more from this thinker | more from this text
Full Idea
Cantor's first innovation was to treat cardinality as strictly a matter of one-to-one correspondence, so that the question of whether two infinite sets are or aren't of the same size suddenly makes sense.
Gist of Idea
Cardinality strictly concerns one-one correspondence, to test infinite sameness of size
Source
report of George Cantor (works [1880]) by Penelope Maddy - Naturalism in Mathematics I.1
Book Ref
Maddy,Penelope: 'Naturalism in Mathematics' [OUP 2000], p.17
A Reaction
It makes sense, except that all sets which are infinite but countable can be put into one-to-one correspondence with one another. What's that all about, then?
| 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] |