Fundamental Property of Integers

[ChuckleFox]

So there is a proof that the sum of any two even numbers is an even number.

2k + 2l = 2(k +l)

We have written the sum as 2 times an integer. Therefore the sum of any two even numbers is an even number.

An essential part of this proof is that k + l is an integer. How do we know this? Is it an assumed property of integers, an axiom, or is there a proof out there that this is true?

 

[alfred_Tarski]

Closure axiom

 

[micromass]

So there is a proof that the sum of any two even numbers is an even number.

2k + 2l = 2(k +l)

We have written the sum as 2 times an integer. Therefore the sum of any two even numbers is an even number.

An essential part of this proof is that k + l is an integer. How do we know this? Is it an assumed property of integers, an axiom, or is there a proof out there that this is true?

How did you define integers, and how did you define +? The answer to your problem relies on this.

 

[ChuckleFox]

How did you define integers, and how did you define +? The answer to your problem relies on this.

We didn't define anything. We just used our intuitive understanding of integers and addition. I assume this makes what we were doing uselessly vague?

Your answer makes me think that integers and addition can be defined in different ways. Do different theories exist depending on how we define integers and addition? For example, if we define integers in two different ways, does this lead to two different number theories? I mean this in the same way that different axioms lead to different geometries.

 

[micromass]

We didn't define anything. We just used our intuitive understanding of integers and addition. I assume this makes what we were doing uselessly vague?

No, it's not uselessly vague. It is probably fine for what your course/book needs. It's not always possibly to do things starting from the axioms, and it would probably be overkill here. However, it does mean that you'll have to take some things for granted, for example that the sum of two integers is an integer. But since that is very intuitive anyway, it shouldn't be a problem.

Your answer makes me think that integers and addition can be defined in different ways.

Yes. Do different theories exist depending on how we define integers and addition? For example, if we define integers in two different ways, does this lead to two different number theories? I mean this in the same way that different axioms lead to different geometries.

Now, we can prove that all theories are "isomorphic" (this means that they are the same for all practical purposes). We can alter the axioms a bit and get a new theory, but we don't call this the integers anymore.