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, thenM = 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 xand y in S - can be proved by induction (which is based on Peano'sPostulate #4

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