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:
\((a,b).(c,d)=(ac+bd, ad+bc)\)
\(ab=c\)
\(c_1=a_1b_1+a_2b_2\)
\(c_2=a_2b_1+a_1b_2\)
\(a-b=c\)
\(c_1=a_1+b_2\)
\(c_2=a_2+b_1\)