Algebra/Arithmetic/Numerical Axioms

Numerical Axioms
It is possible to define a regular set of numbers in a formal fashion. The set of Peano axioms define the series of numbers known as the natural numbers. They are as follows:


 * 1) There is a whole number 0.
 * 2) Every natural number a has a successor, denoted by a + 1.
 * 3) There is no natural number whose successor is 0.
 * 4) Distinct natural numbers have distinct successors: if a <> b, then a + 1 <> b + 1.
 * 5) If a property is possessed by 0 and also by the successor of every natural number it is possessed by, then it is possessed by all natural numbers.

Let us attempt to motivate these axioms. We want these axioms to eliminate any set which is not the natural numbers. E.g., any set fulfilling the above should at least be infinite.

The first two are obvious properties of the natural numbers (and of integers) as we know them. Note that some prefer to use 1 as the lowest number. The reason for choosing zero has root in [set theory], in which the first natural number is chosen as the empty set $$\emptyset$$.

The 3rd axiom prevents circularity. If this axiom was not included, defining $$0 + 1 = 0$$ would trivially fulfill the remaining axioms --- prove this for yourself by considering each remaining axiom!

The 4th prevents a partial loop. Consider a the set $$\{0,1\}$$ and set $$0+1=1$$ and $$1+1=1$$. This set fulfills every axiom but the 4th --- prove this for yourself.

The 5th is sometimes called the induction axiom. It ensures that the set is connected, i.e. that we can reach any number by using the 2nd axiom repeatedly on 0. An example of a set that fulfills every axiom but the 5th is $$\left\{0, \frac{1}{2}, 1, \frac{3}{2}, \dots\right\}$$ with the usual meaning of +1.

From this we can deduce the existence of a series of quantities like this:


 * 0
 * 0 + 1
 * 0 + 1 + 1
 * 0 + 1 + 1 + 1
 * 0 + 1 + 1 + 1 + 1
 * 0 + 1 + 1 + 1 + 1 + 1
 * and so on

where '0' is a constant and the first natural number and '1' is a constant natural number equivalent to the difference in value between two consecutive natural numbers.

This set is sufficient for counting. However, it is inconvenient to refer to a large natural number as '0' followed by the requisite large number of '+ 1' expressions. Due to this, each of the natural numbers is given a label, and to make the labelling easier another axiom is introduced:

 '0 + 1' is equivalent to '1'.

Thus the series of natural numbers may be written so for some brevity:


 * 0
 * 1
 * 1 + 1
 * 1 + 1 + 1
 * 1 + 1 + 1 + 1
 * 1 + 1 + 1 + 1 + 1
 * and so on

Once this is done, giving each quantity its own label is trivial. And so the series of natural numbers can then be written:


 * 0
 * 1
 * 2
 * 3
 * 4
 * 5
 * and so on