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I am going to read field theory next month. I want to know if there's any book or paper where I can avail a thorugh treatment of the five number systems. I mean from the development of Peano axioms with induction and recursion theory to integers, ordered fields and convergence of sequences, Cauchy and Dedekind completeness properties etc.

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Try "The Number System" by H. A. Thurston and "Foundations of Analysis" by Edmund Landau, I've read both books and I find that they suit most of your requirements, Thurston provides an explanatory as well as a systematic treatment for the construction of the five number systems, however, he uses the idea of a Cauchy sequence to construct the real number system from the rationals and not the idea of cuts as proposed by Dedekind, Landau, on the other hand, uses the idea of cuts for the construction of the reals. Note that, Thurston considers 0 to be a natural number, or rather, a "whole number" as he calls it, while Landau's natural numbers start with 1.

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