# Proof that the sum of two integers is an integer [closed]

How do you prove that the sum of any two integers is always an integer? Even though it is as obvious as it gets, proving it is actually tricky, because doing so involves some circular reasoning. This makes me wonder if it is just an axiom, or if there is a proof, lying somewhere deeper into mathematics.

• What definition of integer are you starting from? How is addiotion definied in the context where you think you need a proof? Commented Sep 5 at 1:51
• Need a definition of integers first. And a definition of addition second. But if you are using anything akin to the Peano postulates it follows by induction. Commented Sep 5 at 1:55
• Dang those down-voters. This is actually a really intelligent question. A difficult one to answer without going into the many ways to define the integers but it shows an accute awareness of the subtleties of mathematical construction. Commented Sep 5 at 2:05
• As others have mentioned, this is meaningless without more context. It's trivial given the right set of axioms, but specifying those axioms is highly non trivial.
– lulu
Commented Sep 5 at 2:32
• Depends on how you define things, but at heart, this is an axiom. Usually, we start with natural numbers, define integers later. The axiom applies to sums of natural numbers, not integers. But most of the result is in the natural numbers. Commented Sep 5 at 2:36