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.
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.
