No, as #[0,1] is aleph one, whilst #Z+ is aleph null. I.e. the real numbers are a denser, uncountable infinity. The reason the author says the issue is at infinitely many layers, is because this would be alephnull^2, which is provably aleph one.
To better understand this without needing to understand Cantor’s diagonalisation method, simply ask yourself, if you label the first passenger of your coach 1, then what will the second be labelled? (Remembering, of course, that infinitesimal numbers do not exist in the reals, but rather the Hyperreals and Surreals, and other sets of this nature). Indeed, only the first and last passenger is given an explicit real number, as the reals are not countable and hence not bijective to aleph null.