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Hello I am a high school student currently reading through Calculus by Spivak which has been recommended by many people on this site. I was slightly disappointed by the first chapter in which properties of numbers are discussed. I value rigour highly and don't want to study learn calculus until I understand the abstract and axiomatic aspects of how numbers work. The author, for example, does not construct $\mathbb{N}$ from Peano's axioms, nor does he construct $\mathbb{Z}$ and $\mathbb{Q}$ as, respectively, equivalence classes of ordered pairs of natural numbers and integers. He doesn't talk about the field/ring axioms either. He also makes various assumptions which are bizarre considering how he is otherwise meticulous (closure under addition and multiplication, for example).

This leads me to my question: can someone suggest free, online references (class notes will suffice) which discuss the construction of various types of numbers from a rigorous standpoint?

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  • $\begingroup$ Landau's Foundations of Analysis has what you want. I don't know where you can get it for free, but in less than a minute I found a used copy for $7 with free shipping in the USA. $\endgroup$ – bof Dec 4 '15 at 8:59
  • $\begingroup$ You can see also : ITTAY WEISS, The Real Numbers - a survey of constructions. $\endgroup$ – Mauro ALLEGRANZA Dec 4 '15 at 11:55

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