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- Jan 7th 2008, 02:59 AM #1

- Jan 7th 2008, 04:11 AM #2

- Jan 7th 2008, 08:54 AM #3

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- Jan 7th 2008, 09:20 AM #4
Having seen this before, if you were not given a set of axioms, then Peano's axioms may be helpful for this. They 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.

- Jan 7th 2008, 09:36 AM #5

- Jan 7th 2008, 10:46 AM #6
The way I have seen it done is thus:

As a matter of notation, we write 1' = 2, 2' = 3, etc. We define

addition in S as follows:

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

Now to prove that 1 + 1 = 2.

From 1: 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.

- Jan 7th 2008, 08:52 PM #7
Alfred North Whitehead and Bertrand Russell wrote Principia Mathematica and published it in three volumes in the years 1910-1913. In it they laid the foundation of modern mathematics. On page 362, they finally got around to proving that 1 + 1 = 2.

On the other hand, 1 + 1 = 3 for 1 sufficiently large

- Jan 7th 2008, 08:59 PM #8

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- Jan 7th 2008, 11:09 PM #9

- Jan 7th 2008, 11:41 PM #10

- Jan 8th 2008, 01:37 AM #11

- Jan 8th 2008, 05:10 AM #12

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- Jan 8th 2008, 05:26 AM #13

- Jan 8th 2008, 09:48 AM #14

- Jan 8th 2008, 08:48 PM #15