The book is divided into two parts, with the first exploring notions leading to concepts of actual infinity, concrete but infinite mathematical values. After an exploration of number systems, this part discusses set theory, cardinal numbers, and ordinal numbers, transfinite arithmetic, and the existence of different infinite sizes of sets. Topics used to illustrate these concepts include Hilbert's paradox of the Grand Hotel, Cantor's diagonal argument,[4] and the unprovability of the continuum hypothesis.[2]
The second part concerns mathematics related to the idea of potential infinity, the assignment of finite values to the results of infinite processes including growth rates, limits, and infinite series.[4][2] This part also discusses Zeno's paradoxes, Dedekind cuts,[2] the dimensions of spaces, and the possibility of spaces of infinite dimensions, with a mention of higher category theory,[4] Cheng's research specialty.[1][2]
The mathematics is frequently lightened and made accessible with personal experiences and stories,[3][6][7] involving such subjects as the Loch Ness Monster, puff pastry, boating, dance contests, shoes,[3] "Legos, the iPod Shuffle, snorkeling, Battenberg cakes and Winnie-the-Pooh".[6]
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