Survey of Set Theory

Set theory is the standard foundational discipline for mathematics, with sets being used to define number, functions, structures like groups and topological spaces, etc.

In this discussion, we will review the axioms and basic properties of:

  • Zermelo-Frenkel - the standard axiomization of the theory of sets
  • NBG - ZF extended to include "very large" classes
  • Morse-Kelley - An NBG-like system with a slightly stronger comprehension axiom
  • NF - an unorthodox theory that disproves choice, proves infinity, is non-wellfounded, and generally nuts
  • ML - A strengthening of NF with proper classes, with which NF is equiconsistent

Discussion will begin with a brief discussion of set theory's origins, aims, and basic concepts, followed by a quick recap of Russel's and Cantor's paradoxes. After that we will move on to discuss specific axiom systems (primarily ZF and NF, and their variants as time permits) and how they avoid the paradoxes and provide the sets important to mathematics (natural numbers, ordered pairs, and relations). Where possible, fundamental differences in how the theories handle their subject matter will be discussed.

If we manage to run through the above, I hope to include some brief comments on large cardinal axioms, and certain oddities in the relationship between NF and ML.

Note: I'm not going to lie, I totally heart NF and will probably be a little heavier in coverage on that system.

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  • Jon C.

    I enjoyed it also. One question I would have is why bother trying to formulate the nature numbers in terms of sets containing the empty set rather than just positing new objects, like category theory does.

    April 8, 2013

    • Sister E.

      Well, category theory doesn't have a canonical ontology, so it has no unique account of the natural numbers. The reason you don't use something like a set of urelements for the natural numbers is A) it's inelegant to posit a new type of object if only one will suffice, and B) that urelements weaken extensionality (otherwise your urelements are just the empty set), and the wikipedia article on NF shows how much they weaken it in the case of NF+urelements (NFU). I would flip the question and ask "why add new objects to the theory when you've got everything you need in terms of pure sets?"

      Interestingly, Quine gets around the issue of urelements by suggesting the use of sets that are their own sole member (or that satisfy x = {x}) which have the advantage of not tampering with extensionality. These sorts of sets are, unsurprisingly, called "Quine atoms".

      April 8, 2013

    • Sister E.

      (Though even in the case of Quine atoms, supposing that you have a set full of Quine atoms that meets the right criteria would need to be added as a new axiom, as in the case of urelements. Again, there's elegance to being spare about both our ontology and our axioms, as the VN or Frege numbers are provided for by our other axioms.)

      April 8, 2013

  • Nile

    Thanks for the great overview, Alice!

    April 8, 2013

    • Sister E.

      You're welcome, and thank all y'all for the turnout, and for tolerating my hate of curly braces! I had a lot of fun!

      April 8, 2013

  • Raghu

    Aah Looks like I missed it :( I will look forward for the future events!

    April 7, 2013

  • Sister E.

    If anyone was really hoping to find out a lot about MK, you may be disappointed. I'm probably going to cover it very briefly, since I can't find a whole lot about it. It's probably best to construe this as a compare and contrast between ZF and NF as a pair of theories that violate each others' basic intuitions.

    March 17, 2013

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