Hell, it took Russell and Whitehead, in *Principia Mathematica*, some 300 pages to prove that 1+1=2!

So don't feel bad if you couldn't do it so easily.

That was possible, because of the bisection of the angle. Since trisection is impossible using compass and straight edge (in affine geometry), it would have taken a lot more to prove that 2+1=3 (you need affine geometry plus a metric). I will prefer my practical approach, thanks. In number theory, mathematical induction (successor function) is an axiom, it isn't attempted to be proven.

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Addition is a function that maps two natural numbers (two elements of N) to another one. It is defined recursively as:

a+0=a, (1)

a+S(b)=S(a+b). (2)

For example:

a+1=a+S(0) by definition

=S(a+0) using (2)

=S(a), using (1)

a+2=a+S(1) by definition

=S(a+1) using (2)

=S(S(a)) using a+1=S(a)

a+3=a+S(2) by definition

=S(a+2) using (2)

=S(S(S(a))) using a+2=S(S(a))

etc.

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Basically assuming that ... "a commutative monoid with identity element 0" exists. Most mathematicians agree that it does, but not all ... those who reject infinity for example.