Setton, on 17 March 2013 - 12:08 AM, said:

Yes.

P1. 1 is in N.

P2. If x is in N, then its "successor" x' is in N.

P3. There is no x such that x' = 1.

P4. If x isn't 1, then there is a y in N such that y' = x.

P5. 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'

2 is in N by P1, P2, and the definition of 2.

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.

This is a mental construct. It requires an education in algebra to even begin to understand. It proves nothing unless you believe it does, and/or have the knowledge to understand and appreciate it. Unless you believe algebra is real and workable and that one can substitute letters for numbers and make them equate and balance out, then it, again, proves nothing.

Here is a simpler constuct. Here is

**one** pea. Look at it and observe it. See how it is sitting by itslef Now I will add another pea. Look at the two peas together. That grouping is what humans call two. Now pull the peas apart again. See how we have one, and then another one? Push them together again. There is the configuration we call two. Do this as many times as you like, and you will always observe the same pattern of groupings. One, one, then two.

Hence we can observe that, when we place one pea with another one pea, we have what we call in English two peas.

Does this explanation also require belief? Sure to accpet it as PROOF, you have to believe me when I say that pushing two singular peas together will AWAYS give you two peas. But it something anyone can experiment with and find true for themselves.

**Edited by Mr Walker, 19 March 2013 - 09:45 AM.**

You are a child of the universe, no less than the trees and the stars; you have a right to be here. And whether or not it is clear to you, no doubt the universe is unfolding as it should.

Therefore be at peace with God, whatever you conceive Him to be, and whatever your labors and aspirations, in the noisy confusion of life keep peace with your soul.

With all its sham, drudgery and broken dreams, it is still a beautiful world..

Be cheerful.

Strive to be happy.