We demonstrate how the unusual mathematics of transfinite numbers, in particular, a nearly perfect realization of Hilbert's famous hotel paradox, manifests in the propagation of light through fractional vortex plates. It is shown how a fractional vortex plate can be used, in principle, to create any number of open rooms, i.e., topological charges, simultaneously. Fractional vortex plates are therefore demonstrated to create a singularity of topological charge, in which the vortex state is completely undefined and in fact arbitrary. These results hint that transfinite mathematics is much more common and important to optical systems than previously imagined. (C) 2016 Optical Society of America