Construction of the Number Systems .... Natural, Integers, Rationals and Reals

[Math Amateur]

At present I am trying to understand the construction of the number systems ... natural, integers, rationals and reals ...

What do members of MHBs think is the clearest, most detailed, most rigorous and best treatment of number systems in a textbook or in online notes ... ?

NOTE: I am currently using Ethan D Bloch: "The Real Numbers and Real Analysis" ... ... where the coverage is detailed ... and proofs in particular are detailed and in full ... but some of the explanations are not particularly clear ... ...Peter

 

[Evgeny.Makarov]

I heard "Foundations of Analysis" by Edmund Landau is good. It was written in the first half of the 20th century, and it is rather elementary.

Constructions of different classes of numbers are often studied in set theory, which is a part of mathematical logic. But the exposition there start with sets and axioms rather than natural numbers. Then one shows how to construct natural numbers, how to define functions by recursion and only then goes to arithmetic operations and other classes of numbers. These are more details than one usually needs for calculus. Natural numbers and Peano axioms ($x+(y+1)=(x+y)+1$, etc.) are studied in proof theory, which is also a part of mathematical logic. (Peano arithmetic is a favorite logical theory.) But again, things studied there are usually not needed in calculus.

I would think that the first chapters of many calculus textbooks contain information about the construction of numbers. However, some textbooks, such as "Mathematical Analysis" by Vladimir Zorich, use axiomatic approach. They introduce real numbers as a set satisfying some axioms and other classes of numbers as subsets of real numbers.

 

[Ackbach]

I heard "Foundations of Analysis" by Edmund Landau is good. It was written in the first half of the 20th century, and it is rather elementary.

Constructions of different classes of numbers are often studied in set theory, which is a part of mathematical logic. But the exposition there start with sets and axioms rather than natural numbers. Then one shows how to construct natural numbers, how to define functions by recursion and only then goes to arithmetic operations and other classes of numbers. These are more details than one usually needs for calculus. Natural numbers and Peano axioms ($x+(y+1)=(x+y)+1$, etc.) are studied in proof theory, which is also a part of mathematical logic. (Peano arithmetic is a favorite logical theory.) But again, things studied there are usually not needed in calculus.

I would think that the first chapters of many calculus textbooks contain information about the construction of numbers. However, some textbooks, such as "Mathematical Analysis" by Vladimir Zorich, use axiomatic approach. They introduce real numbers as a set satisfying some axioms and other classes of numbers as subsets of real numbers.

Totally agree on the Landau recommendation. Very accessible, clear and direct. Highly recommended!