Thread: Quick question...

How does one go about rigorously proving that the sum of two integers is an integer?

The answer is likely simple, but I ask this because I have a feeling the subject is axiomatic in nature.

Start with a precise definition of integer.

Um... I can't really find a mathematically-precise definition via search engine, as all of the ones I find are by intuitive approach, even in Wikipedia.

Is it fair to say that an integer is a number that is either a natural number (starting at 0) or its negative?

Originally Posted by epidecus

Um... I can't really find a mathematically-precise definition via search engine, as all of the ones I find are by intuitive approach, even in Wikipedia.

Is it fair to say that an integer is a number that is either a natural number (starting at 0) or its negative?

Now you need a precise definition of natural number.

Hi epidicus

The way the integers are defined axiomatically is that they are closed under all the basic arithmetic operations. So there is nothing to prove in that space, as if you have something that isn't closed under addition it can't be the integers.

If you want to prove that something like the integers actually is constructable in set theory, you have to start by defining the natural numbers as some set construction and then slowly pulling yourself up the ladder, showing at each step that all the axioms we want to hold are true.

Assuming you have defined the natural numbers , one way of constructing the integers is as an equivalence class on where if .

Originally Posted by mathman

Originally Posted by epidecus

Um... I can't really find a mathematically-precise definition via search engine, as all of the ones I find are by intuitive approach, even in Wikipedia.

Is it fair to say that an integer is a number that is either a natural number (starting at 0) or its negative?

Now you need a precise definition of natural number.

And this is where the axioms come in, I think. I'm gonna have to get used to all this axiological proofing.

Originally Posted by river_rat

Hi epidicus

The way the integers are defined axiomatically is that they are closed under all the basic arithmetic operations. So there is nothing to prove in that space, as if you have something that isn't closed under addition it can't be the integers.

Ah, thanks. I found "axiom of closure". So if I ever have to prove that the sum of two integers is an integer, I can just cite the axiom of closure (which I guess, as you said, wouldn't really be "proving").

What I find confusing is this... Several axioms, including this one, don't really sound like axioms to me, at least not at first. Not that I'm doubting the system in anyway; I know this is just how it is. But I can't really get myself to understand why. To say "the set of integers is defined to be closed under these operations" sounds more like a property that you would observe after defining the integers themselves. In other words, I'm taking this more like an observational postulate, rather than a laid-out axiom. But it's probably just because I'm not that well-versed at all in the foundational mathematics. Do you get what I'm saying?