Math series 01: Construction of Real Numbers

Hello everyone. I would like to start a math topic series here with the hope of helping who needs.

I will start with constructing real numbers. Many prefer starting with natural numbers but since i am not an expert of set theory i will construct the rest number sets with the help of real number.

Real numbers is not just a set. It is a structure that includes addition, multiplication and inequality. Not to dive into much complexity and confuse readers i also prefer adding zero and one in this structure. We will discuss zero and one much more detailed when we start doing some group theory.

Real numbers is a structure, , that satisfies the following axioms. (Note that, if not mentioned spesifically, every element in the structure satisfies these axioms)

A1:
A2:
A3: For every there exists such that

(A3) Observe that by using A1 and A2 for a given , is unique. And we will denote that as .

A4:

Now axioms for multiplication. We will denote as

M1:
M2:
M3: For a given different than zero, there exist such that (Also is unique and we will denote )
M4:
AM: and

Observe followings;

i)

ii) , where is not equal to zero.

iii)

iv)If

v)

vi)

And last part of axioms.

O1:

O2:

O3:

O4:

AO:

MO:

Note that rational number structer also satisfies these axioms. So we need a specific axiom that characterize real numbers.

Definition: Let be a subset of Real numbers. If, for every element in , there exists a real number, say , such that then we say is upper bound of A

Sup axiom : Any proper(not empty) subset of real numbers, that has upper bound, has a least upper bound.

We call that least upper bound a supremum or sup. So;

. Please generate some examples. Note that, supremum of a set should not necessarily be in that set.

So we say, a sturcture is a Real Number structure if it satisfies these axioms.

Next time i am planning to show an example of an arbitrary irrational number is belonging to this system by using these axioms.
Thats all for now. Thanks for reading :)