counting infinity is strictly smaller than uncountable infinity
Infinity is not one thing. Georg Cantor showed in 1874 that some infinities are genuinely larger than others. The integers, the fractions, and the even numbers are all equally infinite. But the real numbers form a strictly larger infinity, and no list can ever contain all of them.
Cantor's diagonal argument: why the reals cannot be listed
Sizes of infinity: a strict hierarchy
The natural numbers, integers, and rationals are all countably infinite: they can all be put in a one-to-one correspondence with each other. The real numbers are uncountably infinite: a strictly larger infinity. Between these two sizes, the Continuum Hypothesis asks whether there is anything in between.
Hilbert's Hotel: a hotel with infinitely many rooms, all full, always has room
Cantor proved in 1874 that not all infinities are equal. Natural numbers, integers, and rationals are countably infinite: they can be listed. Real numbers are uncountably infinite: no complete list exists, proved by the diagonal argument. Cantor's theorem shows the power set of any set has strictly larger cardinality than the set, generating an infinite hierarchy of infinities. The Continuum Hypothesis, that no infinity lies between the integers and reals, was proved independent of standard set theory.