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Latest revision as of 18:00, 19 November 2004
The "natural numbers" are the numbers we use for counting things. Generally, they're defined as the positive integers: 1, 2, 3, ... That definition, while it gives the correct idea, is unfortunately rather circular, since the integers are defined in terms of the natural numbers.
More specifically, the "counting numbers" are the positive integers. The "whole numbers" are the nonnegative integers: 0, 1, 2, 3, ... The "natural numbers" can refer to either set.
Also called the "finite ordinals", the natural numbers (starting with 0) are also the initial segment of the set of all ordinal numbers. They are used to count sets of things that are well ordered.
The "natural numbers" may also be defined as the Peano integers: the set of numbers which satisfy the five Peano axioms.
For a more concrete definition of the natural numbers, as well as a (limited) proof of the consistency of the Peano axioms, we can exhibit a model of the nonnegative integers. The foundation of the model is the object we will use as zero, along with a "successor" function, which we will call succ. Given any number, the successor function provides us with the next number; we use it to inductively define all the rest of the natural numbers.
- <math>0 \equiv \phi \ \ \ (where\ \phi\ represents\ the\ empty\ set)</math>
- <math>succ(x) \equiv \{x\} \cup x </math>
and then
- 1 = succ(0)
- 2 = succ(1)
- ...
We define a predecessor function, pred(x), implicitly, in terms of succ(x):
- <math>x = pred(y) \equiv y = succ(x)</math>
We observe, without proof, that pred(x) is uniquely defined for all x>0, but pred(0) is not defined.
We also need the concept of the "nth" successor. We define it inductively:
- <math>succ(x,n) \equiv \begin{cases} x, & \mbox{if } n = 0 \\ succ(succ(x,pred(n))), & \mbox{if }n \neq 0 \end{cases}</math>
We can then define a comparison operator, "<math>\leq</math>", as
- <math>x \leq y \equiv \exists n \left ( succ(x,n) = y \right )</math>
Addition is straightforward:
- <math>x + y \equiv succ(x,y)</math>
Multiplication must once again be defined inductively.
- <math>x \cdot y \equiv \begin{cases} 0, & \mbox{if } y = 0 \\ x + (x \cdot pred(y)), & \mbox{if } y \neq 0\end{cases}</math>
The construction of the model is now complete.
Within this model, it's straightforward to prove the Peano axioms as theorems, thus showing, first, that the model does correctly represent the Peano integers, and, second, that the axioms of Peano arithmetic are consistent (assuming the axioms of set theory are consistent).
The natural numbers are important not just in themselves, but because they are the building blocks from which we can construct a model of the integers, the rational numbers, and the real numbers, upon which most of the rest of mathematics is based.
It would be nice to say something about the history of the natural numbers here. In these troubled times it's worth remembering that these are properly called the "Arabic numbers", and they were not invented in Europe, but I'm not sufficiently sure of my facts to say much more than that. --SAL 03:49, 19 Nov 2004 (UTC)
