Can you prove that 1+1=2 or is it an axiom?

[Unredeemed]

Can you prove that 1+1=2 or is it an axiom?

 

[Unknot]

Well, use peano axioms. Look it up yourself!

 

[Hurkyl]

Can you prove that 1+1=2 or is it an axiom?

Most commonly, that (or something very similar) is taken as the definition of the symbol '2'.

 

[Unredeemed]

Well, use peano axioms. Look it up yourself!

Could you say that:

x+1=S(x)

and, therefore
1+1=S(1)
S(1)=2

Would that work as a proof?

 

[Unknot]

I wouldn't call that a proof, unless you were proving it a la Russell/Whitehead.

As Hurkyl said, it probably is a definition more than anything else. 2 is the number next to 1.

 

[Unredeemed]

Okay, thanks :)

 

[CRGreathouse]

I consider S(S(0)) to be the definition of "2". As such, "1 + 1 = 2" unpacks as "S(0) + S(0) = S(S(0))". From the definition of addition, S(0) + S(0) is S(S(0) + 0). Also from the definition of addition, S(0) + 0 is S(0). Thus S(0) + S(0) = S(S(0)) and so 1 + 1 = 2.