$n \in \{0,1,2,...\}$ (which determines the energy) is non-negative, while the quantum number $m = n - n' \in {{\mathbb{Z}}}$ (which determines the angular momentum) is an arbitrary integer. The infinite degeneracy of the Landau levels is now obvious because states with the same $n$ but different values of $n'$ have the same energy.

One may wonder why in subsection 2.3 we found an infinite degeneracy labeled by the continuous momentum $p_y$ and now we only find a countable variety of degenerate states (labeled by the integer $m$). This apparent discrepancy is due to the implicit consideration of two different Hilbert spaces. While the states in the discrete variety labeled by $m$ are normalizable in the usual sense, the continuous variety of plane wave states labeled by $p_y$ is normalized to $\delta$-functions and thus belongs to an extended Hilbert space.

It is remarkable that a quantum mechanical system containing just a single particle has an even infinitely degenerate ground state. The existence of infinitely degenerate ground states is usually associated with the spontaneous breakdown of a continuous global symmetry in systems with infinitely many degrees of freedom. Does the infinite degeneracy of the single-particle Landau levels have anything to do 