Jesse said > So, although the set of all well-defined finite descriptions must clearly > be > "countable" in the traditional sense where arbitrary mappings are allowed, > it is not countable if only finite-describable mappings are allowed, > although it can easily be shown to be smaller than another countable set, > namely the set of all finite descriptions without regard for whether they > are "well-defined" or not

Doesn't this unintuitive result show that you are doomed if you only believe in the platonic existence of mathematical objects with finite descriptions? Was this the point you were making, despite saying earlier you were skeptical of the existence of objects without a finite description? - David