The Joy of Sets

Posted by jrosenhouse on September 13, 2012
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I have a guest post up over at the blog of Oxford University Press, discussing a few amusing tidbits from set theory. The post was inspired by this earlier post, in which I mentioned the bizarre criticism of set theory served up by a publisher of Christian home schooling materials.

In my new post I discuss Russell’s paradox, another classic paradox whose name I do not know. and finish with a set-theory-based “proof” that all counting numbers are interesting. (More precisely, I show that the set of boring counting numbers is empty.) Enjoy!

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Comments

  1. #1 Glenn Branch
    September 13, 2012

    The one whose name you don’t know is the Berry paradox (first discussed in print by Russell, I believe).

  2. #2 Jason Rosenhouse
    September 13, 2012

    Thanks for the information!

  3. #3 Thony C.
    September 14, 2012

    The Russell paradox should be correctly called the Zermelo-Russell paradox.

  4. #4 Blaine
    September 14, 2012

    Fallacies of self-reference…that’s why it’s never polite to talk about oneself.

    Type theory and the Lambda calculus were created to solve this problem. As Godel said, set theory and the theory of types “permits the derivation of modern mathematics and at the same time avoid all known paradoxes.”

  5. #5 melior
    September 15, 2012

    Fun with fuzzy definitions!
    Here are some more, see if you can spot the paradoxes just waiting to be assigned someone’s name. Claim yours today!
    Sa == The set of all ideas too abstract to be included as an element of a set without resulting in either that set or its complement leading to a self-referential paradox.
    Ra == The set of all ideas just abstract enough that every living person except Bertrand Russell considers them insufficiently precise to be enumerated as a member of a set.
    Ca == The set of animal shapes visible in the clouds in the sky.
    ~Ca = The complement of that set.

  6. #6 James Sweet
    September 16, 2012

    Small criticism: Writing for a general audience like this (or probably for any audience, but more crucially for a general audience), I think it’s advisable to analogize Russell’s paradox to “the barber who shaves every man in town who doesn’t shave himself”. Speaking solely for myself, when I first read Russell’s paradox, I was like, “Wait, wut?”, and then read the barber example, and I immediately understood both. I would have gotten Russell’s paradox eventually, but the barber thing made it a lot easier to latch onto.

  7. #7 Kevin
    September 20, 2012

    I apologize if commenting was intentionally closed on the last post, but I am unable to add to it. I wish to comment that, even as a non-materialist, I do not see the issue with using the word “crisis” in the context of evolutionary theory. The word is used in mathematics, but it does not make people doubt whether they have received the right change.

    The question should be: is evolutionary theory sufficiently strong that like math it has ineluctably become part of general knowledge, regardless of current academic developments? An anecdotal answer would be that, while it is possible to become, say, a successful rocket scientist without subscribing to evolutionary theory, the same is not true of mathematics.