Suppose you are given two sets.  One contains 2 items and the other 3.  You form a single set and count them 1 to 5.  This is addition.  What is wrong with this?

Addition from Peano Axioms is consistent with that view, but gives an axiomatic basis for it.  The point of axioms is to have statements in math that lets us make definitions, state theorems and prove the theorems all based on the axioms without adding obvious statements.

Physical counting is not math.  When we take the above approach and ask what is the math version of it, we get methods of defining addition that are within the framework of the Peano Axioms.

We can define S(i,j) as you start at i and then repeat succession j times, S(i,j) is the result.  This definition corresponds to the two sets and just count them definition we started the post with.  We can think of this as taking two head segments from 0 to i and 0 to j and aligning the 0 of the second with the i of the first.  We then get a mapping from 0 to N(i,j) with the N(i,j) corresponding to the translated j.  This works as a pair of number lines.  Take the first number line as 0 to i and put the 0 of the second number line above the i of the first.  Then N(i,j) is under the j of the second number line.

We have to justify the S(i,j) definition using the Recursion Theorem in the Peano Axiom framework.  When we do order first, we can define S(i,j;n) for horizons n from 0 onwards.  We require that i and j are less than or equal to n for each horizon function n.   Where they overlap, the horizon functions are consistent.  We simply append the new addition facts at the end of each horizon to get the next one, copying the rest of the function.  The horizon functions are consistent on the overlaps.   We then take the union of the ordered pairs from each horizon function discarding duplicates.  This gives us the infinite horizon addition function S(i,j).

Want this a bit more slowly with examples?  Buy the book.  Want this stated as theorems with proofs?  Buy the book.