If is often taken for granted that on a rotating disk it is possible to operate a global 3+1 splitting of spacetime such that both lengths and time intervals are uniquely defined in terms of measurements performed by real rods and real clocks at rest on the platform. This paper shows that this assumption, although widespread and apparently trivial, leads to an anisotropy of the velocity of two light beams traveling in opposite directions along the rim of the disk, which (...) in turn implies some recently pointed out paradoxical consequences undermining the self-consistency of the special theory of relativity (SRT). A correct application of the SRT solves the problem and recovers complete internal consistency for the theory. As an immediate consequence, it is shown that the Sagnac effect only depends on the nonhomogeneity of time of the platform and has nothing to do with any anisotropy of the speed of light along the rim of the disk, contrary to an incorrect but widely held idea. (shrink)