Prove: 1+1=2
Starting from what basis? In Peano's axioms for the natural number system, in which "successor" is taken as a primitive notion, "2" is defined as the "successor" of 1 and then "a+ 1" is defined to be the successor of a. From that, immediately, 1+ 1 is the successor of 1, 2.
Peano's axioms: The natural numbers is a set of objects, N, (called "numbers") together with a "successor function", s(n), such that:
1) There is a unique member of the set (called "1") such that the successor function maps N one to one and onto N-{1}.
2) If A is a subset of N such that 1 is in A and, whenever x is in A, the successor of x, s(x), is in A, then A= N.
Once we have that we define "2" to be the successor of "1", "3" to be the successor of "2", "4" to be the successor of "3", etc.
We define "a+ b" by:
1) a+ 1 is s(a).
2) if b is not 1, then there exist c such that b= s(c). In that case, a+ b= s(a+ c).
From (1), 1+ 1= s(1)= 2.
A little more interesting is that 2+ 2= 4. Since 2 is NOT 1, 2= s(1) so 2+ 2= s(2+ 1). 2+ 1= s(2)= 3 so 2+ 2= s(3)= 4.
Without knowing your mathematical background and hence the type of proof expected, asking for a proof of 1 + 1 = 2 is a waste of time. eg. An axiomatic derivation belongs in the Logic subforum (see Russell and Whitehead) and you would be advised to go to a library, find their book and read it.
Thread closed.