The notion of cardinality, as now understood, was formulated by Georg Cantor, the originator of set theory, in 1874–1884. Cardinality can be used to compare an aspect of finite sets; e.g. the sets {1,2,3} and {4,5,6} are not equal, but have the same cardinality, namely three (this is established by the existence of a one-to-one correspondence between the two sets; e.g. {1->4, 2->5, 3->6}).
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Anonymous2013-08-31 23:33
If 2 ≤ κ and 1 ≤ μ and at least one of them is infinite, then:
Max (κ, 2μ) ≤ κμ ≤ Max (2κ, 2μ).
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Anonymous2013-09-01 0:19
One of the needs of any physical and scientific theory is to give usable formulas that correspond to or at least approximate reality. As an example if any object of infinite gravitational mass were to exist, any usage of the formula to calculate the gravitational force would lead to an infinite result, which would be of no benefit since the result would be always the same regardless of the position and the mass of the other object.
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Anonymous2013-09-01 1:04
This assumption gives rise to paradoxes, the simplest and best known of which are Russell's paradox and the Burali-Forti paradox. Axiomatic set theory was originally devised to rid set theory of such paradoxes.
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Anonymous2013-09-01 1:50
Within set theory, many collections of sets turn out to be proper classes. Examples include the class of all sets, the class of all ordinal numbers, and the class of all cardinal numbers.
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Anonymous2013-09-01 2:35
Since X isn't measurable for any rotation-invariant countably additive finite measure on S, finding an algorithm to select a point in each orbit requires the axiom of choice. See non-measurable set for more details.
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Anonymous2013-09-01 3:20
On the other hand, other foundational descriptions of category theory are considerably stronger, and an identical category-theoretic statement of choice may be stronger than the standard formulation, à la class theory, mentioned above.