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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1See https://math.stackexchange.com/questions/987564/books-that-follow-axiomatic-approach. – lhf May 07 '17 at 19:20
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1See e.g. Solomon Feferman, The Number Systems: Foundations of Algebra and Analysis. – Mauro ALLEGRANZA May 07 '17 at 19:36
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Can you tell me where can I download this book? – user398623 May 07 '17 at 19:40
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Maybe usefule also: Melvin Fitting, Numbers (1990). – Mauro ALLEGRANZA May 08 '17 at 07:53
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I posted a list of 9 such book in my answer to Is the real number structure unique?. See also the 12 related items (mostly papers) given in this 17 June 2006 sci.math post (an additional item is in this 18 June 2006 sci.math post). – Dave L. Renfro Apr 29 '20 at 08:29
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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.
