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?

take 2 apples and count them, write it down
write the + symbol as you are going to get another 2 apples and see what they make
take another set of 2 apples and count them, write it down next to the + as this set of apples is the one you are going to put with the others
write equals as you are finnished getting the apples
place thew apples together and count them...

1,2,3,4

2+2=4