So I am going over the Completeness Axiom, and I have a question, First I will explain what I think my answer is and my logic , and I am waiting for feedback thank you.
So the question is about the Examples that the lecturer gave,
As you can see in the picture it says for integers, not dense but without gaps and for rationals it says dense but with gaps.
I am having trouble with understanding this,
I understand the fact that rationals have gaps, because there are numbers like square root of something or number e or pi, numbers that can't be represented with rationals, therefore they are not rationals so there are a lot of gaps on the number line.
But how come Z is without gaps and not dense. I mean I understand the fact that it's not dense because you can't put a number between 1 and 2 or any two integers.
But why is it without gaps? Is it because numbers like 1/2 are not defined in Z? According to the definition that they gave us an integer is in the form of n-m when n and m is naturals, so you can't really talk about fractions in Z because it has no meaning, I guess. Is my logic correct?