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Full Idea
For every transfinite cardinal there is an infinite collection of transfinite ordinals, although the cardinal number of all ordinals is the same as or less than that of all cardinals.
Gist of Idea
For every transfinite cardinal there is an infinite collection of transfinite ordinals
Source
Bertrand Russell (The Principles of Mathematics [1903], §290)
Book Ref
Russell,Bertrand: 'Principles of Mathematics' [Routledge 1992], p.312
A Reaction
Sort that one out, and you are beginning to get to grips with the world of the transfinite! Sounds like there are more ordinals than cardinals, and more cardinals than ordinals.
| 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] |