"This statement is false" is one example.
Fuck!
Epimenides paradox: "Epimenides The Cretan Says, "All Cretans are Liars"
by frankiespeakin 39 Replies latest jw friends
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Eustace
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King Solomon
I love the Monty Hall Paradox, a door-picking paradox (which is counter-intuitive):
http://en.wikipedia.org/wiki/Monty_Hall_problem
And to play it:
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frankiespeakin
Eust,
""You can be a liar and still tell the truth from time to time.
There, I solved a paradox that has wracked the minds of theologians for millenia.""
That is one way to break it,, if we get out of absolutes. If we stay with just absolutes we cannot solve it.
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frankiespeakin
I think an amazing thing is our psyche can point out that aboslutes cause problem where a classical computer will just keep crunching the yes nos to infinity in a paradox loop were as our psyches we no how to rise beyond it and jump out.
Maybe Quantum computer can do the same thing using up and down spin to unravel even more harder computations.
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james_woods
Maybe. I hate those danged C++ logical errors, btw.
Maybe Quantum computer can do the same thing using up and down spin to unravel even more harder computations.
Every such system must contain such paradox - Kurt Goedel.
Very important mathematical principle here.
Read Douglas Hofstadter's book - Goedel, Escher, Bach.
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frankiespeakin
"Ah ha! Moments" many/majority of us have them when as if a flash of light something that has been puzzleing us and in a flash we got it.
Achimedes had them doing geometetry all the time he ran out in the street naked he was so high.
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frankiespeakin
This is refering to natural numbers but what about the quantum world and super positions doesn't that take us out of that loop of natural numbers and into probabilities which are not absolutes??
http://en.wikipedia.org/wiki/G%C3%B6del's_incompleteness_theorems
Meaning of the first incompleteness theorem
Gödel's first incompleteness theorem shows that any consistent effective formal system that includes enough of the theory of the natural numbers is incomplete: there are true statements expressible in its language that are unprovable. Thus no formal system (satisfying the hypotheses of the theorem) that aims to characterize the natural numbers can actually do so, as there will be true number-theoretical statements which that system cannot prove. This fact is sometimes thought to have severe consequences for the program oflogicism proposed by Gottlob Frege and Bertrand Russell, which aimed to define the natural numbers in terms of logic (Hellman 1981, p. 451–468). Bob Hale and Crispin Wrightargue that it is not a problem for logicism because the incompleteness theorems apply equally to second order logic as they do to arithmetic. They argue that only those who believe that the natural numbers are to be defined in terms of first order logic have this problem.
Relation to the liar paradox
The liar paradox is the sentence "This sentence is false." An analysis of the liar sentence shows that it cannot be true (for then, as it asserts, it is false), nor can it be false (for then, it is true). A Gödel sentence G for a theory T makes a similar assertion to the liar sentence, but with truth replaced by provability: G says "G is not provable in the theory T." The analysis of the truth and provability of G is a formalized version of the analysis of the truth of the liar sentence.
It is not possible to replace "not provable" with "false" in a Gödel sentence because the predicate "Q is the Gödel number of a false formula" cannot be represented as a formula of arithmetic. This result, known as Tarski's undefinability theorem, was discovered independently by Gödel (when he was working on the proof of the incompleteness theorem) and byAlfred Tarski.
Discussion and implications
The incompleteness results affect the philosophy of mathematics, particularly versions of formalism, which use a single system formal logic to define their principles. One can paraphrase the first theorem as saying the following:
- An all-encompassing axiomatic system can never be found that is able to prove all mathematical truths, but no falsehoods.
On the other hand, from a strict formalist perspective this paraphrase would be considered meaningless because it presupposes that mathematical "truth" and "falsehood" are well-defined in an absolute sense, rather than relative to each formal system.
The following rephrasing of the second theorem is even more unsettling to the foundations of mathematics:
- If an axiomatic system can be proven to be consistent from within itself, then it is inconsistent.
Therefore, to establish the consistency of a system S, one needs to use some other system T, but a proof in T is not completely convincing unless T's consistency has already been established without using S.
Theories such as Peano arithmetic, for which any computably enumerable consistent extension is incomplete, are called essentially undecidable or essentially incomplete.
Relationship with computability
The incompleteness theorem is closely related to several results about undecidable sets in recursion theory.
Stephen Cole Kleene (1943) presented a proof of Gödel's incompleteness theorem using basic results of computability theory. One such result shows that the halting problem is undecidable: there is no computer program that can correctly determine, given a program P as input, whether P eventually halts when run with a particular given input. Kleene showed that the existence of a complete effective theory of arithmetic with certain consistency properties would force the halting problem to be decidable, a contradiction. This method of proof has also been presented by Shoenfield (1967, p. 132); Charlesworth (1980); and Hopcroft and Ullman (1979).
Franzén (2005, p. 73) explains how Matiyasevich's solution to Hilbert's 10th problem can be used to obtain a proof to Gödel's first incompleteness theorem. Matiyasevich proved that there is no algorithm that, given a multivariate polynomial p(x 1 , x 2 ,...,x k ) with integer coefficients, determines whether there is an integer solution to the equation p = 0. Because polynomials with integer coefficients, and integers themselves, are directly expressible in the language of arithmetic, if a multivariate integer polynomial equation p = 0 does have a solution in the integers then any sufficiently strong theory of arithmetic T will prove this. Moreover, if the theory T is ω-consistent, then it will never prove that a particular polynomial equation has a solution when in fact there is no solution in the integers. Thus, if T were complete and ω-consistent, it would be possible to determine algorithmically whether a polynomial equation has a solution by merely enumerating proofs of T until either "p has a solution" or "p has no solution" is found, in contradiction to Matiyasevich's theorem. Moreover, for each consistent effectively generated theory T, it is possible to effectively generate a multivariate polynomial p over the integers such that the equation p = 0 has no solutions over the integers, but the lack of solutions cannot be proved in T (Davis 2006:416, Jones 1980).
Smorynski (1977, p. 842) shows how the existence of recursively inseparable sets can be used to prove the first incompleteness theorem. This proof is often extended to show that systems such as Peano arithmetic are essentially undecidable (see Kleene 1967, p. 274).
Chaitin's incompleteness theorem gives a different method of producing independent sentences, based on Kolmogorov complexity. Like the proof presented by Kleene that was mentioned above, Chaitin's theorem only applies to theories with the additional property that all their axioms are true in the standard model of the natural numbers. Gödel's incompleteness theorem is distinguished by its applicability to consistent theories that nonetheless include statements that are false in the standard model; these theories are known as ω-inconsistent .
Minds and machines
Main article: Mechanism (philosophy)#Gödelian arguments
Authors including J. R. Lucas have debated what, if anything, Gödel's incompleteness theorems imply about human intelligence. Much of the debate centers on whether the human mind is equivalent to a Turing machine, or by the Church–Turing thesis , any finite machine at all. If it is, and if the machine is consistent, then Gödel's incompleteness theorems would apply to it.
Hilary Putnam (1960) suggested that while Gödel's theorems cannot be applied to humans, since they make mistakes and are therefore inconsistent, it may be applied to the human faculty of science or mathematics in general. Assuming that it is consistent, either its consistency cannot be proved or it cannot be represented by a Turing machine.
Avi Wigderson (2010) has proposed that the concept of mathematical "knowability" should be based on computational complexity rather than logical decidability. He writes that "when knowability is interpreted by modern standards, namely via computational complexity, the Gödel phenomena are very much with us."
Paraconsistent logic
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King Solomon
What is it with running around naked after a great find?
Same thing supposedly happened with George Smith, and Epic of Gilgamesh:
Then, in 1872, George Smith, a British Museum curator, realized that the tablet he was reading contained a Flood myth similar to the one he knew from the Bible. Smith, of course, had stumbled on what we now know as the dialogue between Gilgamesh and Utnapishtim. Although he knew nothing else of the Gilgamesh story (because nobody did), Smith was immediately so overwhelmed by this discovery that he laid the tablet down on the table, stripped off, and ran, seemingly half-naked, around the room, much to the astonishment of his fellow-scholars.
Not sure how well-documented that account is, but it sounds an awful lot like someone was having fun saying Smith acted like the wild-animal Enkidu before he was domesticated by Shamat, in a process any newlywed wife understands: after a week of sex, she made him bathe, wear clothes (covered his nakedness), eat proper human food AND taught him to drink beer (something to keep in mind, fellas, when wifey complains about the drinking: blame it on Shamat).
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james_woods
Quantum increases uncertainty, not decreases it.
Shroedinger's Quantum Cat joke-parable said that the Cat was neither alive nor dead, or else BOTH alive and dead.
Which itself is perhaps the ultimate paradox.
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frankiespeakin
James,
I find the fact that or psyche can rise about a paradox of absolutes and see the distinction an indication that our psyche is occurring at a quantum and classical phsysics level were it can jump back and forth between absolutes and less absolutes.
Creativity often being the results and with it a sort of elations of varying degrees.
