this post was submitted on 02 Oct 2025
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For every integer, there are an infinite number of real numbers until the next integer. So you can't make a 1:1 correspondence. They're both infinite, but this shows that the reals are more infinite. (and yeah, as other people mentioned, it's the 1:1 correspondence, countability, that matters more than the infinite quantity of the Real numbers)
There are infinitely many rational numbers between any two integers (or any two rationals), yet the rationals are still countable, so this reasoning doesn't hold.
The only simple intuition for the uncountability of the reals I know of is Cantor's diagonal argument.
You can assign each rational number a single unique integer though if you use a simple algorithm. So the 1:1 correspondence holds up (though both are still infinite)
There are also an infinite number of rationale between two integers, but the rationals are still countable and therefore have the same cardinality as the naturals and integers.
Makes sense, thanks!