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Thread: Naive Set Theory

  1. #1
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    Naive Set Theory

    As the title suggests I've gone and treated myself to Halmos' little book.

    Before I discuss my problem I just want to point out a tiny detail that made me think a little. I was immediately introduced with the axiom of extension. I thought that this is a bit of an exotic name for an axiom? After a little research I realised that I can use set and extension interchangeably; watch this


    Axiom of sets: Two extensions are equal iff they have the same elements.

    Anyway, back to my main problem, Halmos states in chapter 1

    Suppose, for instance, that we consider human beings instead of sets, and that, if x and A are human beings, we write x is a member of A whenever x is an ancestor of A. (The ancestors of a human being are his parents, his parents' parents, their parents, etc., etc.) The analogue of the axiom of extension would say here that if two human beings are equal, then they have the same ancestors (this is the "only if" part, and it is true), and also that if two human beings have the same ancestors, then they are equal (this is the "if" part, and it is false).

    Now I breakdown his argument as follows

    1) If two human beings are equal, then they have the same ancestors (TRUE)

    2) If two human beings have the same ancestors, then they are equal (FALSE)

    Now, correct me if I'm wrong, but isn't 1) the "if" part and 2) the "only-if" part? I thought the "if" part is the forward-method and the "only-if" part the backward method. Has Halmos got it back-to-front here or is it me?
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  2. #2
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    Re: Naive Set Theory

    I don't see that Halmos did phrase that as "if and only if". If the statement is "two human beings they have the same ancestors if and only if they are equal" (with human being "A" equal to human being "B" meaning that "A" and "B" are names for the same person) then is true that the "if" part is "if they have the same ancestors then they are equal" (which is, as you say, is false- my brother and I have the same ancestors but are not the same person). And the "only if" part is "if they are equal then they have the same ancestor", which is true.
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