2+2=4...?

I'm asking in all seriousness, but what level/area of mathematics is needed prove simple statements like 2+2=4 and what not? Can it be proved with elementary math, or do you have to get into axioms and other crazy shit?

>>19
Define 2. Define apples. Define count. Define write it down. Define see what they make. Define equals. Define place them together.

Etc.

Also, fail.