Talk:Cantor's paradox

message: I've given this article a general cleanup, but I think it could use a little more:
# The formal statement of Cantor's paradox isn't so great.
# Some context explaining why Cantor's paradox is significant would be very useful here.
If we can fix those two things then I think the  tag can be taken away. Tim Pierce 06:30, 12 December 2005 (UTC)
replies: text: I've done a major revision on the page and I think I've addressed your concerns, so I'm excising the  tag. Ryan Reich 17:16, 21 January 2006 (UTC)
thread_title: Need more context

message: "The fact that NBG set theory resolves the paradox is therefore a point in its favor as a suitable replacement."
What is NBG set theory?  This should either be a link, or it should be removed.  Surely axiomatic set theory is the standard 'replacement' for naive set theory anyway. InformationSpace 00:38, 4 May 2007 (UTC)
replies: text: Von Neumann–Bernays–Gödel set theory, but, as well-known, ZFC also solves the paradox in the same manner, and you can deal with classes in ZFC (but not as objects of the theory). The sentence has to be removed. 81.57.95.102 12:41, 8 May 2007 (UTC)
thread_title:  NBG set theory 

message: Although there is a connection between Cantor's paradox and the Burali-Forti paradox, the latter isn't necessary to understand the former. Indeed, you don't need the ordinals to verify that the cardinals constitute a proper class. I've revised the article to make this clear. (Lorenzo Traldi (talk) 13:06, 10 April 2011 (UTC)
thread_title:  Ordinals and cardinals 

message: I think this article would be better if it included a single statement of Cantor's paradox. According to Britannica, "Cantor's paradox" refers to the fact that a hypothetical universal set U would be both larger and strictly smaller than its power-set. That is, the paradox is in the juxtaposition of the inequalities |U|<2U from Cantor's theorem and |U|>=2U from the fact that U is universal. (Naturally we conclude that such a set U cannot exist, just as we do with Russell's paradox.) I hesitate to make such a major edit without hearing from an author though. (Lorenzo Traldi (talk) 17:34, 10 April 2011 (UTC)
replies: text: Exactly. I found the article confusing too, without a statement of the paradox itself and had to consult wolfram mathworld. voidnature 06:07, 16 June 2011 (UTC)
replies: text: I agree. According to the article, the fact that there is no greatest cardinal is the "paradox". This in itself is not a paradox at all, and I'm extremely sceptical that Cantor or anybody else would have considered it such - the fact that there is no greatest natural number does not cause concern, so why would this? There is only a paradox when this is considered in conjunction with a contemplation of the "set of all sets".--86.178.22.211 (talk) 17:01, 19 October 2013 (UTC)

text: Nine (!) years later this has not yet been fixed :-( I'm hesitant to edit the article myself, but I really think the actual paradox should be stated. 2A02:8109:A380:B58:49F3:8809:DDC6:7CBE (talk) 15:47, 4 June 2020 (UTC)
thread_title:  What's the paradox? 

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thread_title:  External links modified 

message: "Thus, not only are there infinitely many infinities, but this infinity is larger than any of the infinities it enumerates."
I bet it's infinitely larger, too!  am I right?
92.23.191.106 (talk) 13:31, 3 April 2017 (UTC)
thread_title:  Makes perfect sense... 

message: Looking online I'm finding two different definitions of Cantor's paradox:
# The set of all infinite cardinals is not a set.
# The set of all sets is not a set.
They are very similar, and indeed, closely related. But which one is the official paradox that we want to discuss?  — Preceding unsigned comment added by 65.92.68.49 (talk) 13:41, 6 April 2020 (UTC)
thread_title:  Which is it?