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A countable set is an infinite set that can be put into a one-to one correspondence with the counting numbers. In other words, it is possible to arrange all of the elements of the set in a sequence with a first element, a second element, and so on.

Georg Cantor proved that the rational numbers are countable, but the real numbers are not.

One proof (not Cantor's) that the rationals are countable: Choose any rational number, write it out in its simplest form. Whatever number you wrote is represented by a finitely long string of either the numerals or the division slash (possibly preceded the negative sign). If you consider a base-eleven number system with the division slash as the eleventh numeral, then whatever rational number you just wrote out corresponds directly and unambiguously to one specific integer. Since there exists a mapping scheme that assigns any arbitrary rational number to a specific integer, the rational numbers are countable.

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Q: What is countable set in measure theory?
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Yes. the set of rational numbers is a countable set which can be generated from repeatedly taking countable union, countable intersection and countable complement, etc. Therefore, it is a Borel Set.


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