Subject: Re: Is 1+1=2 an axiom?
Date: Tue, 09 Mar 2000 09:44:05 -0500
From: Alex Bogomolny
One thing you have to realize is that there is nothing in this world that is inherently demonstrable or not. Statements A,B,C follow from axioms a,b,c. But then quite often a,b,c can be deduced from A,B,C - and roles are reversed.
So, if you want to take 1+1=2 as an axiom, that's completely up to you. Be aware, though, that saying that is an easy part. For, before you declare 1+1=2 an axiom, you'll have to endow "+" and "=" with some meaning. You'll have to think about the other natural numbers: how you define them. Or even what is 2. How helpful your axiom will be in your attempts to answer these questions.
Therefore, although you are at liberty to declare 1+1=2 an axiom, you may ponder the wisdom of such an action. Unless you tell me how is it going to be used in the context of natural numbers development, I'll be questioning the wisdom of such a step. There are Peano's axioms that reduce natural numbers to very basic principles. I am completely satisfied with them. 1+1=2 in Peano's system is not an axiom but a derivable theorem.
To sum up, declaring 1+1=2 an axiom, as plain and as simple as it is, in itself is meaningless unless it is set in a context of additional axioms and term definitions. You should go the whole length to see whether 1+1=2 is a convenient axiom or not. I, personally, would not do that.
All the best,