Is there a scientific reason why you cannot find a one-to-one correspondence between the integers and the rational numbers no matter how hard you try?
I know there is one for positive integers and positive rationals
I don't know about negatives, though
This is one of the things that I've never been able to wrap my head around. I can get the whole finding a number that exists in irrationals but can't in rationals thing, but simply numbering them? This is one case where it hurts my head because no matter how many numbers are between whatever arbitrary numbers you have from the rationals, you'll still have enough integers to cover those because the integers don't end. I get how each time you iterate to the next rational number, the integers get further and further behind; however, the integers will eventually get to that point...
To me, saying you can't create a one to one for this is like saying that just because a car is travelling at 10mph, it can't pass the same 100 miles that the car going 1000mph was going.