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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]
     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.
     From: report of George Cantor (works [1880]) by Penelope Maddy - Naturalism in Mathematics I.1
     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?
You can't get a new transfinite cardinal from an old one just by adding finite numbers to it [Russell]
     Full Idea: It must not be supposed that we can obtain a new transfinite cardinal by merely adding one to it, or even by adding any finite number, or aleph-0. On the contrary, such puny weapons cannot disturb the transfinite cardinals.
     From: Bertrand Russell (The Principles of Mathematics [1903], §288)
     A reaction: If you add one, the original cardinal would be a subset of the new one, and infinite numbers have their subsets equal to the whole, so you have gone nowhere. You begin to wonder whether transfinite cardinals are numbers at all.
For every transfinite cardinal there is an infinite collection of transfinite ordinals [Russell]
     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.
     From: Bertrand Russell (The Principles of Mathematics [1903], §290)
     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.
Very large sets should be studied in an 'if-then' spirit [Putnam]
     Full Idea: Sets of a very high type or very high cardinality (higher than the continuum, for example), should today be investigated in an 'if-then' spirit.
     From: Hilary Putnam (The Philosophy of Logic [1971], p.347), quoted by Penelope Maddy - Naturalism in Mathematics
     A reaction: Quine says the large sets should be regarded as 'uninterpreted'.
First-order logic can't discriminate between one infinite cardinal and another [Hodges,W]
     Full Idea: First-order logic is hopeless for discriminating between one infinite cardinal and another.
     From: Wilfrid Hodges (Model Theory [2005], 4)
     A reaction: This seems rather significant, since mathematics largely relies on first-order logic for its metatheory. Personally I'm tempted to Ockham's Razor out all these super-infinities, but mathematicians seem to make use of them.
Infinity has degrees, and large cardinals are the heart of set theory [Maddy]
     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.
     From: Penelope Maddy (Naturalism in Mathematics [1997], I.1)
     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.
For any cardinal there is always a larger one (so there is no set of all sets) [Maddy]
     Full Idea: By the mid 1890s Cantor was aware that there could be no set of all sets, as its cardinal number would have to be the largest cardinal number, while his own theorem shows that for any cardinal there is a larger.
     From: Penelope Maddy (Naturalism in Mathematics [1997], I.1)
     A reaction: There is always a larger cardinal because of the power set axiom. Some people regard that with suspicion.
An 'inaccessible' cardinal cannot be reached by union sets or power sets [Maddy]
     Full Idea: An 'inaccessible' cardinal is one that cannot be reached by taking unions of small collections of smaller sets or by taking power sets.
     From: Penelope Maddy (Naturalism in Mathematics [1997], I.5)
     A reaction: They were introduced by Hausdorff in 1908.
There are at least eleven types of large cardinal, of increasing logical strength [Koellner]
     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).
     From: Peter Koellner (On the Question of Absolute Undecidability [2006], 1.4)
     A reaction: [I don't understand how cardinals can have 'logical strength', but I pass it on anyway]
Paradox: there is no largest cardinal, but the class of everything seems to be the largest [Lavine]
     Full Idea: The paradox of the largest cardinal ('Cantor's Paradox') says the diagonal argument shows there is no largest cardinal, but the class of all individuals (including the classes) must be the largest cardinal number.
     From: Shaughan Lavine (Understanding the Infinite [1994], III.5)