I know that the question of cardinalities does not come up in the usual discussions or explanations of the Casimir effect, as in http://en.wikipedia.org/wiki/Casimir_effect, and invoking different cardinalities in an argument as given in is questionable. However, it raises the question whether the statement in the latter that there are an infinite number of possible wavelengths both in and outside the region between the plates can be valid. I put my musings in the form of statements, but that is in the form of trial balloons, and would be happy to have them shot down by showing me the faults in the assuredly naive arguments. I emphasize that I am talking not about the possible number of waves, but the possible number of wavelengths. First, between the plates, the possible wavelengths would follow the harmonic sequence up to the point at which the wavelength would get down to the Planck scale, so that any wavelength smaller would make the virtual particle collapse into a black hole.... hence, this limit means that there would be a finite number of possible wavelengths between the plates. As far as outside the plate, there are two points: one, the statement in that latter source that the number of wavelengths is the infinity of the continuum seems strange: the fact that energy is quantized would seem to indicate that the number of possible wavelengths would be countable (the infinity of the natural numbers). Then, and here I am on very shaky ground, would the size of the universe at any one moment pose an upper bound for the size of a possible wavelength, bringing the possibilities back down to the finite range?
Thanks for any indications.