brown in my case).
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Colin JM0397
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Technically speaking, the grass isn't green... or brown.
It merely reflects that very narrow wavelength of the electromagnetic spectrum that you can perceive.

Your brain reconstructs the electromagnetic energy into a picture in your head... We all agree that said picture in our heads is “green”.

[Razz]

-- Add --
Which brings us back to the "there is no spoon" point. If truth and reality is what we perceive, then the universe resides squarely in our heads.

[ October 19, 2006, 02:38 PM: Message edited by: jm0397 ]

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rightleft22
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“This is likely why we rely on others to tell us our value as a human being”

And why so many people are miserable and or depressed.

“If truth and reality is what we perceive, then the universe resides squarely in our heads.”

And then foolish if we allow other universes to define our value.

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kenmeer livermaile
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"Your brain reconstructs the electromagnetic energy into a picture in your head... We all agree that said picture in our heads is “green”.

[Razz]"

But, giving the possibility of different outcomes via inconclusive operations of matching axioms, this consensual 'green' you cite is arbitrary.

If MY head starts seeing purple grass via Godelian unproveble trueness, it's purple, dammit, and that's TRUE whether you or I can disprove/prove it or not.

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Kent
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"Prove to me that I don't see dead people," eh Kenmeer?

Science can only test what it can test. Those we people who insist in untestable axioms being true we put in asylums or make them leaders of religions.

[ October 19, 2006, 03:09 PM: Message edited by: Kent ]

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kenmeer livermaile
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Kenmeer for Prez:

AH SEE THANGS!

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Tom Curtis
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Wayward:

quote:
How about the concept of "real?"

Does the universe itself follow an axiomatic system of "basic laws" which combine to produce what we see about us? If so, can't we then divide what we see as being real ("true") and fantasy ("false")? And if both of these apply, doesn't then Godel Theorem apply to reality, too?

Does this mean that even reality is inconsistent?

No. It merely means that some truths about the universe were not inherent in the initial conditions (axioms).

From this it follows that if the initial conditions are the only determinants, determinism is false (but if a God is allowed to interfere, perhaps it isn't).

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Tom Curtis
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jm:

quote:
Technically speaking, the grass isn't green... or brown.
It merely reflects that very narrow wavelength of the electromagnetic spectrum that you can perceive.

"X is green" means "x reflects a particular pattern of electro-magnetic radiation in white light, or x emmits a particular pattern of electro-magnetic radiation". (That's the very short version as I have to rush of to work.)
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pickled shuttlecock
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quote:
Originally posted by Kent:
"Prove to me that I don't see dead people," eh Kenmeer?

Science can only test what it can test. Those we people who insist in untestable axioms being true we put in asylums or make them leaders of religions.

You believe your mother loves you. Off to the asylum with you.

In other words, your assertion is basically a load of crap. I'd use politer terms, but I'm deadly sick of this whole line of argument (science is the only way to ALL TRUTH!) and all of its implied straw men and ad hominem, so I'm bowing out of this once-interesting discussion now with this last statement.

Have fun on your new science vs. religion thread.

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Kent
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PS, I have always been interested in how we know what we know (epistemology) which always boils down to who we give credence to, because accepting axioms of existence is a choice we all make (and you are correct that in this crowd it usually gravitates towards theists vs. atheists, ie. the great and terrible questions).

I do not agree that science is the only way to all truth, but it is the way to describe and explain most truths I know (or think I know as it may be). I likely miscommunicated my point, which is that science can only test those things it knows how to test, not that it can test all truth. Most of us are hopeful that one day it will be able to test everything (all truth), but according to Gödel that is likely impossible. Many of us are determinists here, I think, but Gödel must be arguing against it as I understand it; and I can see no logical flaws.

So my question then is, if determinism is false, isn't anything possible?

[ October 19, 2006, 04:49 PM: Message edited by: Kent ]

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Wayward Son
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quote:
So my question then is, if determinism is false, isn't anything possible?
Simple answer: no.

Godel's theorem proved that not everything that is imaginable can be shown to be either true or false using logic. But that does not mean that anything can be true. True and false statements can still be logically deduced. And just because something cannot be proven to be false does not mean that it isn't.

But, conversely, something that cannot be proven to be true may still be true. I believe that is what "incompleteness" means.

quote:
From this it follows that if the initial conditions are the only determinants, determinism is false (but if a God is allowed to interfere, perhaps it isn't).
For Godel's theorem to be applied to the real world, there would need to be a parallel world--a "map" of the world, in the mathematical sense (IIRC). So, actually, we would need God, or at least a Heaven or Hell, to apply the theorem to reality.

For my own, ahem, humble speculation on this, see The Monitor of Dorean Gates.

[ October 19, 2006, 04:58 PM: Message edited by: Wayward Son ]

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Kent
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That was a very fun short story Wayward! You have obviously been thinking about this for years.

So what is the "divine" map or measuring stick? In your story it was a computer monitoring millions of sensors which comes the closest to omniscience that we know of ("we" being those I give credence to [Big Grin] ).

Give me a slogan, an axiom, or a creed or something to help me make sense of it all!

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Wayward Son
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For a slogan, all I can do is lead you to that great philosopher of our time, Robin Williams:

"Reality. What a concept!" [Smile]

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kenmeer livermaile
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"So my question then is, if determinism is false, isn't anything possible?"

Well now, that would be a determination, you see. An extrapolation of the axiom "determinism is false". And Godel says...

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foliated
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@pickled shuttlecock and kent:

quote:
Originally posted by Kent:
So, it all comes down to faith?
More or less. Or intimidation - that works, too.

Modern mathematics is built on set theory, which is built on predicate logic, which is built on truth tables. Faith version: "We believe these truth tables represent reality, therefore all that is derived from them is correct." Intimidation version: "Any fool can see these truth tables represent reality, therefore all that is derived from them is correct."

Then there's the version that's an amalgam of what my teachers told me when this issue came up: "yeah, it could be that we're missing something here. but 1) if we are, a whole lot of really smart people (gauss, newton, riemann, euclid, dirichlet, etc etc) were off, too, so you're not in bad company. 2) for something like 4 millenia, we've been fixing things every time we've found problems, and have always come up with something better and more beautiful. So if someone ever does find something fundamentally wrong with, say, the notion of subset, I'm sure we'll be able to think of something. [Smile] "

I paraphrase, of course.

anyway, I'm kind of uncomfortable with this whole "represent reality" business ... what exactly do you mean, pickled shuttlecock? Of course, it may be that I'm just missing something. Is predicate logic based on truth tables?

kent's comment was kind of cryptic to me, too. which means that this whole thing could be an inadvertent distractogram....in which case, I apologize.

Edited. Too many smileys.

[ October 20, 2006, 01:20 AM: Message edited by: foliated ]

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