Defining the integers and rationals

Question

What axiomatic system is most commonly used by modern mathematicians to describe the properties of the integers and the rationals?

Properties like

$a+0=a$,

$a*1=a$,

$a+b=b+a$,

Also given these axioms, can I show that a contradiction cant be derived from them, such as $0=1$.. etc

Answer 1

See "Foundations of Analysis" by Landau. Starts with the Peano axioms for the integers, then fractions, then reals via Dedekind cuts, the complex numbers.

There is also a book titles "Numbers" (iirc). It's a yellow paperback I have in storage somewhere.

Answer 2

The Peano's Axioms are used to define the set of natural numbers.

The axioms are

1. There exists a non-empty set $P$. In particular $0 \in P$.

2. There exists a successor mapping $s$ from $P$ into itself

3. The sucessor mapping is injective.

4. The sucessor mapping is not surjective. In particular $0$ is not in the image of sucessor mapping.

5. For any subset $A \subset P$ which has an element of P which is the successor of no element, such that it has the successor of every number in it, is the same set as $P$.

   The set P is then the set of natural numbers.

Now, the set of integers can be defined by the following equivalence relation on $\mathbb N^2$.$$ (a,b) \sim (c,d) \Leftrightarrow a+d=c+b $$ The equivalence classes of this relation are called integers.

Similarly, one can define the set of rational numberby the following equivalence relation on $\mathbb Z^2$. $$ (a,b) \sim (c,d) \Leftrightarrow ad=cb $$ The equivalenence classes of this relation are called rational numbers.

Getting real numbers from rationals is more complicated. That can be done using cauchy sequences or dedekind cuts.

The following is done using cauchy sequences. Let $R$ repesent the set of all cauchy sequences in $\mathbb Q$.Cauchy sequences can be added (subtracted) and multiplied using the following rule. $$(a_n) +(b_n)=(a_n+b_n)$$ $$(a_n)*(b_n)=(a_n*b_n)$$

Now condsider an equivalence relation on $R$ in which cauchy sequences are said to be equivalent if their difference tends to zero.The set of the equivalence classes 0f this equivalence relation is the set of real numbers $\mathbb R$.

Answer 3

May I suggest one reference? Check out The Number Systems: Foundations of Algebra and Analysis by Solomon Feferman.

If you click on the link, you can preview this book, to see, e.g., if the Table of Contents, etc., seem promising in answering your questions. Perhaps you can obtain it at a library, or through inter-library loan.

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For "quicker" access to some discussion of your interest, see:

$(1)$ Number (Wikipedia)

It begins by describing Natural Numbers, referring both to set theory, and to the Peano axioms.

Then it moves to integers, where you'll find a link to the integer entry, which describes how one can construct integers from natural numbers.

Then it moves to the rational numbers, where you'll find a link to the rational number entry, which describes how one can construct rational numbers from the integers..., ...etc.

Also see the entry on numbers that follows:

$(2)$ Number (Encyclopedia of Mathematics)

In particular, scroll down the entry, until you find the following passages:

"Throughout the 19th century, and into the early 20th century, deep changes were taking place in mathematics. Conceptions about the objects and the aims of mathematics were changing. The axiomatic method of constructing mathematics on set-theoretic foundations was gradually taking shape. In this context, every mathematical theory is the study of some algebraic system. In other words, it is the study of a set with distinguished relations, in particular algebraic operations, satisfying some predetermined conditions, or axioms.

From this point of view every number system is an algebraic system. For the definition of concrete number systems it is convenient to use the notion of an "extension of an algebraic system" . This notion makes precise in a natural way the principle of permanence of formal computing laws, which was formulated above..."

You can read then how the numbers, along with appropriate operations and elements like an additive identity (and/or multiplicative identity), etc, comprise algebraic systems and extensions of algebraic systems, with "axioms" related to each system/extended from predecessor systems.

The details, and additional links and references, are left for the reader to explore!