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James_K comments on Contrived infinite-torture scenarios: July 2010 - Less Wrong
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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.