The interesting thing here is that it's much harder to put this problem properly into mathematical terms than it is to solve it. The whole insight here is that

The interesting thing here is that it's much harder to put this problem properly into mathematical terms than it is to solve it. The whole insight here is that you a "description" of number is just some finite sequence of symbols from a finite alphabet. Now, if you understand why cardinality of continuum is greater than aleph null, it's totally straightforward to show that there are only countably many descriptions, but continuum many reals, it's the kind of problem you give freshmen students on their first encounter with cardinalities. Thus, the big achievement of this guy is to actually come up with idea of indescribable numbers by himself and interpreting this question mathematically. In other words, the questions are usually more important than answers. There's one minor, but nevertheless important mistake made by the author. Author says : >The infinity just a step bigger than [aleph null] is Aleph one, the infinity of real numbers: The infinity of an impossibly dense line of numbers While cardinality of reals is certainly larger than cardinality of integers, it is not true that it's just one step larger. Funny thing is that it's not false either: it was proved by Cantor and Cohen that it's impossible to prove or disprove that aleph one is cardinality of continuum. This is the famous continuum hypothesis.