A set X is Dedekind-infinite if there exists a proper subset Y of X with |X| = |Y|, and Dedekind-finite if such a subset doesn't exist. The finite cardinals are just the natural numbers, i.e., a set X is finite if and only if |X| = |n| = n for some natural number n. Any other set is infinite. Assuming the axiom of choice, it can be proved that the Dedekind notions correspond to the standard ones. It can also be proved that the cardinal ℵ0 (aleph null or aleph-0, where aleph is the first letter in the Hebrew alphabet, represented ℵ) of the set of natural numbers is the smallest infinite cardinal, i.e. that any infinite set has a subset of cardinality ℵ0. The next larger cardinal is denoted by ℵ1 and so on. For every ordinal α there is a cardinal number ℵα, and this list exhausts all infinite cardinal numbers.
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Anonymous2013-09-01 16:12
Descriptive set theory is the study of subsets of the real line and, more generally, subsets of Polish spaces. It begins with the study of pointclasses in the Borel hierarchy and extends to the study of more complex hierarchies such as the projective hierarchy and the Wadge hierarchy.