Re: Proper classes

Dave Seaman wrote: 
Although it is possible to extend the notion of a "function" so that its
"domain" and "codomain" are allowed to be proper classes, my recollection
is that the concept is not particularly interesting from the viewpoint of
cardinality, since all proper classes would turn out to have the same
"cardinality" in this extended sense.

Not necessarily. Early attempts to formulate the limitation of size principle (by Jordan and others) failed precisely because there is no apparent way to conclude that e.g. the class of ordinals and V can be placed in bijection (this is equivalent to global choice). Without global choice you can have incomparable proper classes, as can be seen rather trivially: take a model of ZFC in which there is no definable well-ordering of V and add on top of it the definable subsets as classes and you'll have a model of NBG in which there is no bijection between V and the class of ordinals. I'll have to think a bit about MK, but I'd be (at least mildly) surprised if you could prove that all proper classes are the same size in it...

--
Aatu Koskensilta (aatu.koskensilta@xxxxxxxxx)

"Wovon man nicht sprechen kann, darüber muss man schweigen"
 - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
.

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