Some infinities are bigger than others. This is a mathematical fact, and you can prove it yourself in the next two minutes.
Imagine a dark theater. You can't count the seats or the people, but if every person is in a seat and no seat is empty, you know: same number of each.
This trick -- pairing things up with nothing left over -- is the only tool you need. Mathematicians call it a bijection. It works even when the sets are infinite.
Multiply each natural by 2 to get an even. Divide each even by 2 to get back. Perfect pairing, nothing left over.
The integers stretch in both directions: ..., -3, -2, -1, 0, 1, 2, 3, ... but you can zigzag through them all.
Every fraction (p/q) can sit in an infinite grid. Columns are the numerator, rows the denominator. A zigzag path visits every cell, skipping duplicates like 2/2 = 1/1.
Naturals, evens, integers, rationals -- all the same size. Any set you can list in order (1st, 2nd, 3rd, ...) is called countable, and its size is aleph-null.
At this point you might think every infinity is the same. Cantor proved that wrong.
Pretend someone can list "every" real number between 0 and 1. We'll read one digit from each row -- the 1st digit of row 1, the 2nd digit of row 2, and so on -- and change each one to build a number that can't be anywhere on their list.
Our number disagrees with row 1 at position 1, row 2 at position 2, row 3 at position 3, and so on forever. It can't match any row on the list.
But the list was supposed to contain every real. We just proved it doesn't. And this trick works against any list, no matter how cleverly it's arranged.
The real numbers are fundamentally unlistable. No bijection with the naturals is possible.
There are more real numbers between 0 and 1 than there are natural numbers -- even though both sets are infinite. The reals have a strictly larger cardinality called c, the continuum.
And it never stops. For any infinite set, its power set is strictly bigger. There's an endless staircase of infinities, each inconceivably larger than the last. Georg Cantor proved this in the 1870s, and it became the foundation of modern set theory.