The infinite set plays an important role in mathematical analysis. Infinite sets may be divided by cardinality into countable infinite sets and uncountable infinite sets. Not all infinities are the same! For example, the rational number set is countably infinite, since every fraction can be placed in a one-to-one correspondence with the natural number set. On the other hand, the real number set is uncountable (proof).

A good definition of an infinite set is:

A set is infinite if we can remove some of its elements without reducing its size.

Clever, isn't it?

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