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Full Idea
The paradox of the largest cardinal ('Cantor's Paradox') says the diagonal argument shows there is no largest cardinal, but the class of all individuals (including the classes) must be the largest cardinal number.
Gist of Idea
Paradox: there is no largest cardinal, but the class of everything seems to be the largest
Source
Shaughan Lavine (Understanding the Infinite [1994], III.5)
Book Ref
Lavine,Shaughan: 'Understanding the Infinite' [Harvard 1994], p.62
Related Ideas
Idea 15917 Paradox: the class of all ordinals is well-ordered, so must have an ordinal as type - giving a bigger ordinal [Lavine]
Idea 11015 Cantor's diagonal argument proved you can't list all decimal numbers between 0 and 1 [Cantor, by Read]
| 18173 | Cardinality strictly concerns one-one correspondence, to test infinite sameness of size [Cantor, by Maddy] |
| 14138 | You can't get a new transfinite cardinal from an old one just by adding finite numbers to it [Russell] |
| 14140 | For every transfinite cardinal there is an infinite collection of transfinite ordinals [Russell] |
| 18200 | Very large sets should be studied in an 'if-then' spirit [Putnam] |
| 10480 | First-order logic can't discriminate between one infinite cardinal and another [Hodges,W] |
| 18175 | For any cardinal there is always a larger one (so there is no set of all sets) [Maddy] |
| 18172 | Infinity has degrees, and large cardinals are the heart of set theory [Maddy] |
| 18196 | An 'inaccessible' cardinal cannot be reached by union sets or power sets [Maddy] |
| 17890 | There are at least eleven types of large cardinal, of increasing logical strength [Koellner] |
| 15918 | Paradox: there is no largest cardinal, but the class of everything seems to be the largest [Lavine] |