Question

The sum of two integers is always an integer ?

Answer 1

Define 0 and the operation +1
State as an axiom that x in N => x+1 in N
State as an axiom that 0 in N
Now you know 1 = 0 +1; 2 = 1 +1 = 0 +1 +1; etc

Then work your way up and define x + y = x + 1 + 1 + 1 + 1....etc
You're done!

The opening assumptions are basically axioms, you can replace them with set theory axioms if you like but it doesn't make much difference....

Answer 2

Yes,the sum of two integers is always a integer