Using the Peano axioms, which are often used as the basis for arithmetic, you first define a successor function, often denoted as •' and the number 0. The natural numbers (including 0) then are defined by repeated application of the successor function (of course, you also first need to define what equality is):

0 = 0
1 := 0'
2 := 1' = 0''

etc

Addition, denoted by •+• , is then recursively defined via

a + 0 = a
a + b' = (a+b)'

which quickly gives you that 1+1=2. But that requires you to thake these axioms for granted. Mathematicians proved it with fewer assumptions, but the proof got a tad verbose