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Truth, Incompleteness and the Gödelian Way
New York Times ^ | 2005 | Edward Rothstein

Posted on 05/21/2005 2:42:12 AM PDT by infocats

Is there a more powerful modern Trinity? These reigning deities proclaim humanity's inability to thoroughly explain the world. They have been the touchstones of modernity, their presence an unwelcome burden at first, and later, in the name of postmodernism, welcome company.

Their rule has also been affirmed by their once-sworn enemy: science. Three major discoveries in the 20th century even took on their names. Albert Einstein's famous Theory (Relativity), Kurt Gödel's famous Theorem (Incompleteness) and Werner Heisenberg's famous Principle (Uncertainty) declared that, henceforth, even science would be postmodern.

(Excerpt) Read more at r-s-r.org ...


TOPICS: Philosophy
KEYWORDS: incompleteness; mathematics; relativity; uncertainty
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To: Mitchell; D-fendr
it's independent of the axioms of set theory
and we're waiting for further insight
into whether it's true or false.

I don't quite understand this.
If it's independent of the axioms of set theory
then we are free to accept it as either true or false.

Sort of like
if I don't like playing chess tonight
I can always look for a bridge party.

41 posted on 05/21/2005 7:37:59 PM PDT by Allan
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To: Mitchell; JasonC
Your confusing the theorems with the subject that they're intended to be about.

Your -> You're

Sorry for the typo.

By the way, the reason your argument doesn't work is that you don't "hit a snag once you have sets of cardinality greater than the integers and start having theorems about each element of them." In fact, most of these elements cannot be individually defined via a formula of set theory. If a real number, say, is undefinable in set theory, there is no way to write a sentence of set theory that talks about it specifically. You can write a sentence of set theory that applies to it (because it says, for example, that "every real number has some property"), but you can't write a sentence that distinguishes it from all other real numbers.

The full picture is trickier than this, since you can't define "definability" in set theory, but that's the idea.

42 posted on 05/21/2005 7:48:33 PM PDT by Mitchell
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To: Mitchell
If it were about chess, it would be of no great significance :-).

Why do you dismiss the game of chess so lightly?

OK, maybe chess is not sufficiently complex
but I think it is a reasonable hunch to say
every proof in any finite axiomatic system can be reduced
to finding the correct move in some position in the game of GO.

If this seems outlandish
don't forget that Conway has shown
that his game of LIFE is sufficiently complex
that it can produce a Turing machine.

43 posted on 05/21/2005 7:49:56 PM PDT by Allan
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To: D-fendr
Sort of like Christianity without the salvation part.

I take it you're talking about socialism.

44 posted on 05/21/2005 8:06:35 PM PDT by Allan
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To: Allan

IMO, it would be closer to salvation without God.

I think I'd rather define socialism as salvation by man - or the theory that man can be perfected on earth (by the right institutions and systems).

Socialism and utopianism share alot as I see them.


45 posted on 05/21/2005 10:48:22 PM PDT by D-fendr
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To: Nick Danger; M Kehoe; Matchett-PI; oldglory; mcmuffin
"...if a car is moving at a constant 60 mph (no acceleration) wouldn't it have a force were it to slam into me? You are confusing force with energy. F=ma where m, the mass of the object a, the acceleration But E=mc**2 where m, the mass of the object c, this is what happens when you stand in the road. c is squared because you would c stars 2 if you got hit by a car. I hope this clears things up.

Farking engineers - we're all the same.

Play some Ray Charles music, and watch the movie 'Ray'. It's pretty good and deserved some Oscars. I'm sorta on Rays' side, 'cause I've been enjoying his music since 1952.

R.I.P., Ray Charles Robinson, and thank you for the music..........Barry/gonzo

Hey Dave - we're drowning in a sea of FReepers now!!!!!!!!!!!!!!!!!!!

46 posted on 05/21/2005 11:01:54 PM PDT by gonzo (Arm everyone! An armned society is a polite society...)
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To: Allan


I don't think its a matter of complexity, but one of "proof" and knowing all the statements within the system using the system itself, the validity of an axiomatic system to be "complete" in that sense. Something beyond the system is needed, and then the incompleteness is encountered again.

I can't really illustrate using your chess analogy effectively off the top of my head, maybe someone else will.


47 posted on 05/21/2005 11:03:54 PM PDT by D-fendr
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To: Allan
I had written:

... it's independent of the axioms of set theory and we're waiting for further insight into whether it's true or false.

You replied:

I don't quite understand this. If it's independent of the axioms of set theory then we are free to accept it as either true or false.

Sort of like if I don't like playing chess tonight I can always look for a bridge party.

This is an interesting question. There's clearly a sense in which one can do exactly as you say. If a sentence is independent of set theory, both the sentence and its negation are consistent with set theory, so you can develop two different versions of mathematics without hitting a contradiction in either one. (This is much like Euclidean and non-Euclidean geometries.)

But people generally want their mathematics to mean something, not just to be a formal game with symbols. (Consistency is just a formal, syntactic property.) The axioms of set theory are intended to be true of the mathematical universe as a whole, whatever that might consist of.

Plenty of axiomatizations are consistent but are of no particular interest. For example, by Gödel, assuming that ZF is consistent, the following is also a consistent theory: ZF together with the sentence "ZF is inconsistent". But this isn't really a very interesting theory, and no one would propose adopting it as a base for mathematics. Why? Because it's not satisfied by the actual mathematical universe (since ZF really is consistent).

Why do people take Zermelo-Fraenkel set theory (possibly with the Axiom of Choice, depending on your taste) as being a description of the general accepted principles of mathematics? ZF isn't set in stone, after all. It's just that people know what mathematical principles they accept, and ZF seems to embody them.

In fact, when Zermelo first compiled his axiomatization of set theory, he didn't include what is now known as the Replacement Axiom Schema. Fraenkel pointed this out, and it's reported that Zermelo agreed that it should be there and said that he just forgot to put it in.

The thing is that mathematicians have a good idea as to what the mathematical universe looks like, and they want to use an axiom system that correctly describes that universe and that lets them prove as much as possible. By Gödel's Incompleteness Theorem, no consistent axiomatization of mathematics can be complete, so there's always the possibility of realizing that there are mathematical principles that you would accept but that haven't been included in the theory, and then adding them in.

From a formal perspective, one can add a new axiom "ZF is consistent". (The consistency of ZF is widely accepted, although it's not provable within ZF.) One can then add the consistency of the new theory. And one can then add the consistency of that new theory, and so on. Each one of these axioms can be seen informally to be true, but cannot be proven from the axioms admitted before it.

But it's more interesting to look at one's intuition about the mathematical universe and use that to find new axioms that one can argue are true in the mathematical universe but that ZF doesn't prove. This would be very much like the situation with the Replacement Axiom Schema -- you may think that you have a good axiom system, but a little reflection indicates that the mathematical universe is richer in structure than the axiom system requires, so you decide to add a new axiom.

The basic intuition of set theory is that the universe of sets ought to be as large as possible and as rich as possible, since it's supposed to contain all conceivable mathematical objects. Several of the axioms of ZF can be thought of as expressing (some small part of) this idea.

Many axioms that go beyond ZF are known, and people have varying amounts of confidence in whether these axioms are true in the universe of sets. Most of these axioms are so-called large cardinal axioms; a large cardinal axiom requires the existence of sets larger than can be proven to exist without that axiom.

A simple example of a large cardinal axiom is the axiom of inaccessibility, which states that there exists a "strongly inaccessible" cardinal. [This is an uncountable cardinal kappa so large that: (1) the union of fewer than kappa-many sets of size less than kappa must have size less than kappa; and (2) the power set of any set of size less than kappa must have size less than kappa.] One gets the feeling that if the universe is to be as large as possible, then strongly inaccessible cardinals ought to exist. If they don't exist, it means you just didn't include everything you could have. It turns out that the existence of a strongly inaccessible cardinal proves the consistency of ZF. It follows that one can't prove, assuming just ZF, that strongly inaccessible cardinals exist. But one's intuition says that they should exist; ZF is just too weak to prove it.

And so on to larger and larger cardinals.... Mahlo cardinals, weakly compact cardinals, Ramsey cardinals, measurable cardinals, strongly compact cardinals, supercompact cardinals, huge cardinals, ..., the existence of these being progressively more powerful assumptions. Strongly inaccessible cardinals are, in fact, very small in the hierarchy of large cardinals.

48 posted on 05/22/2005 1:25:32 AM PDT by Mitchell
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To: Allan
Why do you dismiss the game of chess so lightly?

OK, maybe chess is not sufficiently complex but I think it is a reasonable hunch to say every proof in any finite axiomatic system can be reduced to finding the correct move in some position in the game of GO.

If this seems outlandish don't forget that Conway has shown that his game of LIFE is sufficiently complex that it can produce a Turing machine.

It's certainly possible for a game to be complex enough to embed mathematics in. If so, however, the interest in the game would be due primarily to the fact that you can embed mathematics in it, not for the game itself. (Well, the game might be interesting or entertaining as a technical problem anyway, but you had asked why Gödel's incompleteness theorem is considered profound. I don't think any analysis of chess or Go or whatever could be considered profound -- again, unless you could embed something of independent interest in it, in which case that would be the reason for considering the result to be profound.)

Incidentally, I don't think that Go is complex enough for what you're suggesting, since it's played on a finite board, unlike Conway's game of life. (I would guess that Go might be PSPACE-complete, which is pretty complicated, but nowhere near complicated enough to embed a universal Turing machine in.)

49 posted on 05/22/2005 11:23:45 AM PDT by Mitchell
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To: Mitchell
Incidentally, I don't think that Go is complex enough for what you're suggesting, since it's played on a finite board.

True. The game would have to be modified in some way
either by allowing for an arbitrarily large board
or else by allowing for the use
of an arbitrarily large number of stones
(The latter would be preferable if one actually wanted to play it)

However
as it stands
GO is a fiendishly complicated game
I've only played it a few times
and I don't see how anyone could master the strategy.

50 posted on 05/22/2005 11:50:58 AM PDT by Allan
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To: Mitchell

btw could you explain in a few words
what PSPACE complete means?


51 posted on 05/22/2005 11:52:36 AM PDT by Allan
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To: Allan
A decision problem is in PSPACE if there is a polynomial p(x) such that the problem can be solved by a Turing machine that is guaranteed to halt and that, when presented with an input of size x, requires the use of at most p(x) tape cells for the computation. [If you like, you can replace "Turing machine" with "computer", and "size x" or "x tape cells" with "x bytes". That yields an equivalent definition.]

A problem is PSPACE-complete if it's in PSPACE and if every problem in PSPACE can be reduced to it (via a computation bounded by polynomial time in the size of the input).

Asking whether a game like Go is in PSPACE isn't really an accurate phrasing, since the concept of PSPACE only applies to problems with arbitrarily large inputs. The conjecture is that Go played on an arbitrary-sized board is in PSPACE; the problem to be solved would be something like: given a Go position on any size board, determine whether it's a winning position for White or for Black (or if it's a tie -- I forget, does Go have drawn games?). If there is some halfway-reasonable limit (polynomial in the size of the board) on the number of moves in a game of Go, then Go is in PSPACE. I don't know if there is such a limit or not. (But even if not, it's solvable in exponential time, so it's still far short of universal Turing-machine complexity.)

52 posted on 05/22/2005 12:59:48 PM PDT by Mitchell
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To: Allan
The game would have to be modified in some way either by allowing for an arbitrarily large board or else by allowing for the use of an arbitrarily large number of stones

This would need to be done in such a way that a game itself can grow arbitrarily large over time. Just playing Go on an arbitrarily-large nXn board (where n is fixed at the beginning of any particular game) wouldn't be sufficient.

But if you allowed the board to grow over time in the course of playing a particular game (the way it does in Conway's Game of Life, for instance), that would change things completely. As if Go wasn't complicated enough on a fixed-size board!

53 posted on 05/22/2005 1:04:42 PM PDT by Mitchell
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To: Mitchell
But people generally want their mathematics to mean something
not just to be a formal game with symbols.

That is an extra-ordinarily Platonic statment.
In order for a transfinite number 'to mean something'
it must in some sense 'exist' .

But in what way can an 'entity'
which never conceivably could be used by we mortals
to 'measure' or 'describe' something here down on earth
be said to exist??

I am far from being a materialist
but I am unable in any way to visualize transfinite numbers
as any thing other than an interesting and rich idea game.

54 posted on 05/23/2005 1:09:00 AM PDT by Allan
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To: Mitchell
For example, by Gödel
assuming that ZF is consistent
the following is also a consistent theory:
ZF together with the sentence "ZF is inconsistent".
But this isn't really a very interesting theory
and no one would propose adopting it as a base for mathematics.
Why?
Because it's not satisfied by the actual mathematical universe
.

I would rephrase this as follows:

The game obtained by adjoining "ZF is inconsistent" to ZF is not interesting.

(Just as chess was not interesting back in the days
when a Queen and Bishop only could move 1 square at a time)

(since ZF really is consistent)

Once again
what do you mean by the word really?
Are you saying sets really do exist
(in some Platonic realm)?

55 posted on 05/23/2005 1:36:39 AM PDT by Allan
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To: JasonC; Allan
What comes after postmodernism?

If post modernism states that truth doesn't exist, wouldn't it be logical to either come full circle or simply become nonexistent? Even given the fallibist tradition of truth without certainty, doesn't the fact that there is uncertainty allow for the existence of God to account for that uncertainty or am I being to simplistic? And doesn't postmodern belief that even science isn't truth actually then support the existence of a higher truth, a higher power? Or would you first need to prove the existence of a truth?....Just some thoughts.

56 posted on 05/27/2005 5:09:08 PM PDT by Archon of the East ("universal executive power of the law of nature")
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To: muir_redwoods
how profoundly he changed everything

Yes, but it has taken fifty years for him to not be completely ignored. His influence is just beginning.

57 posted on 05/27/2005 5:11:53 PM PDT by RightWhale (These problems would not exist if we had had a moon base all along)
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To: Socratic

You're confusing force and energy.

I recall the formula for kinetic energy as 1/2m*V-squared.


58 posted on 05/27/2005 5:19:53 PM PDT by BeHoldAPaleHorse
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To: BeHoldAPaleHorse

Thanks


59 posted on 05/27/2005 6:25:52 PM PDT by Socratic (There are methods and meth-heads. Life is about choice.)
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