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Vladimir_Nesov comments on Contrived infinite-torture scenarios: July 2010 - Less Wrong
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Comments (188)
Countable infinities are no fun.
There's more than one countable infinity?
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?
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.
Somewhere in this thread there's been a mix-up between countable and uncountable.
There's only one countable infinity (at least if you're talking about cardinal numbers), and it's much more fun than the uncountable infinities (if by ‘fun’ you mean what is easy to understand). As Sniffnoy correctly states, there are many, many uncountable infinities, in fact too many to be numbered even by an uncountable infinity! (In the math biz, we say that the uncountable infinities form a ‘proper class’. Proper classes are related to Russell's Paradox, if you like that sort of thing.)
Compared to the uncountable infinities, countable infinity is much more comprehensible, although it is still true that you cannot answer every question about it. And even if the universe continues forever, we are still talking about a countable sort of infinity.