Definition talk:Universe (Set Theory)

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Hello,

I just stumbled across this wiki accidentally, and I'm currently writing up notes on this very subject. So I thought I would discuss some rather esoteric stuff that I didn't think would be worth writing on the main page of the wiki.

First, there is no definition of a "universe" for sets...it's just a fuzzy, emotional concept. I mentioned that Michael Shulman more or less defined it as a model for ZFC set theory. This is probably the best definition to have. Why? Well, there are many different notions of a universe.

For example, there is a Grothendieck universe which consists of a "nice set" (yes, a God honest set!) which has the axioms one would expect from the universe. There are many different Grothendieck universes (the empty set, the set of finite ordinals, etc). The only place this is really discussed in any detail is

  • Bourbaki, Nicolas (1972). "Univers". In Michael Artin, Alexandre Grothendieck, Jean-Louis Verdier, eds. (in French). Séminaire de Géométrie Algébrique du Bois Marie - 1963-64 - Théorie des topos et cohomologie étale des schémas - (SGA 4) - vol. 1 (Lecture notes in mathematics 269). Berlin; New York: Springer-Verlag. pp. 185–217.

Then there is the von Neumann universe. This formalizes the intuition of the "class of all sets" in NBG set theory, I think. It is constructed recursively, and I will refer the interested reader to Wikipedia for more details---it is more or less accurate, but one should be cautious with Wikipedia's statement "This collection, which is formalized by Zermelo–Fraenkel set theory (ZFC)..." since I think they mean the collection consists of all sets satisfying the ZFC axioms.

At any rate, there is a third universe (Godel's universe, a.k.a. the constructible universe) which is constructed out of simple sets. I confess I don't know much about it, nor do I care enough to learn about it.

Second, with the extension of ZFC conservatively to include classes results in von Neumann–Bernays–Gödel set theory (NBG). The best reference for this is:

  • Mendelson, Elliott, 1997. An Introduction to Mathematical Logic, 4th ed. London: Chapman & Hall. ISBN 0412808307. Pp. 225–86 contain the classic textbook treatment of NBG, showing how it does what we expect of set theory, by grounding relations, order theory, ordinal numbers, transfinite numbers, etc.

It is the basic "follow your nose" generalization of most concepts after a rather cute trick: the basic item we study is a class. But if a class is a member of another, then it is a set. So we usually refer to a "proper class" to mean a class that is not a set. Thus we may have the proper class of all sets, which is conventionally referred to as the "universe" in NBG set theory.

The only exception is the axiom for comprehension: we do not permit quantifiers over classes. Why? Because that would basically enable statements involving the "universe of classes".

We also demand that a class is a set if and only if it is not bijective with the proper class of all sets. This axiom implies the axiom of global choice (equivalently: the class of all sets is well-ordered) because the class of ordinals is not a set; hence there exists a bijection between the ordinals and the universe.

So if you think about it for a minute, this proper class of all sets is a model for ZFC set theory and thus really does deserve the name "universe".

At any rate, that's all I have to say about that.

Best, Pqnelson 21:12, 15 October 2011 (CDT)