The next wave of excitement in set theory came around 1900, when it was discovered that Cantorian set theory gave rise to several contradictions, called antinomies or paradoxes. Bertrand Russell and Ernst Zermelo independently found the simplest and best known paradox, now called Russell's paradox: consider "the set of all sets that are not members of themselves", which leads to a contradiction since it must be a member of itself, and not a member of itself. In 1899 Cantor had himself posed the question "What is the cardinal number of the set of all sets?", and obtained a related paradox. Russell used his paradox as a theme in his 1903 review of continental mathematics in his The Principles of Mathematics.
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Anonymous2013-08-31 23:19
Multiplication distributes over addition: κ·(μ + ν) = κ·μ + κ·ν and (μ + ν)·κ = μ·κ + ν·κ.
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Anonymous2013-09-01 0:04
Cantor defined two kinds of infinite numbers: ordinal numbers and cardinal numbers. Ordinal numbers may be identified with well-ordered sets, or counting carried on to any stopping point, including points after an infinite number have already been counted.
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Anonymous2013-09-01 0:50
A derived binary relation between two sets is the subset relation, also called set inclusion. If all the members of set A are also members of set B, then A is a subset of B, denoted A ⊆ B. For example, {1,2} is a subset of {1,2,3} , but {1,4} is not.
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Anonymous2013-09-01 1:35
These are invariants in the sense that any two isomorphic models of set theory must give the same cardinal for each invariant. Many cardinal invariants have been studied, and the relationships between them are often complex and related to axioms of set theory.
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Anonymous2013-09-01 2:20
The negation of the axiom can thus be expressed as:
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Anonymous2013-09-01 3:06
The Cartesian product of any family of nonempty sets is nonempty.