I am trying to define an embedding whose range includes classes. Is there a coherent way of assigning "cardinality" to proper classes?
1 Answer
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Erin, there is no need to do this. I do not know of any practical reasons for doing it. And, of course, "cardinality" has to be properly interpreted to make some sense of the word.
In extensions of set theory where classes are allowed (not just formally as in ZFC, but as actual objects as in MK or GB), sometimes it is suggested to add an axiom (due to Von Neumann, I believe) stating that any two classes are in bijection with one another. Under this axiom, the "cardinality" of a proper class would be ORD, the class of all ordinals. (By the way, by class forcing, given any proper class, one can add a bijection between the class and ORD without adding sets, so this assumption bears no implications for set theory proper.)
Without assuming Von Neumann's axiom, or the axiom of choice, I know of no sensible way of making sense of this notion, as now we could have some proper classes that are "thinner" than others, or even incomparable. Of course, we could study models where this happens (for example, work in ZF, assume there is a strong inaccessible $\kappa$, and consider $V_\kappa$ as the universe of sets, and $Def(V_\kappa)$ in Gödel's sense (or even $V_{\kappa+1}$) as the collection of classes).
answered Oct 31, 2010 at 4:24
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1$\begingroup$ So one can simply force global choice? $\endgroup$– Asaf Karagila ♦Commented Mar 8, 2012 at 8:37
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4$\begingroup$ Yes, one can force global choice. Andres's assertion that this won't add sets and is therefore conservative over set theory presupposes, of course, local choice in the set theory. $\endgroup$ Commented Mar 8, 2012 at 14:20
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2$\begingroup$ Just to drive the point home, if choice fails, it may not be possible to force it back. For example, one cannot force (local) choice over Gitik's model where all (well-ordered) cardinals have cofinality $\omega$. $\endgroup$ Commented Mar 8, 2012 at 14:54
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$\begingroup$ But, if we extend the definition of cardinal to bijections among elements of two entities (not necessarily sets, but also proper classes), why can't we define the cardinality of the proper class of the set of all sets? It would be useful in the sense that now you do have the largest possible cardinal number (it would not be inconsistent with cantor's theorem because the proper class in not a set so there is not such thing a the power set of a class). Why is this not useful? $\endgroup$ Commented Mar 30, 2013 at 16:24
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$\begingroup$ Do this "largest" cardinal have a name? $\endgroup$ Commented Mar 30, 2013 at 16:26