axiomatic sets 20 (integers)

In preceding post, we have defined negative numbers using ordered pairs of $0$ and a natural number $n$.

\[ \mathbb{N}^-=\left\{(0,n) : n\in\mathbb{N},n\ne 0 \right\}  \]

Therefore, integers $\mathbb{Z}$ will be simply constructed by $\mathbb{N}\cup\mathbb{N}^-$ .

However, such integers are complicated as we always have to concern the difference of positive numbers and negative numbers. Because integers are made by two kinds of sets (0, positive numbers $1,2,3,\cdots$ and negative numbers $(0,1),(0,2),(0,3),\cdots$) .

We shall introduce an equivalent relation "$\sim$" to any ordered pair of two natural numbers.

For any natural numbers $k,h,n,m\in\mathbb{N}$ , $(k,h)\sim (n,m)$ if and only if $k+m=h+n$ .

That is,
\[ (1,0)\sim (2,1)\sim (3,2)\sim (4,3)\sim\cdots  , \]
\[ (2,0)\sim (3,1)\sim (4,2)\sim (5,3)\sim\cdots  , \]
\[ (0,0)\sim (1,1)\sim (2,2)\sim (3,3)\sim\cdots  , \]
\[ (0,1)\sim (1,2)\sim (2,3)\sim (3,4)\sim\cdots  , \]
$\cdots,\cdots$ .

We will write $[(k,h)]$  as meaning
\[ [(k,h)]=\left\{(n,m) : n,m\in\mathbb{N}, (k,h)\sim(n,m)  \right\} .  \]
For given $k,h$ , $[(k,h)]$  becomes a class of sets.

For example,
\[ [(1,0)]=\left\{(1,0),(2,1),(3,2),(4,3),\cdots  \right\} , \]
\[ [(2,0)]=\left\{(2,0),(3,1),(4,2),(5,3),\cdots  \right\} , \]
\[ [(0,0)]=\left\{(0,0),(1,1),(2,2),(3,3),\cdots  \right\} , \]
\[ [(0,1)]=\left\{(0,1),(1,2),(2,3),(3,4),\cdots  \right\} , \]
$\cdots,\cdots$ .

As a matter of course, for example,
\[ [(1,0)]=[(2,1)]=[(3,2)]=[(4,3)]=\cdots  .    \]

You will have to find $[(1,0)]$ as the intuitive natural number $1$ and
$[(2,0)]$  as $2$, $[(0,0)]$  as $0$ , $[(0,1)]$  as $-1$ , $\cdots\cdots$ .

We make $\mathbb{Z}$  denote the set of all equivalent classes with respect to $\sim$ .
The elements of $\mathbb{Z}$  will be called integers.

\[  \mathbb{Z}=\left\{ [(0,0)],[(1,0)],[(0,1)],[(2,0)],[(0,2)],\cdots  \right\} \]

You will see intuitively it means
\[   \mathbb{Z}=\left\{  0,1,-1,2,-2,\cdots \right\} . \]

Also, you can do define
\[  \mathbb{Z}=\left\{ [(0,0)],[(2,1)],[(2,3)],[(4,2)],[(3,5)],\cdots  \right\}  . \]

axiomatic sets 19 (negative numbers)

We want to extend natural numbers to integers.

Integers, as you know well, are constructed by natural numbers (which are positive numbers and 0) and negative numbers.

In axiomtaic set theory, all objects must be sets which are collections of defined something.

A set can not become negative and always be positive because those exist here and there.

Thus, we will identify some sort of sets same as negative numbers.

Most simple way for negative numbers is the ordered pair $(0,n)$ ,where $n$  is a natural number and not $0$ .

That is to say, $(0,1)$  means $-1$ , and $(0,2)$  is $-2$ ,･･･, $(0,n)$  is $-n$, and so on.

For this definition, we have to accept the following lemma. But it is very natural.

If natural numbers $m,n$ are $m\lt n$ ,then there is a natural number $i$ such that $m+i=n$ .

We will state an "$i$"  "$n-m$" .

Next, we want to use this rule in the case when $m\ge n$ ,
because it is very useful and we do not need to think about the magnitude relation of $m,n$ .

Especially, if $n=0$ ,then $i=-m$ omitting $n$ .

Then, we will define the ordered pair $(0,m)$ as $-m$. ( $(0,0)=0$  and If $n=m$ ,then $i=0$ )

This is one of definitions of negative numbers.

(Might you have any questions?  OK, I will continue.)