The theory of infinite sets is my favorite subject in mathematics. The results are very counterintuitive - where would you get intuition about infinite things? As such, they are quite delightful; at least they are to me. This is a theoretical subject, the kind of math done by mathematicians, rather than the applied subjects taught in secondary school and the first two years of college. Sometimes a little theory is covered in those courses, but usually not much. Some high schools teach significant theory in geometry, but that comes before most students have developed much mathematical maturity, and I'm afraid the enjoyment and significance of theory are lost on most.
The main topics of the book are (1) countably infinite sets of "words", (2) countably infinite sets of numbers, (3) uncountably infinite sets of strings, numbers, functions, and sets, including Cantor's theorem, (4) paradoxes of set theory, including Russell's paradox, and (5) computable functions.
Along the way related topics are touched on, including base numeral systems, representations of rational and irrational numbers, polynomial equations and algebraic numbers, transcendental numbers, the cubic formula, the irrationality of the square root of two and other integers, functions (onto, 1-1), self-descriptive sequences, discrete dynamical systems, including chaotic systems, semantic paradoxes, Euclid’s theorem, the twin-prime conjecture, limits, continuity, the Fibonacci sequence, inductive/recursive definitions, and proofs by induction.
There are about 100 problems with solutions to almost all.
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