This seems a bad way to think about things—except maybe for someone who’s just been introduced to formal set theory—especially as proper classes are precisely those classes that are too big to be sets.
This isn’t right—aleph numbers are indexed by all ordinals, not just natural numbers. What’s equivalent to AC is that the aleph numbers cover all infinite cardinals.
Yes, there are, but there is only one Eliezer Yudkowsky.
ETA: (This was obviously a joke, protesting the nitpick.)
Has anyone counted how many uncountable infinites there are?
No, because there’s an uncountable infinity of uncountable infinities.
Not than anyone could have actually counted them even were there a countable infinity of them.
The class of all uncountable infinities is not a set, so it can’t be an uncountable infinity.
This seems a bad way to think about things—except maybe for someone who’s just been introduced to formal set theory—especially as proper classes are precisely those classes that are too big to be sets.
Doesn’t the countable-uncountable distinction, or something similar, apply for proper classes?
As it turns out, proper classes are actually all the same size, larger than any set.
Thanks for the correction :)
No. For example, the power set of a proper class is another proper class that is bigger.
No, the power set (power class?) of a proper class doesn’t exist. Well, assuming we’re talking about NBG set theory—what did you have in mind?
oops...I was confusing NBG with MK.
M, I don’t know anything about MK.
Zermelo and Cantor did. They concluded there were countably many, which turned out to be equivalent to the Axiom of Choice.
This isn’t right—aleph numbers are indexed by all ordinals, not just natural numbers. What’s equivalent to AC is that the aleph numbers cover all infinite cardinals.