Why the principle of counting does not match with our common sense Principle of counting says that 
"the number of odd integers, which is the same as the number of even integers, is also the same as the number of integers overall."
This does not match with my common sense (I am not a mathematician, but a CS student).
Can some people here could help me to reach a mathematicians level of thinking for this problem. I have searched net a lot (Wikipedia also)

Because sets of numbers can be infinitely divisible. See this Reddit comment.
I think his intuition comes from the fact that the world is discrete in practice. You have 2x more atoms in [0, 2cm] than in [0, 1cm]. If you are not looking at something made of atoms, let's say you have 2x more Planck lengths in [0, 2cm] than in [0, 1cm]. See what I mean?
OP's intuition can be correct for physical things in our world, but mathematics go beyond that, with rational numbers being infinitely divisible. As soon as there is a limit to how much you can divide things, even if it's one million digits after the decimal point, OP's intuition is valid.
