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  #1  
Old 05-10-2008, 04:19 AM
nightdreamer nightdreamer is offline
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Mathematicians are in deep trouble for 2 reasons

The australian philosopher colin leslie dean points out Mathematicians are
in deep shit for 2 reasons

1) skolem discovered a paradox which makes set theory inconsistent

of which freankel and most mathematicians at the time saw

http://www.math.ucla.edu/~asl/bsl/0602/0602-001.ps.

Quote:
Neither have the books yet been closed on the antinomy, nor has agreementon its significance and possible solution yet been reached." –
(Abraham
Fraenkel)

and that

"most mathematicians followed fraenkels skepticiam

[in regard to skolem relativism attempt at resolution - which is at
present is not accepted]


(John von Neumanns states

"At present we can do no more than note that we have one more reason here
to entertain reservations about set theory and that for the time being no
way of rehabilitating this theory is known."

of which a few mathematician also agreed
now rather than solving the paradox before moving on with set theory
mathematicians just ignored it and used set theory for all sorts of
proofs

Now mathematicians are in deep shit for there is now so much invested in
set theory that the skolem paradox threatens the very foundations of
mathematics

so some mathematician now try to argue away the paradox by saying it is
not a contradiction
but
skolems paradox want go away it is at present unable to be disproved
and modern maths is buried so much in shit for useing set theory they cant
get out

2)
mathematician have so much invested in godels incompleteness theorem
much maths is reliant on it
but at the time godel wrote his theorem he had no idea of what truth was
as peter smith the Cambridge expert on Godel admitts

Quote:
Gödel didn't rely on the notion
of truth
but truth is central to his theorem
as peter smith kindly tellls us

http://assets.cambridge.org/97805218...40_excerpt.pdf

Quote:
Godel did is find a general method that enabled him to take any theory T
strong enough to capture a modest amount of basic arithmetic and
construct a corresponding arithmetical sentence GT which encodes the claim ‘The sentenceGT itself is unprovable in theory T’. So G T is true if and only
if T can’t prove it

If we can locate GT

, a Godel sentence for our favourite nicely ax-
iomatized theory of arithmetic T, and can argue that G T is
true-but-unprovable,

and godels theorem is

http://en.wikipedia.org/wiki/G%C3%B6...s_theorems#Fir...

Quote:
Gödel's first incompleteness theorem, perhaps the single most celebrated result in mathematical logic, states that:

For any consistent formal, recursively enumerable theory that proves
basic arithmetical truths, an arithmetical statement that is true, but not provable in the theory, can be constructed.1 That is, any effectively
generated theory capable of expressing elementary arithmetic cannot be
both consistent and complete.
you see godel referes to true statement
but Gödel didn't rely on the notion
of truth



now because Gödel didn't rely on the notion
of truth he cant tell us what true statements are
thus his theorem is meaningless

this puts mathematicians in deep shit because all the modern idea derived
from godels theorem have no epistemological or mathematical worth for we
dont know what true statement are



without a notion of truth we dont know what makes those statements true
thus the theorem is meaningless

and modern mathematics is in deep shit for useing a meaningless theorem
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  #2  
Old 05-10-2008, 04:37 AM
nightdreamer nightdreamer is offline
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EXTENTION TO SKOLEMS PARADOX

1) skolem discovered a paradox which makes set theory inconsistent
http://en.wikipedia.org/wiki/Skolem's_paradox

THE PARADOX IS

Quote:
Using the Löwenheim-Skolem Theorem, we can get a model of ZF set theory which contains only a countable number of objects. However, it must contain the aforementioned uncountable sets. This appears to be a contradiction, since the uncountable sets are subsets of the (countable) domain of the model
freankel and most mathematicians at the time saw this paradox as a contradiction /antinomy

http://www.math.ucla.edu/~asl/bsl/0602/0602-001.ps.

Quote:
Neither have the books yet been closed on the antinomy, nor has agreementon its significance and possible solution yet been reached." –
(Abraham
Fraenkel)

and that

"most mathematicians followed fraenkels skepticiam

skolem attempted a solution of this paradox - which is at
present is not accepted]


(John von Neumanns states

"At present we can do no more than note that we have one more reason here
to entertain reservations about set theory and that for the time being no
way of rehabilitating this theory is known."

of which a few mathematician also agreed
skolems relativistic solution had the affrect of destroying set theory -of which he himself noted
http://www.math.ucla.edu/~asl/bsl/0602/0602-001.ps

Quote:
"I believed that it was so clear that axiomatization in terms of sets was not a satisfactory ultimate foundation of mathematics that mathematicians would, for the most part, not be very much concerned with it. But in recent times I have seen to my surprise that so many mathematicians think that these axioms of set theory provide the ideal foundation for mathematics; therefore it seemed to me that the time had come for a critique." – ([[Skolem]"The Bulletin of symbolic logic" Vol.6, no 2. June 2000, pp. 147
Peter Suber points out the problem with Skolems relativism IE IT DESTROYS SET THEORY

http://www.earlham.edu/~peters/cours...skol.htm#amb3]



Quote:


This means that there simply are no sets whose cardinality is absolutely uncountable. For many, this view guts set theory, arithmetic, and analysis. It is also clearly incompatible with mathematical Platonism which holds that the real numbers exist, and are really uncountable, independently of what can be proved about them.
now rather than solving the paradox before moving on with set theory
mathematicians just ignored it and used set theory for all sorts of
proofs

Now mathematicians are in deep shit for there is now so much invested in
set theory that the skolem paradox threatens the very foundations of
mathematics

so some mathematician now try to argue away the paradox by saying it is
not a contradiction
but
skolems paradox want go away it is at present unable to be disproved
and modern maths is buried so much in shit for useing set theory they cant
get out


so we have
either the paradox means set theory ZFC is inconsistent
or
set theory is destroyed
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  #3  
Old 05-11-2008, 05:51 PM
Mike Dubbeld Mike Dubbeld is offline
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I don't know the specfics of your articles above but I knew long ago from Morris Kline Mathematics The Loss of Certainty and other sources the so-called crises in mathematics. They are ignored as you say and not only that but progress in mathematics has accelerated in the last century - more was done in the last century than the previous century (Professor Steven Goldman Science in the Twentieth Century lecture series for The Teaching Company lectures 12 and 13. The concensus seems to be the problem will go away on its own with new developments and as a practical matter, the so-called crises in mathematics are not crises in science because scientific truth is not the same as mathematical truth.

I will save your post for reference the next time I get around that way on Set Theory and Godel.

Mike Dubbeld
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  #4  
Old 05-11-2008, 08:14 PM
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airmikee airmikee is offline
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How can you say they're in deep shit, using Wikipedia as a reference for your post, and not include this tidbit from the same page?

Quote:
The "paradox" is viewed by most logicians as something intriguing, but not a paradox in the sense of being a logical contradiction (i.e., a paradox in the same sense as the Banach–Tarski paradox rather than the sense in Russell's paradox). Timothy Bays has argued in detail that there is nothing in the Löwenheim-Skolem theorem, or even "in the vicinity" of the theorem, that is self-contradictory.
It's an intriguing paradox, but not self contradictory.
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  #5  
Old 05-11-2008, 10:15 PM
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Mathematicians are in deep trouble for 2 reasons

1) they are boring!
2) they need to get laid!

but i hope they don't give up the good work, i have no doubt's they benifit my life in some way, so way to go Mathematicians, just explain again, what the fuck is algibra for, and how do you spell it again?
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  #6  
Old 05-12-2008, 12:31 AM
nightdreamer nightdreamer is offline
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Quote:
It's an intriguing paradox, but not self contradictory.
read the complete wiki article
most mathematician at the time saw it as a contradiction paradox
read suber skolem and von neumann etc same wiki article

even they say it is a paradox

most logivian only say it is not a paradox based on skolems attempted solution but even skolem said his solution destroyed set theory-thats why many say his attempt is not accepted,
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  #7  
Old 05-12-2008, 01:58 AM
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airmikee airmikee is offline
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I did read the whole article.

Quote:
Most mathematicians agree that the Skolem paradox creates no contradiction. But that does not mean they agree on how to resolve it. [The Löwenheim-Skolem Theorem, http://www.earlham.edu/~peters/cours...skol.htm#amb3]
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Old 05-12-2008, 04:41 AM
nightdreamer nightdreamer is offline
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Quote:
Most mathematicians agree that the Skolem paradox creates no contradiction. But that does not mean they agree on how to resolve it. [The Löwenheim-Skolem Theorem, http://www.earlham.edu/~peters/cours...skol.htm#amb3]

it says also

Quote:
Peter Suber on the contrary argues there are a number of contradictions that result from the skolem paradox and that mathematicians claim skolems paradox is not a contradiction but they dont know how to prove it is not a contradiction
in other words
they say it is not a contradiction
but
cannot resolve the contradiction

Quote:
Peter Suber argues that the skolem paradox is a paradox in the ancient sense [ibid]


Insofar as this is a paradox it is called Skolem's paradox. It is at least a paradox in the ancient sense: an astonishing and implausible result. Is it a paradox in the modern sense, making contradiction apparently unavoidable?

Now Suber shows that a reading of LST gives us a serious contradiction

One reading of LST holds that it proves that the cardinality of the real numbers is the same as the cardinality of the rationals, namely, countable. (The two kinds of number could still differ in other ways, just as the naturals and rationals do despite their equal cardinality.) On this reading, the Skolem paradox would create a serious contradiction, for we have Cantor's proof, whose premises and reasoning are at least as strong as those for LST, that the set of reals has a greater cardinality than the set of rationals. [ibid]
even skolem said it was a contradiction
but
could not resolve according to many


Quote:
Suber goes on to pount out contradictions due to Skolems paradox in non-relativistic accounts

If we want to insist on the non-relativity of our set theoretic notions, and if we hold that our formal systems to date fail fully to capture the secret of the real numbers, then we must choose between the unattractive options (1) that the theory of real numbers is inconsistent, hence has no model, and (2) that the secret of the real numbers cannot be captured by any first-order formal system, i.e. that every attempt will fail either by having no model or by "incurring" a merely countable model. LST puts us to the choice between inconsistency and non-categoricity. If we discard the first of these, then we are left with a view that implies that our notions of uncountable infinities, including the continuum, cannot be fully formalized. As John Myhill put it, in LST we have proved an insurmountable limitation of formalization itself. [ibid]

Last edited by nightdreamer; 05-12-2008 at 04:50 AM.
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  #9  
Old 05-12-2008, 04:52 AM
nightdreamer nightdreamer is offline
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There are a number of contradictions that result from the skolem paradox as pointed out by suber

Quote:
If all the models of a system are isomorphic with one another, we call the system categorical. LST proves that systems with uncountable models also have countable models; this means that the domains of the two models have different cardinalities, which is enough to prevent isomorphism. Hence, consistent first-order systems, including systems of arithmetic, are non-categorical. We might have thought that, even if a vast system of uninterpreted marks on paper were susceptible of two or more coherent interpretations, or even two or more models, at least they would all be "equivalent" or "isomorphic" to each other, in effect using different terms for the same things. But non-categoricity upsets this expectation. Consistent systems will always have non-isomorphic or qualitatively different models. LST proves in a very particular way that no first-order formal system of any size can specify the reals uniquely. It proves that no description of the real numbers (in a first-order theory) is categorical. Very Very Serious Incurable Ambiguity: Upward and Downward LST If the intended model of a first-order theory has a cardinality of 1, then we have to put up with its "shadow" model with a cardinality of 0. But it could be worse. These are only two cardinalities. The range of the ambiguity from this point of view is narrow. Let us say that degree of non-categoricity is 2, since there are only 2 different cardinalities involved. But it is worse. A variation of LST called the "downward" LST proves that if a first-order theory has a model of any transfinite cardinality, x, then it also has a model of every transfinite cardinal y, when y > x. Since there are infinitely many infinite cardinalities, this means there are first-order theories with arbitrarily many LST shadow models. The degree of non-categoricity can be any countable number. There is one more blow. A variation of LST called the "upward" LST proves that if a first-order theory has a model of any infinite cardinality, then it has models of any arbitrary infinite cardinality, hence every infinite cardinality. The degree of non-categoricity can be any infinite number. A variation of upward LST has been proved for first-order theories with identity: if such a theory has a "normal" model of any infinite cardinality, then it has normal models of any, hence every, infinite cardinality. [ibid]

Suber notes that mathematician claim skolems paradox is not a contradiction but they dont know how to prove it is not a contradiction
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Old 05-12-2008, 01:56 PM
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I still don't see how mathematicians are in deep trouble because of this.. there are countless unsolvable problems in science, but that doesn't make it inaccurate, just incomplete.
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Old 05-12-2008, 03:35 PM
Bikerman Bikerman is offline
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It is a bit different though. Maths is (largely) a closed system of logic which doesn't rely on empirical evidence and must be self-consistent. If an inconsistency (paradox) can be shown to exist in set theory (which is what we are talking about here) then it calls into question the whole notion.
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Old 05-12-2008, 07:42 PM
Mike Dubbeld Mike Dubbeld is offline
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In the early part of the 20'th century (1929) Hilbert formulated 23 (famous) outstanding problems in mathematics and challenged mathematicians to solve them. One of them was attempting to show mathematics was a complete and consistent system on all accounts. Godel showed not even something as simple as arithmetic can be proven to be both complete and consistent. Thats an example of a problem in mathematics. Another example is that it was taken for granted for 2000 years that Euclidean geometry was the be-all-end-all geometry but that was flushed down the tubes in the 19'th century leading to having to deal with just what is the ultimate set of axioms/self-evident truths that require no proofs? Then along came chaos.....

The bottom line is if you do not have certainty in mathematics? Where do you get certainty. In the universe NOTHING whatsoever is certain except to x number of decimal places as agreed on by population x at time t. The ONLY reality there is for anything whatsoever for things in the universe (as opposed to abstract ideas like mathematics where 1 + 1 = 2 was true before the universe came into being and will still be true when it is gone). So much for the so called 'ad populum' fallacy which is not a fallacy at all. Apart from agreement (popularity) of minds on something being what it is to x number of decimal places nothing in the universe has any reality at all. Agreement by minds for things in the universe to x number of decimals as empirical truth is the ONLY reality available. But 1 + 1 = 2 is true regardless of a beholder of it but this (abstract/metaphysical) mathematical truth is arithmetic and as I already said, Godel showed that even something as simple as arithmetic is not certain in the sense it cannot be proven to be certain as a complete closed consistent system. On the other hand, almost EVERYTHING YOU KNOW cannot be proven to be true another consequence of what Godel showed. Nightdreamer thinks he has shown Godel to be wrong but so do about 20,000 other AI nuts. I went through Godels work as well as Alan Turing (who basically did the same thing algorithmically) and I agree with Godel as does Stephen Hawking and Roger Penrose.

Other 'problems' in mathematics are interpretations of what mathematics means. There is a Philosophy of Mathematics just like there is a Philosophy of Science. Formalism, Logical Positivism, Intuitionism. I don't really care about pure mathematics. If you can't apply mathematics to things in the universe it is basically useless to me so I don't really think much of these so-called problems. For instance only a mathematician goes into existential despair over things like infinity. To me as long as set theory is useful it serves as a tool. (In a town whose barber shaves all heads that don't shaves themselves, who shaves the barber? Mother/set of all sets-Frege/Russell/Whitehead/Principia Mathematicia/ etc yippee dippee) I don't know about Nightdreamer's skolems paradox though.

Below is one of the best web pages on Godel (first 2 paragraphs below)--
http://users.ox.ac.uk/~jrlucas/mmg.html

Minds, Machines and Gödel
First published in Philosophy, XXXVI, 1961, pp.(112)-(127); reprinted in The Modeling of Mind, Kenneth M.Sayre and Frederick J.Crosson, eds., Notre Dame Press, 1963, pp.[269]-[270]; and Minds and Machines, ed. Alan Ross Anderson, Prentice-Hall, 1954, pp.{43}-{59}.

"Gödel's theorem seems to me to prove that Mechanism is false, that is, that minds cannot be explained as machines. So also has it seemed to many other people: almost every mathematical logician I have put the matter to has confessed to similar thoughts, but has felt reluctant to commit himself definitely until he could see the whole argument set out, with all objections fully stated and properly met.1 This I attempt to do.

Gödel's theorem states that in any consistent system which is strong enough to produce simple arithmetic there are formulae which cannot {44} be proved-in-the-system, but which we can see to be true. Essentially, we consider the formula which says, in effect, "This formula is unprovable-in-the-system". If this formula were provable-in-the-system, we should have a contradiction: for if it were provablein-the-system, then it would not be unprovable-in-the-system, so that "This formula is unprovable-in-the-system" would be false: equally, if it were provable-in-the-system, then it would not be false, but would be true, since in any consistent system nothing false can be provedin-the-system, but only truths. So the formula "This formula is unprovable-in-the-system" is not provable-in-the-system, but unprovablein-the-system. Further, if the formula "This formula is unprovablein- the-system" is unprovable-in-the-system, then it is true that that [256] formula is unprovable-in-the-system, that is, "This formula is unprovable-in-the-system" is true."

The best books on Godel I have ever seen is Godel a Life of Logic by John L. Casti and Werner DePauli and Roger Penrose books The Emperor's New Mind and Shadows of the Mind which in their own way trashes AI nuts. We are not what we are conscious of/we are not minds any more than we are our toaster or our cars. Consciousness is one thing and the mind or brain something entirely different.

Mike Dubbeld
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Old 05-12-2008, 08:18 PM
Bikerman Bikerman is offline
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Whilst I agree with some of this, I find the assertion that consciousness is 'apart from' mind/brain to be insupportable.
What Penrose (and others) have deduced from Godel is that (human?) consciousness is not entirely algorithmic. He deduces from this that AI will not, in its present formulation, be able to mimic the abilities of the conscious mind. It's an interesting hypothesis but it is certainly not a done deal.
Neither is he saying that consciousness is not a product of the mind/brain - in fact he is saying quite the opposite. His hypothesis is that quantum effects are a necessary part of consciousness. He thinks that certain 'micro-tubules' in the brain are small enough to display quantum effects - decoherence/superposition etc - and that these effects are necessary for what we regard as human consciousness.
http://findarticles.com/p/articles/m..._15447461/pg_1
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Old 05-12-2008, 08:53 PM
Bikerman Bikerman is offline
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PS - there is an interesting treatment of the Penrose hypothesis here
http://www.valdostamuseum.org/hamsmith/QuanCon.html

And here are some critiques of Penrose's hypothesis
http://psyche.cs.monash.edu.au/v2/ps...mcdermott.html
http://www.1729.com/consciousness/godel.html

Critique of Godel's theorem
http://gamahucherpress.yellowgum.com...phy/GODEL5.pdf

Last edited by Bikerman; 05-12-2008 at 09:02 PM.
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Old 05-12-2008, 09:20 PM
SpudWithKnife SpudWithKnife is offline
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Mile Dubbled can wax poetic with mathematics and science, but when it comes to beating David Blaine's underwater stunt, his yoga let's him down. Tut tut.
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