You can set up a one to one correspondence between all the points on a line (or in a square or in a cube) and all the clock faces a working clock can produce but you cannot do the same with all possible clock faces.
Perhaps I can help explain why this is wrong by giving a constructive counter-example. Correct me if I’m wrong, but by “the set of all clock faces” you mean the set of all positions the two hands of a clock could take. You can specify the position of the clock hands by stating the angles they make relative to any fixed position—say the 12 position for concreteness. Suppose the angles are expressed in radians and take values in the set [0, 2*pi). Multiply each angle by the conversion factor “1 rotation per 2*pi radians” to map the angles into the set [0, 1). Now you can express any specific “clock face” by a point in the unit square. Then you can return to my original point about space-filling curves showing that this has the same cardinality as a line segment.
Perhaps I can help explain why this is wrong by giving a constructive counter-example. Correct me if I’m wrong, but by “the set of all clock faces” you mean the set of all positions the two hands of a clock could take. You can specify the position of the clock hands by stating the angles they make relative to any fixed position—say the 12 position for concreteness. Suppose the angles are expressed in radians and take values in the set [0, 2*pi). Multiply each angle by the conversion factor “1 rotation per 2*pi radians” to map the angles into the set [0, 1). Now you can express any specific “clock face” by a point in the unit square. Then you can return to my original point about space-filling curves showing that this has the same cardinality as a line segment.