Science and Randomness

[Mark 13:13]

I googled all the way to Peano axioms too, but I was on my mobile, so I'd take some half an hour to reply with that...
Let's see the important ones for the 1+1=2 discussion.

1. 0 is a natural number.

6. For every natural number n, S(n) is a natural number.

Peano's original formulation of the axioms used 1 instead of 0 as the "first" natural number. This choice is arbitrary, as axiom 1 does not endow the constant 0 with any additional properties. However, because 0 is the additive identity in arithmetic, most modern formulations of the Peano axioms start from 0. Axioms 1 and 6 define a unary representation of the natural numbers: the number 1 can be defined as S(0), 2 as S(S(0)) (which is also S(1)), and, in general, any natural number n as Sn(0). The next two axioms define the properties of this representation.

7. For every natural number n, S(n) = 0 is false. That is, there is no natural number whose successor is 0.

[...]

Addition is the function + : N × N → N (written in the usual infix notation, mapping two elements of N to another element of N), defined recursively as:
a+0=a,
a+S(b) = S(a+b)

Now, do the math!

hock: i know where you are at so i've tried to not get into anything to complicated, but knowing some of the members of the forum they will be able to settle the issue and explain it simply without getting too confusing.