Bilangan alef: Perbedaan antara revisi

'''ω₁''' is actually a useful concept, if somewhat exotic-sounding. An example application is "closing" with respect to countable operations; e.g., trying to explicitly describe the [[sigma-algebra|σ-algebra]] generated by an arbitrary collection of subsets (see e. g. [[Borel hierarchy]]). This is harder than most explicit descriptions of "generation" in algebra ([[vector space]]s, [[group theory|group]]s, etc.) because in those cases we only have to close with respect to finite operations—sums, products, and the like. The process involves defining, for each countable ordinal, via [[transfinite induction]], a set by "throwing in" all possible countable unions and complements, and taking the union of all that over all of '''ω₁'''.

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:<math>2^{\aleph_0}=\aleph_1.</math>