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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / i. Cardinal infinity

[infinity as a collection of transcendent size]

10 ideas
Cardinality strictly concerns one-one correspondence, to test infinite sameness of size [Cantor, by Maddy]
You can't get a new transfinite cardinal from an old one just by adding finite numbers to it [Russell]
For every transfinite cardinal there is an infinite collection of transfinite ordinals [Russell]
Very large sets should be studied in an 'if-then' spirit [Putnam]
First-order logic can't discriminate between one infinite cardinal and another [Hodges,W]
For any cardinal there is always a larger one (so there is no set of all sets) [Maddy]
Infinity has degrees, and large cardinals are the heart of set theory [Maddy]
An 'inaccessible' cardinal cannot be reached by union sets or power sets [Maddy]
There are at least eleven types of large cardinal, of increasing logical strength [Koellner]
Paradox: there is no largest cardinal, but the class of everything seems to be the largest [Lavine]