A brief
google search reveals that the
natural numbers can be defined by the
Peano axioms as such:
Let N be the set of natural numbers.
- There exists a natural number which we will call 1.
- For all x in N, there exists a successor, called x + 1.
- For all x in N, 1 != x + 1.
- For all x and y in N, x+1=y+1 => x=y.
- Mathematical Induction works.
Except for the last bullet point,
Wikipedia has a more English explanation than the one given here. No $650 dollars needed,
go buy yourself a new pair of pants!
If we were talking about
fields, there exists one where 1 + 1 = 1.
However, we're talking about the natural numbers. So, for the natural numbers, 1 + 1 != 1. We can
define a number, 2, to be the successor of 1. So, 1 + 1 = 2.
QED.
Back in school, we called this powerful technique
proof by definition.