Tom McCabe: The category of Hilbert spaces includes spaces of both finite and infinite dimension, so it presumably includes both countable and uncountable infinities.

mitchell porter: But the Hilbert space of a quantum field, naively, ought to have uncountable dimension, because there are continuum-many degrees of freedom.

Given any cardinal number, there exists a Hilbert space with that orthogonal dimension. Note, however, that even if the dimension is uncountable, individual elements are still given by linear combinations with countably many terms. In other words, only countably many dimensions are used "at one time" in specifying an element.

Thus, a Hilbert space formalism cannot accommodate "uncountably many degrees of freedom" in the sense people mean here. Which is okay, because I don't think that's what you need anyway.