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The definition for a countable set is: A non-empty set $S$ is countable when its elements can be a arranged in a sequence; that is, when $S$ is of the form: $$S= \{s_1,s_2,s_3,...\}$$ The sequence can be finite or infinite.

But how can an infinite set be countable?

Edited 17/03/19

The definition was from the book 'The Lebesgue Integral for Undergraduates' by William Johnston. The term 'countable' in this book has been used in the following context: 'sets are countable when the elements of that set can be arranged as a sequence'. The definition I believe is for the cardinality of a set.

My question is how can an infinite set be arranged if the elements seemingly go on forever?