So I got asked a question earlier on these forums to prove that 1+1=2 and I think this is a correct proof and I was wondering if there was any other math buffs here who could check it for me.

Thanks!

The proof starts from the Peano Postulates, which define the natural

numbers N. N is the smallest set satisfying these postulates:

1) 1 is in N.

2) If x is in N, then its "successor" x' is in N.

3) There is no x such that x' = 1.

4) If x isn't 1, then there is a y in N such that y' = x.

5) If S is a subset of N, 1 is in S, and the implication

(x in S => x' in S) holds, then S = N.

Then you have to define addition recursively:

Def: Let a and b be in N. If b = 1, then define a + b = a'

(using P1 and P2). If b isn't 1, then let c' = b, with c in N

(using P4), and define a + b = (a + c)'.

Then you have to define 2:

Def: 2 = 1'

Theorem: 1 + 1 = 2

Proof: Use the first part of the definition of + with a = b = 1.

Then 1 + 1 = 1' = 2 Q.E.D.