Rigorous book developing number systems and set theory

Asked Dec 06 '21 at 03:52

Active Jul 31 '22 at 06:33

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Baby Rudins chapter 1 assumes familiarity with the arithmetic of natural numbers and rational numbers. Rudin constructs the reals and complex numbers from the rational numbers.

For curiosity's sake, I would like to see treatment from either an axiomatization of natural numbers (Peano) or construction (via ZFC). Is there a rigorous book that constructs the number systems along with the necessary objects like functions, relations, cardinalities in a comprehensive and rigorous fashion? Looking for a rigorous style more than a sketch or chatty style.

asked Dec 06 '21 at 03:52

iart

1

You might look at "Naive Set Theory" by Paul Halmos. – morrowmh Dec 06 '21 at 03:54
I remember Amann & Escher's book Analysis I had a careful construction of the complex numbers starting from Peano's axioms, as well as background on set theoretic material. – Brian Shin Dec 06 '21 at 04:37
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Enderton's Elements of Set Theory builds the naturals from Peano's axioms, and then constructs the integers, rationals, and real. It also develops functions and relations (and recursion) prior to that. Later chapters investigate cardinals and ordinals. – Ben West Dec 06 '21 at 05:19
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I also recommend @Ben West's suggestion of Enderton's book (see here). See also Landau's Foundations of Analysis and the books listed in this answer. Finally, for alternative approaches to the construction of the reals from the rationals, see this answer. – Dave L. Renfro Dec 06 '21 at 06:35
Thanks all! Will check these out :) – iart Dec 06 '21 at 06:37

Rigorous book developing number systems and set theory

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