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I am looking for a book that discusses the concepts of infinity like the actual infinite and related concepts. Any suggestions?

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I only read an excerpt, but it was recommended by someone I trust: The Infinite by A.W. Moore. – Git Gud Jun 1 '14 at 0:00

For a nice introduction to the many different notions of infinity in mathematics see Rudy Rucker's book: Infinity and the Mind. Unlike many other popularizations, this is written by someone who did a Ph.D. on the topic. Rucker goes to great lengths to make the presentation faithful to the mathematics but still accessible to an educated layperson. See this answer for an excerpt on Alephs.

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