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prove that 1+1=2

The proof starts from the Peano Postulates, which define the natural 
numbers N. N is the smallest set satisfying these postulates:

  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.

Note: There is an alternate formulation of the Peano Postulates which 
replaces 1 with 0 in P1, P3, P4, and P5. Then you have to change the 
definition of addition to this:
  Def: Let a and b be in N. If b = 0, then define a + b = a.
       If b isn't 0, then let c' = b, with c in N, and define
       a + b = (a + c)'.

You also have to define 1 = 0', and 2 = 1'. Then the proof of the 
Theorem above is a little different:

Proof: Use the second part of the definition of + first:
       1 + 1 = (1 + 0)'
       Now use the first part of the definition of + on the sum in
       parentheses:  1 + 1 = (1)' = 1' = 2  Q.E.D.