Proof that 1 + 1 = 2 Using Peano's Postulates

Date: 09/12/2002 at 11:38:08
From: Janrik Öberg
Subject: How to prove that 1+1=2.

How do I prove that 1 + 1 = 2?

Could you please explain this, or is it too difficult? (For me to 
understand or for you to explain.) I've read a paper proving it,
but unfortunately I only understand about half of it.

-nb5

Date: 09/12/2002 at 12:25:33
From: Doctor Jerry
Subject: Re: How to prove that 1+1=2.

Hi Janrik,

Peano's Postulates are:

1. Let S be a set such that for each element x of S there exists a 
   unique element x' of S.

2. There is an element in S, we shall call it 1, such that for every 
   element x of S, 1 is not equal to x'.

3. If x and y are elements of S such that x' = y', then x = y.

4. If M is any subset of S such that 1 is an element of M, and for 
   every element x of M, the element x' is also an element of M, then
   M = S.

Just as a matter of notation, we write 1' = 2, 2' = 3, etc. We define 
addition in S as follows:

   (a1)  x + 1 = x'
   (a2)  x + y' = (x + y)'

The element x + y is called the sum of x and y.

Now to prove that 1 + 1 = 2.

From (a1), with x = 1, we see that 1 + 1 = 1' = 2.

Standard properties of addition - for example, x + y = y + x for all x
and y in S - can be proved by induction (which is based on Peano's
Postulate #4.

If the above proof seems too easy, we can try to show that 2 + 2 = 4.

2 + 2 = 2 + 1' = 3' = 4.

- Doctor Jerry, The Math Forum
  http://mathforum.org/dr.math/