Assuming the axiom of choice and, given an infinite cardinal σ and a cardinal μ, there exists a cardinal κ such that μ + κ = σ if and only if μ ≤ σ. It will be unique (and equal to σ) if and only if μ < σ.
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Anonymous2013-08-31 8:52
\sum_{i=0}^{\infty} \, f(i) = \infty means that the sum of the infinite series diverges in the specific sense that the partial sums grow without bound.
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Anonymous2013-08-31 9:37
Modern understanding of infinity began in 1867–71, with Cantor's work on number theory. An 1872 meeting between Cantor and Richard Dedekind influenced Cantor's thinking and culminated in Cantor's 1874 paper.
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Anonymous2013-08-31 10:23
For example, the existence of sufficiently large cardinals implies that there is an inner model satisfying the axiom of determinacy (and thus not satisfying the axiom of choice).
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Anonymous2013-08-31 11:08
Given any set X of pairwise disjoint non-empty sets, there exists at least one set C that contains exactly one element in common with each of the sets in X.