There are many foundations to set theory, ZFC, NBG, SEAR, to name a few, and while they differ in how sets, classes, and higher-order collections are represented as mathematical objects, they all attempt to provide useful tools to mathematicians without being glaringly inconsistent.

Note that many set theories stop at the set, or only weakly partially define classes. But for those that go further, I have the following question:

What kinds of operations are well-defined when working with sets, classes, conglomerates, and yet higher order collections? Or, what kinds of operations are commonly assumed to be well-defined in fields such as category theory, etc.?

Here are some operations which I am curious about.

1) Applying the axiom of choice to the class of all sets, or to the conglomerate of all classes, and so on.

2) Given two classes $A$ and $B$, forming the set of classes $\{A,B\}$. Similarly for sets of conglomerates, etc. Of course, the set of all classes should be disallowed by Russell's paradox.

3) Given two classes $A$ and $B$ and a set $C$, make a "set-class function" mapping $C$ into $A$, a "class-class function" mapping $A$ into $B$, and a "class-set function" mapping $A$ into $C$. Similarly for any pair of higher-order collections of any type.

4) Regarding the cardinality of a set, class, conglomerate (etc) as a way to make equivalence classes of sets, classes, conglomerates (etc), respectively.

5) Regarding the cardinality of sets, classes, conglomerates, and higher-order collections as a way to define an equivalence class among all mathematical objects.

6) Forming the set of all "class-class functions" (or is it a class?)

I'm the least confident that #'s 5-6 will be acceptable, but the rest seem reasonable to my untrained eyes.

  • $\begingroup$ The axioms are what you say they are. $\endgroup$ – Bruce Westbury Jan 28 '13 at 7:34
  • $\begingroup$ Whoops, you're right. It's probably too late for me to be asking this. Editing question to clarify that by "axiomatic" i mean "implied by the axioms" $\endgroup$ – Benjamin Braun Jan 28 '13 at 7:35
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    $\begingroup$ Indeed. The formalism puts restrictions on what we can do. At any rate, in set theory, there are no "mathematical objects"; there are only sets (and possibly classes if you work in a theory like NBG). If you want to be able to treat proper classes like any other "small" mathematical object, then probably you should use something like Grothendieck universes... $\endgroup$ – Zhen Lin Jan 28 '13 at 8:25
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    $\begingroup$ See this question I asked at m.SE: math.stackexchange.com/questions/231087/… $\endgroup$ – David Roberts Jan 28 '13 at 8:58
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    $\begingroup$ @Benjamin No. TG set theory actually gives you a whole infinite hierarchy of "higher-order collections", because you can redefine "set" to mean a member of a fixed universe. $\endgroup$ – Zhen Lin Jan 28 '13 at 9:41

This question is considered in detail in Mike Shulman's Set Theory for Category Theory.

The "big picture" way to think about it is to think that each time you do a power-set-type operation, you are constructing an object of a "higher order". You just then need to postulate high enough orders to do whatever it is you want. $U_0$ is the class of all sets, $U_1$ is the class of all classes of sets, etc.

Once you are forming classes of classes, etc., you are strictly outside what is provably consistent with ZFC, but set theorists routinely consider much, much stronger theories that do not seem to harbor contradiction. So it's probably okay.

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    $\begingroup$ It may very well be function 'classes' that cause one to go to higher orders, because one usually has a subobject classifier $2$, and so given pullbacks any mono of sets/classes/etc is the pullback of $true:1 \to 2$. Without function 'classes' one cannot give the power 'class'$\lbrace X \to 2\rbrace$, but the problem is not restricted to this construction. $\endgroup$ – David Roberts Jan 28 '13 at 23:15
  • $\begingroup$ You're right, function classes is probably a better way to put it. $\endgroup$ – arsmath Jan 29 '13 at 10:23

Regarding ZFC, I don't see any way to make sense of your questions; the language of ZFC doesn't allow you to mention anything called a "class", a "conglomerate" or a "higher order collection".

  • $\begingroup$ Yes, editing question to clarify that I'm interested in the more modern theories, NBG and SEAR, etc, which were developed (as far as I can tell) to provide better support for classes and higher order collections. $\endgroup$ – Benjamin Braun Jan 28 '13 at 7:26
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    $\begingroup$ Benjamin, NBG is not much older than ZF. $\endgroup$ – Asaf Karagila Jan 28 '13 at 9:01
  • $\begingroup$ (Err, the other way around, of course, ZF predates NBG - but not by much...) $\endgroup$ – Asaf Karagila Jan 28 '13 at 11:57

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