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Full Idea
By the mid 1890s Cantor was aware that there could be no set of all sets, as its cardinal number would have to be the largest cardinal number, while his own theorem shows that for any cardinal there is a larger.
Gist of Idea
For any cardinal there is always a larger one (so there is no set of all sets)
Source
Penelope Maddy (Naturalism in Mathematics [1997], I.1)
Book Ref
Maddy,Penelope: 'Naturalism in Mathematics' [OUP 2000], p.17
A Reaction
There is always a larger cardinal because of the power set axiom. Some people regard that with suspicion.
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] |