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Math Help - Russel Paradox

  1. #1
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    Russel Paradox

    Can someone answer this:

    What the Russell Paradox is (a complete proof that at some point you will want to prove that certain assumption leads to a contradictiion). And then Explain why it is so bad that a contradiction can be proved somewhere.( why ncant we just ignore the fact that in some remote corner of set theory this contradiction arises?), And Finally my 3rd question abotu russel paradox iii) Assuming a Statement B, which happens tobe a contradiction, prove that 6=21.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by ruprotein View Post
    Can someone answer this:

    What the Russell Paradox is (a complete proof that at some point you will want to prove that certain assumption leads to a contradictiion).
    See this.

    RonL
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  3. #3
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    what if i asked u the last question, i kinda know what russell paradox is, its a simple theorem im trying to answer this questiont hough

    or the question Assuming a statement C, which happens to be a contradiction prove that 2=7
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  4. #4
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    The article referred to by CaptainBlack is nice. I will however try to discuss it in easier, if perhaps more imprecise terminology.

    Frege's original conception of a set was essentially as follows: anything you can describe is a set. The set of all natural numbers. The set of all elephants. The set of all tea cups. The set of all things that are green. etc, etc.

    So, let R denote "the set of all sets which are not members of themselves." There is nothing a priori in Frege's original conception which would disallow such a thing. Now ask the following question: is R a member of itself?

    However you try to answer this question leads to an inevitable contradiction. Suppose R is a member of itself. Then, by definition, R is not a member of itself (since it contains precisely those sets which are not members of themselves). Conversely, if R is not a member of itself, then by definition it is a member of itself. There is the paradox.

    Now you ask, what is so bad about a paradox in a formal system? This is called inconsistency. And the problem is, you can prove (and disprove) absolutely anything once you've encountered one.

    Suppose you have a statement A, and that you've established the contradiciton that both A and ~A (read NOT A) are true. Let P be any other statement. The statement "A or ~A" is a tautology. Thus "~P implies (A or ~A)" is also a tautology. This is equivalent to, by contraposition, "~(A or ~A) implies P". But "~(A or ~A)" is equivalent to "A and ~A". Thus we have "(A and ~A) implies P". But "A and ~A" is a true statement! And since "(A and ~A) implies P", we must conclude that P is true.

    So we see that a paradox is never self-contained. Once you have both a statement and its negative being true, you can prove absolutely any other statement (and its negative).
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  5. #5
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    It reminds me of this.....
    "In a town of Seville the barber shaves everyone who does not save himself. Who shaves the barber".

    ---

    BubbleBrain I have a question.

    In axiomatic set theory the Russel paradox is settled?
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  6. #6
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    2=7

    ok well nop1 seemed to answer this so maybe u cansee if my proof is correct ore there is some mistake somewhere
    PROOF.
    Step 1. let p(x) be the statement "2=7"
    step 2. then let S be the set such that S = {x: p(x)}
    s3. Assume p(x)
    s4. Then p(x) is such an x such that the set S is true, but we know 2 doesnt equal 7. Then S is not true which results in a contradiction. This is an example of the Russel Paradox.
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  7. #7
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    Brain_103 gave you a very clear discussion of this problem.
    I will just add a few comments.
    If you have access to a mathematics library find NAÏVE SET THEORY by Paul Halmos. That book has one of the clearest discussions of the problem. As Halmos shows that with proper axioms one can prove that “nothing contains everything”. In other words, the whole problem is one of extensionality just as the Brain said.

    To answer TPH’s question: paradoxes are never settled. We just improve the axioms to avoid the problem. In the newer axiom systems, most use ZF or Quine, both avoid the Russell paradox.

    Having read you question a number of times, I still am unsure about #3.
    The sentence, If 6 \in \emptyset then 6 = 21 is a true statement.
    You see a of property of implication is: A false statement implies any statement.
    Thus, if we allow any paradox to remain in the system then any is true.
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  8. #8
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    Quote Originally Posted by Plato View Post

    To answer TPH’s question: paradoxes are never settled. We just improve the axioms to avoid the problem. In the newer axiom systems, most use ZF or Quine, both avoid the Russell paradox.
    I think I understand what you said. I just want to be sure on it. You are saying that the paradoxes are not settled but they do not arise because the ZFC axioms forbid such a set to exist. Thus, we did not settle the paradox rather we just ignore it and it never bothers us because of the consistency of the axioms?
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  9. #9
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    Quote Originally Posted by ThePerfectHacker View Post
    the paradoxes are not settled but they do not arise because the ZFC axioms forbid such a set to exist.
    Yes that is correct. We do not bother because they will not arise.
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  10. #10
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    Quote Originally Posted by Plato View Post
    Yes that is correct. We do not bother because they will not arise.
    Sweet! I cannot wait the day I learn set theory.
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  11. #11
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    Quote Originally Posted by ThePerfectHacker View Post
    It reminds me of this.....
    "In a town of Seville the barber shaves everyone who does not shave himself. Who shaves the barber?".
    I have a simple answer to this.

    Barber's wife!

    Women are not shaving at all so the rule that "barber shaves everyone who does not shave himself" doesn't apply to them...

    I've got another futuristic answer...

    Robot!!!
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  12. #12
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by OReilly View Post
    I have a simple answer to this.

    Barber's wife!

    Women are not shaving at all so the rule that "barber shaves everyone who does not shave himself" doesn't apply to them...

    I've got another futuristic answer...

    Robot!!!
    There is a problem with the "woman" answer.

    The original statement of the paradox says:
    Suppose there is a town with just one male barber; and that every man in the town keeps himself clean-shaven: some by shaving themselves, some by attending the barber. It seems reasonable to imagine that the barber obeys the following rule: He shaves all and only those men who do not shave themselves.

    Under this scenario, we can ask the following question: Does the barber shave himself?
    The problem statement clearly specifies that the barber is a man.

    -Dan
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  13. #13
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    Quote Originally Posted by topsquark View Post
    There is a problem with the "woman" answer.

    The original statement of the paradox says:

    The problem statement clearly specifies that the barber is a man.

    -Dan
    I think my English was misunderstood.

    I didn't mean that barber is female.

    My answer was that barber is shaved by his wife.
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  14. #14
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by OReilly View Post
    I think my English was misunderstood.

    I didn't mean that barber is female.

    My answer was that barber is shaved by his wife.
    Ah, no. I just misunderstood you. However the barber shaves every man that doesn't shave himself. Thus if he doesn't shave himself, the barber (not the barber's wife) must shave him.

    -Dan
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  15. #15
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    Quote Originally Posted by topsquark View Post
    Ah, no. I just misunderstood you. However the barber shaves every man that doesn't shave himself. Thus if he doesn't shave himself, the barber (not the barber's wife) must shave him.

    -Dan
    I know, I was just joking.
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