Well, no you probably haven't. Most people don't.
After all, the Natural Numbers are...um...natural. They're a fundamental part of how we think about numbers. But believe it or not their official definition (originally stated by Peano as a set of axioms) is fairly complex!
The following is quoted from Edmund Landau, Foundations of Analysis, Chelsea, 1951, pp. 1-18 (via http://www.ms.uky.edu/~lee/ma502/notes2/node7.html).
We assume the following to be given:
A set (i.e. totality) of objects called natural numbers, possessing the properties--called axioms--to be listed below.
Before formulating the axioms we make some remarks about the symbols = and ≠ which will be used.
Unless otherwise specified, small italic letters will stand for natural numbers...
If x is given and y is given, then:
either x and y are the same number; this may be written x=y (to be read ``equals'');
or x and y are not the same number; this may be written x≠y ( to be read ``is not equal to'').
Accordingly, the following are true on purely logical grounds:
- x=x for every x.
- If x=y then y=x.
- If x=y, y=z then x=z.
- Axiom 1:
- 1 is a natural number. That is, our set is not empty; it contains an object called 1 (read ``one'').
- Axiom 2:
- For each x there exists exactly one natural number, called the successor of x, which will be denoted by x'.
- Axiom 3:
- We always have x' ≠ 1 . That is, there exists no number whose successor is 1. That is, there exists no number whose successor is 1.
- Axiom 4:
- If x'=y' then x=y. That is, for any given number there exists either no number or exactly one number whose successor is the given number.
- Axiom 5 (Axiom of Induction):
- Let there be given a set M of natural numbers, with the following properties:
- I.
- 1 belongs to M.
- II.
- If x belongs to M then so does x'.
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