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Math Help - Set Theory Question

  1. #1
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    Set Theory Question

    I've started reading the book Set Theroy and Logic by Robert R. Stoll (Dover edition) to keep myself active over the summer break.

    I came accross a weird problem (problem 2.1) early on. It reads as follows:

    "Try to devise a set which is a member of itself."

    I gave it a good college try, but I still cant understand this problem. The way its framed with "try" is starting to make me suspicious if this can be done at all. What do you guys think about this problem?
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    This is very famous paradox. It is known as Russell’s Paradox. It is named for Bertrand Russell who found it as a counter example to the work of Gottlob Frege. Russell’s famous question to Frege was “Can a set of teaspoons be a teaspoon?”

    On the “set of sets” let \Omega  = \left\{ {A:A \not\subset A} \right\}.
    Now for the paradox: Is it true that \Omega  \in \Omega ?
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    What Plato said is an example to the above.
    Another example is "set of all sets".

    However, the problem is these two examples are not "sets".
    Axiomatically these two sets are not allowed to be constructed.

    One can ask a question if it is possible to axiomatically construct a set which is an element of itself? It was shown that it is impossible to construct (nor disprove) a set which is an element of itself using Zermelo-Frankael-Choice axioms (of course dropping the Axiom of Foundation). Thus, if you are looking for an example which is not really a set, then you can use Plato's or my example. However, if you are looking for a real set, then it is not possible to find such a set.
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    Thanks for the help Plato and Hacker. This problem made for a good dinner time discussion, but we couldn't think up those examples. This books approach in sections 1 and 2 of chapter 1 has been to introduce "Cantor's Concept of a set" (name of section 1). I didn't realize there was another approch where these paradoxes do not live. In Stoll's own words, heres a part of the introduction to chapter 1:

    "Specifically, this chapter discusses, within the framework of set theory, three important mathematical concepts: function, equivalence relation, and ordering relation. Sections 3—6 contain the necessary preliminaries, and Sections 1 and 2 describe our point of departure for Cantor’s theory.

    One might question the wisdom of choosing a starting point which is known to lead ultimately to disaster. However, we contended that the important items of this chapter are independent of those features which characterize the Cantorian or “naive” approach to set theory. Indeed, any theory of sets, if it is to serve as a basis for mathematics, will include the principle definitions and theorems appearing in this chapter. Only the methods we employ to obtain some of these results are naive. No irreparable harm results in using such methods; they are standard tools in mathematics."
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    Quote Originally Posted by Jacobsen View Post
    I didn't realize there was another approch where these paradoxes do not live.
    When Georg Cantor first introduced set theory there been problems with it. Like the above examples. Mathematicians made set theory formal, by making it axiomatic. This got rid of all the problems it had. And as a result it became a foundation of mathematics. Kronecker's old quotation, "God made the integers, all else is the work of men", no longer applied. Now it was more like, "God made the sets, all else is the work of men".

    Was your screen name inspired by the Jacobson? As in Basic Algebra?
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    Thanks Hacker, that is very interesting. I'm not familiar with analysis so its hard for me to catch on to many of these subtleties.

    My screen name is my last name. I couldn't come up with anything creative when I singed up here.
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    Quote Originally Posted by Jacobsen View Post
    I've started reading the book Set Theroy and Logic by Robert R. Stoll (Dover edition) to keep myself active over the summer break.

    I came accross a weird problem (problem 2.1) early on. It reads as follows:

    "Try to devise a set which is a member of itself."

    I gave it a good college try, but I still cant understand this problem. The way its framed with "try" is starting to make me suspicious if this can be done at all. What do you guys think about this problem?
    An informal, non-mathematical, example of this is that "the set of all abstract concepts" is itself an abstract concept. However, as others have pointed out, a mathematically consistent definition of a set has to exclude the possibility of a set belonging to itself.
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    Quote Originally Posted by Jacobsen View Post
    I've started reading the book Set Theroy and Logic by Robert R. Stoll (Dover edition) to keep myself active over the summer break.

    I came accross a weird problem (problem 2.1) early on. It reads as follows:

    "Try to devise a set which is a member of itself."

    I gave it a good college try, but I still cant understand this problem. The way its framed with "try" is starting to make me suspicious if this can be done at all. What do you guys think about this problem?
    Hope you continue to read Stoll.
    I'll borrow (with slight modification) comments made by K.Devlin.

    Consider (in a set-theoretical context), modelling items of information in an information-storage device, say for example Stoll's textbook.

    Suppose somewhere burried deep in the text, Stoll actually wrote, "Let A be the set of all sets explicitly referred to in this book". It should be clear that A itself has now been referred to in this book.

    Then we have AeA ('e' denoting set membership).

    More generally, a little thought will give closed loops of set membership (usually referred to as circular), like for example, AeBeCeDeA.

    Of course, Foundation (in ZF) blocks formation of circular sets, or sets having themselves as members. But if you have the time, you might explore what's been called non-well-founded set theory. The work of P.Aczel would be a good place to start.

    Now in light of the above, can you reconsile the following words of J.Lewin.
    (Remember avoidance does not imply non-existence.)

    Mathematicians prefer to avoid sets that are members of themselves. They feel that if we are going to collect some objects together to make a set A, then all of those objects ought to be well known to us before we collect them together. Now what if one of those objects is the set A itself? We would need to know what the set A is even before we have collected its members together to define it.
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    Thanks for those neat examples Piper and Oplag. I really like that quote from J.Lewin. It seems like one of the weird things with sets which contain themselves as members is that you can end up with something that looks like


    A = {x1, x2, ..., A = {x1, x2, ..., A = {x1, x2, ..., A = {...} ..., xn}..., xn} ..., xn}

    which would be a set that contains an infinite amount of sets even when A was thought to be finite. Is this a misguided interpretation or would that be one of the kind of objects we would look at in non-well founded set theory?
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    Quote Originally Posted by Jacobsen View Post
    Thanks for those neat examples Piper and Oplag. I really like that quote from J.Lewin. It seems like one of the weird things with sets which contain themselves as members is that you can end up with something that looks like


    A = {x1, x2, ..., A = {x1, x2, ..., A = {x1, x2, ..., A = {...} ..., xn}..., xn} ..., xn}

    which would be a set that contains an infinite amount of sets even when A was thought to be finite. Is this a misguided interpretation or would that be one of the kind of objects we would look at in non-well founded set theory?
    The notation is a bit confusing, but maybe you're thinking about the particular infinite descending e-chain, AeAeA... .
    Removing Foundation from ZF permits the existence of such sets. They would be objects of study in Aczel's theory.

    The formulation of the theory was driven in part by computer science (theoretical and applied), where self-referencial structures are everywhere.

    To me, what's more interesting is the approach to its development, as compared to the approach of a standard treatment (e.g., ZF). Loosely, you could say that they start at opposite ends of a spectrum. ZF is one of synthesis, Aczel's is one of analysis. One starts with "pure sets" and builds the structures of maths; the other starts with the structures of maths and systematically peals off structure until all that's left is a "smile".

    No reason to expect that the two approaches will arrive at the "same" theory of sets.
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