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April 19, 2012

Why "1+1=2"?


We come across many arguments that attempt to prove or disprove the statement 1+1=2. We can put away many of such attempts under the category called "Crackpot proofs".

Before going forward with any proof, we must ask what it means to say, "1+1=2".

The meanings of natural numbers are normally defined under what we call Peano arithmetic (PA). PA defines natural numbers in terms of five fundamental axioms:


1. 0 is a natural number.
2. Every natural number, n, has a successor s(n).
3. No natural number has 0 as its successor.
4. Every natural number has a unique successor.
5. The axiom of induction.

The statement, 1+1=2, is a direct interpretation of axiom number two. Therefore, it is not decidable (neither can be proved nor disproved) under PA. One can extend PA to argue about the validity of 1+1=2.

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