Climb into Cantor’s Attic, where you will find infinities large and small. We aim to provide a comprehensive resource of information about all notions of mathematical infinity.
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The upper attic
The middle attic
The lower attic
The parlour
The playroom
The library
The cellar
Sources
Cantor's Attic (original site)
Joel David Hamkins blog post about the Attic
Latest working snapshot at the wayback machine
A set is countable when it is equinumerous with a subset of $\omega$. This includes all finite sets, including the empty set, and the infinite countable sets are said to be countably infinite. An uncountable set is a set that is not countable. The existence of uncountable sets is a consequence of Cantor’s observationt that the set of reals is uncountable.
Cantor’s diagonal argument shows that the set of reals is uncountable.
More generally, the power set of any set is a set of strictly larger cardinality.
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