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Full Idea
Some of the standard large cardinals (in order of increasing (logical) strength) are: inaccessible, Mahlo, weakly compact, indescribable, Erdös, measurable, strong, Wodin, supercompact, huge etc. (...and ineffable).
Gist of Idea
There are at least eleven types of large cardinal, of increasing logical strength
Source
Peter Koellner (On the Question of Absolute Undecidability [2006], 1.4)
Book Ref
-: 'Philosophia Mathematica' [-], p.10
A Reaction
[I don't understand how cardinals can have 'logical strength', but I pass it on anyway]
| 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] |