To extend the number line to negative numbers, we define:

\(\forall ab \in \mathbb{N} \exists c (a+c=b)\)

For any pair of numbers there exists a terms which can be added to one to get the other.

For \(1+x=3\) this is another natural number, however for \(3+x=1\) there is no such number.

Integers are defined as the solutions for any pair of natural numbers.

There are an infinite number of ways to write any integer. \(-1\) can be written as \(0-1\), \(1-2\) etc.

The class of these terms form an equivalence class.

Integers can be defined as an ordered pair of natural numbers, where the integer is valued at: \(a-b\).

For example \(-1\) could be shown as:

\(-1= \{ \{ 0 \},\{0,1\}\}\)

\(-1= \{ \{ 5 \},\{5,6\}\}\)

\((a,b)=a-b\)

Natural numbers can be shown as integers by using:

\((n,0)\)

Natural numbers can be converted to integers:

\(\{\{a\},\{a,0\}\}\)

Integers are an ordered pair of naturals.

\(\{\{x\},\{x,y\}\}\)

For example \(-4\) can be:

\(\{\{4\},\{4,8\}\}\)

\(\{\{0\},\{0,8\}\}\)

We extend the ordering to say:

\(\{\{x\},\{x,y\}\}\le \{\{s(x)\},\{s(x),y\}\}\)

\(\{\{x\},\{x,s(y)\}\}\le \{\{x\},\{x,y\}\}\)

So can we define this on an arbitrary pair:

\(\{\{a\},\{a,b\}\}\le \{\{c\},\{c,d\}\}\)

We know that:

\(\{\{a\},\{a,b\}\}=\{\{s(a)\},\{s(a),s(b)\}\}\)

And either of:

\(\{\{a\},\{a,b\}\}=\{\{0\},\{0,A\}\}\)

\(\{\{a\},\{a,b\}\}=\{\{B\},\{B,0\}\}\)

\(\{\{a\},\{a,b\}\}=\{\{0\},\{0,0\}\}\)

As the latter is a case of either of the other \(2\), we consider only the first \(2\).

So we can define:

\(\{\{a\},\{a,b\}\}\le \{\{c\},\{c,d\}\}\)

As any of:

\(1: \{\{0\},\{0,A\}\}\le \{\{0\},\{0,C\}\}\)

\(2: \{\{0\},\{0,A\}\}\le \{\{D\},\{D,0\}\}\)

\(3: \{\{B\},\{B,0\}\}\le \{\{0\},\{0,C\}\}\)

\(4: \{\{B\},\{B,0\}\}\le \{\{D\},\{D,0\}\}\)

Case 1:

\(\{\{0\},\{0,A\}\}\le \{\{0\},\{0,C\}\}\)

Trivial, depends on relative size of \(A\) and \(C\).

Case 2:

\(\{\{0\},\{0,A\}\}\le \{\{D\},\{D,0\}\}\)

We can see that:

\(\{\{D\},\{D,A\}\}\le \{\{D\},\{D,0\}\}\)

And therefore this holds.

Case 3:

\(\{\{B\},\{B,0\}\}\le \{\{0\},\{0,C\}\}\)

We can see that:

\(\{\{B\},\{B,0\}\}\le \{\{B\},\{B,C\}\}\)

And therefore this does not hold.

Case 4:

\(\{\{B\},\{B,0\}\}\le \{\{D\},\{D,0\}\}\)

Trivial, like case 1.

Then we can define addition as:

\((a,b)+(c,d)=(a+c,b+d)\)

Integer addition can then be defined:

\(a+b=\{\{a_1\},\{a_1,a_2\}\}+\{\{b_1\},\{b_1,b_2\}\}\)

\(a+b=\{\{a_1+b_1\},\{a_1+b_1,a_2+b_2\}\}\)

Or:

\(a+b=c\)

\(c_1=a_1+b_1\)

\(c_2=a_2+b_2\)

Similarly, multiplication can be defined as:

\((a,b).(c,d)=(ac+bd, ad+bc)\)

\(ab=c\)

\(c_1=a_1b_1+a_2b_2\)

\(c_2=a_2b_1+a_1b_2\)

\(a-b=c\)

\(c_1=a_1+b_2\)

\(c_2=a_2+b_1\)