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Full Idea
The stunning discovery that infinity comes in different degrees led to the theory of infinite cardinal numbers, the heart of contemporary set theory.
Gist of Idea
Infinity has degrees, and large cardinals are the heart of set theory
Source
Penelope Maddy (Naturalism in Mathematics [1997], I.1)
Book Ref
Maddy,Penelope: 'Naturalism in Mathematics' [OUP 2000], p.16
A Reaction
It occurs to me that these huge cardinals only exist in set theory. If you took away that prop, they would vanish in a puff.
| 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] |