Cantor also developed a large portion of the general theory of cardinal numbers; he proved that there is a smallest transfinite cardinal number (ℵ0, aleph-null) and that for every cardinal number, there is a next-larger cardinal
(ℵ1, ℵ2, ℵ3...)
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Anonymous2013-08-31 23:36
The logarithm of an infinite cardinal number κ is defined as the least cardinal number μ such that κ ≤ 2μ. Logarithms of infinite cardinals are useful in some fields of mathematics, for example in the study of cardinal invariants of topological spaces, though they lack some of the properties that logarithms of positive real numbers possess.
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Anonymous2013-09-01 0:21
This point of view does not mean that infinity cannot be used in physics. For convenience's sake, calculations, equations, theories and approximations often use infinite series, unbounded functions, etc., and may involve infinite quantities.
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Anonymous2013-09-01 1:07
Zermelo set theory, which replaces the axiom schema of replacement with that of separation;
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Anonymous2013-09-01 1:53
For example, Russell's paradox suggests a proof that the class of all sets which do not contain themselves is proper, and the Burali-Forti paradox suggests that the class of all ordinal numbers is proper.
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Anonymous2013-09-01 2:37
A proof requiring the axiom of choice may establish the existence of an object without explicitly defining the object in the language of set theory. For example, while the axiom of choice implies that there is a well-ordering of the real numbers, there are models of set theory with the axiom of choice in which no well-ordering of the reals is definable.
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Anonymous2013-09-01 3:23
If two small categories are weakly equivalent, then they are equivalent.