– Let us construct natural numbers (0,1,2,3..)
– Then from naturals, lets construct integer numbers (…,-2,-1,0,1,2…)
– Then from integers, lets construct rationals (…,-0.3,…,0,…,1/2,..,0.75,..)
– And finally from rationals – real numbers (…,-0.3,…,0,…,1/π,…,1/2,..,0.75,..,e,…,π,…)

Mathematician Guiseppe Peano proposed next axiomatic definition at 1889, based on naive set theory(set, member of the set, and equivalence relation).

In this case, he also proposed such concept as follower, and thus Peano’s system (N,0,*) can be described with next axioms:

1. is set, and
2. For each member there is only one follower n’
3. Member 0 is not follower of any member in
4. If and m’=n’, follows that m=n
5. If , and if , then

It should be noticed, that we can define less(<) relation with natural numbers:

Lets define ~ equicalence relation on natural numbers in next maner:

Let

Therefore

Each rational x can be presented in a form x=a/b where and

And in a same way, like we received integers from naturals, we can

Therefore

Most interesting case in my opinion, first we can observe, that equation with integers 2 and 7 has no integer solution for x: 2x=7

However if then there is problem no more. x=7/2=3.5, which is rational numbers.

But sometimes rational numbers aren’t enough. For exampe x²=2 has no x in rational numbers.

To solve this problem, we define new number, lets call  its

We can use it with rationals to get  more numbers, etc

Thus we got extended rationals –

But that doesn’t give is solution to x²=3, x³=3,x³=9 etc. List is long,but we can define set of algebraic number (well, its not all algebraic numbers, we need complex numbers too).

Algebrain numbers are solution of all polynomials with integer coefficients

Thus we got real alreabric numbers –

Did we get all real numbers? No.  Actually real algebraic numbers are countable set(all n-tuples {for finite} n of countable set is also countable set), and real numbers is not.

Except real alrebraic numbers, there are also transcendentals, here are two of them

π=3.1415.., e=2.7182

Those are exotic examples, but in reality – there is much, much more transcendental numbers than algebraic numbers.

Let us meet, what is called Dedekind cut.

Let there be a rational number set, graphically speaking:

But, our line doesn’t contain anything, except rational numbers!

I.e. 1,-4,⅗ are there, but  π isn’t there. Neither √2.

So, lets make two sets out of rationals. Let

and Let

Ok, what if x=√2?

` `

But.. but.. there is no , so

So, there is hole in rationals. It will, of course, happen with all irrational numbers. √2,√3 etc

Which makes bijection between all real numbers and all possible Dedekind cuts of rational numbers.

To makes things easier, way we define upper and lower sets, it is possible to refer to a cut using only lower set.

Dedekind cuts introduce relations to real numbers, which is ordered set.

So for x and y in real numbers,one thing is true: either x<y,x>y or x=y