Integer

The integers are the ring of non-fractional numbers:

... -3, -2, -1, 0, 1, 2, 3, ...

These are the natural numbers, with the addition of their additive inverses. In other words, the integers extend the natural numbers into an abelian group under addition.

We can construct a model for the integers. Start with the model of the natural numbers. Then construct two "flag objects", using pieces of that model:

<math>F_1 \equiv \{0,2\} = \{\phi,\{\phi,\{\phi\}\}\}</math>

<math>F_2 \equiv \{0,3\} = \{\phi,\{\phi,\{\phi\}, \{\phi, \{\phi\}\}\}\}</math>

Our model of the natural numbers, as laid out on the natural numbers page, consisted of the sets

<math>0 \equiv \Phi = \{\}</math>

<math>1 \equiv \{0\}</math>

<math>2 \equiv \{0,1\}</math>

<math>3 \equiv \{0,1,2\}</math>

...

We can see by inspection that F₁ and F₂ are not equal to any of those. We can, therefore, use the flags to tag two copies of the natural numbers. We now define the objects used in the model of the integers, shown here with primes attached, to distinguish them from the similar objects used in modeling the natural numbers:

<math>0' \equiv \{F_1\} \cup \{0\} = \{F_1, 0\}</math>

<math>1' \equiv \{F_1\} \cup \{1\} = \{F_1, 1\}</math>

<math>2' \equiv \{F_1\} \cup \{2\} = \{F_1, 2\}</math>

...

<math>succ'(\{F_1, x\}) \equiv \{F_1, succ(x)\}</math>

Note that we have also defined the successor function, "succ'", with a prime attached to distinguish it from the "succ" function which operates on the natural numbers.

In the remainder of this page, we will use <math>\mathbb{N}</math> to refer to this set:

<math>\mathbb{N} \equiv \{0', 1', 2', ... \}</math>

We will refer to members of this set as "nonnegative integers".

We can now define the negative integers as:

<math>-1' \equiv \{F_2\} \cup \{1\} = \{F_2, -1\}</math>

<math>-2' \equiv \{F_2\} \cup \{2\} = \{F_2, -2\}</math>

<math>-3' \equiv \{F_2\} \cup \{3\} = \{F_2, -3\}</math>

...

<math>pred'(\{F_2, x\}) \equiv \{F_2, succ(x)\}</math>

The entire set of integers, positive, negative, and zero, will be referred to as I.

We'll need negation later so we define it now:

<math>-(\{F_1,n\}) \equiv \begin{cases}

\{F_1,n\}, \mbox{if } n = 0 \\ \{F_2,n\}, \mbox{if } n \neq 0 \end{cases}</math>

<math>-(\{F_2,n\})\, \equiv \{F_1,n\}</math>

and we define > and <, with respect to zero only, as:

<math>x > 0 \equiv x \in (\mathbb{N} - {0})</math>

<math>x < 0 \equiv x \in (\mathbb{I} - \mathbb{N})</math>

For negative integers -- elements of I - N -- we define the succ' function implicitly, by the relation:

<math>succ'(x) = y \equiv x = pred'(y)</math>

along with the special case

<math>succ'(-1') \equiv 0'</math>

and finally we define pred' for values in <math>\mathbb{N}</math>:

<math>pred'(\{F_1,x\}) \equiv \begin{cases}

\{F_2,1\}, & \mbox{if } x = 0 \\ \{F_1,pred(x)\}, & \mbox{if } x \neq 0 \end{cases}</math>

We can now define the n^th successor to a number inductively, as

<math>succ'(x,n) \equiv \begin{cases}

x, & \mbox{if } n = 0 \\ succ'(succ'(x),pred'(n)), & \mbox{if }n > 0 \\ succ'(pred'(x),succ'(n)), & \mbox{if }n < 0 \\ \end{cases}</math>

With these definitions in hand, we can define a general "<math>\leq</math>" as

<math>x \leq y \equiv \ \exists n \in \mathbb{N} \left ( y = succ'(x, n) \right )</math>

As with the natural numbers we again can define addition and multiplication in the obvious way.

<math>x + y \equiv succ'(x, y)</math>

<math>x \cdot y \equiv \begin{cases}

0, & \mbox{if } y = 0 \\ x + (x \cdot pred'(y)), & \mbox{if } y > 0 \\ - (x + (x \cdot pred'(-y))), & \mbox{if } y < 0 \end{cases}</math>

This completes the basic model: We have comparisons, addition, and multiplication. The natural numbers, N, are contained within our new model, and comparison in the integers is clearly an extension of comparison in the natural numbers. It's also not hard to show that addition and multiplication are also extensions of the same operations on the natural numbers.

We could now go on to prove the axioms which define the integers as theorems within our model, including particularly the fact that each integer has a unique additive inverse, thus showing that I really is a model of the integers.