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I'm a past-graduate in mathematics and familiar with the basics of ordinals and cardinals. My question is: how many infinities are there? There are obviously infinitely many, but since we already know that there are different kinds of infinity, the question remains for me: how many infinitely many? Is this a sensible question at all?

Pi o r
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  • A proper class of them, since we can produce distinct ones for every ordinal. – André Nicolas Aug 19 '14 at 16:15
  • Infinitely many :P – user5402 Aug 19 '14 at 16:17
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    The collection of all cardinals in ZFC forms a proper class and does not form a set. This means roughly speaking that it is larger than any set whose existence ZFC proves. As I recall the usual argument is that if $A$ were the set of all cardinals, then the power set of $A$ would be a cardinal which by Cantor's theorem is not in $A$. It is important that I am talking about ZF(C); in NF(U), the situation is rather different. – Ian Aug 19 '14 at 16:17
  • This is essentially asking how many ordinal numbers there are. But there can't be a number of ordinal numbers, or that number would itself be an ordinal number, larger than any other ordinal number, which is absurd. This is the Burali-Forti paradox. – MJD Aug 19 '14 at 16:22
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    http://math.stackexchange.com/questions/283581/how-many-different-sizes-of-infinity-are-there – user157227 Aug 19 '14 at 16:24
  • If the question is not how many different ordinals/cardinals are there, but rather how many different notions of infinity are there, then please let me know, and I'll reopen so the question can be closed as a different duplicate. – Asaf Karagila Aug 19 '14 at 16:28

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