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How to prove 1+1=2
By Li Zhou on Wednesday, January 16, 2002 - 04:43 pm:

Why 1 plus 1 equals 2? Try to prove it!!

By Kerwin Hui on Wednesday, January 16, 2002 - 05:42 pm:

That depends on how you define things. To prove this statement, you will need to define what you mean by '1', '+1' and '2'. For example, if you take the Peano axioms and ZF set theory as your starting point (there are some confusions between books as to how we define the naturals. We shall take the naturals starting from 1):

The Peano axioms are (numbers here means the natural numbers):
i) there is a number '1', and there is a sucessor function.
ii) If a is a number, then the sucessor of a is also a number
iii) '1' is not a sucessor of any number
iv) If the sucessors of two numbers are equal, then the numbers themselves are equal.
v) (Induction axiom) If a set S contains '1' and also the sucessor of every member of S, then every number is in S.

We could write the axioms more precisely. For example
peano

Now we write f(1)=2, f(2)=3, ... and write a+1 for f(a). Can you see how this statement is a direct consequence of the Peano axioms? Obviously when you extend to the integers, the rationals, the reals or the complexes, you will have to prove that you have not change these properties of naturals.

Kerwin

By Yatir Halevi on Wednesday, January 16, 2002 - 05:43 pm:

This is actually a very hard question.
In the begining of the century Whitehead and Russel tried to prove all the fundamental theorem of arithmetics basing on a few axioms of set theory. It took the 100 pages before they could finally prove that 1+1=2.

Here is a proof that I sketched out, basing on two axioms:
m(a+b)=ma+mb
ma=a+a
Lets say that it is true:
1+1=2
multiplying both sides by 2 we get:
2(1+1)=2×2
2+2=2+2
Q.E.D

Yatir

By Gavin Adams on Wednesday, January 16, 2002 - 05:53 pm:

It's the way the natural numbers are defined

Taken from "Abstract Algebra" by W. E. Deskins:

"The system of natural numbers consists of a set N, two binary operations on N which are called addition and multiplication and are symbolized by + and ×, and the following axioms:

Axiom A. N contains exactly one element, denoted by 1, with the property that a × 1 = 1 × a = a for each a that is an element of N

Axiom B. For each a and each b of N, a×(b + 1) = a × b + a

Axiom C. For each a and each b of N, a + (b + 1) = (a + b) + 1

Axiom D. For each a!=1 in N there is a unique element b of N such that a = b + 1

Axiom E. For given elements a and b of N exactly one of these three equations, a = b, a + x = b, a = b + y where x and y are in N, is valid.

Axiom F. If M is a subset of N with the properties:
(i) 1 is an element of M,
and
(ii) k + 1 is in M whenever the element k (of N) is in M,
then M = N

So we see by axiom 6 that N is defined to have an element (2) that is k + 1 when k = 1. It doesn't really matter WHAT this element is, but it's the way our number system is defined. Of course, this system is VERY useful in DESCRIBING actions, but it is not necessarily true - only consistent.