Measurements of the magnetic field induced in the moon while it is in the geomagnetic tail lobes have been interpreted in terms of lunar magnetic permeability due to free iron content; such studies ignored the possibility that a highly conducting lunar core (Fe or FeS) would exclude magnetic fields with an apparent diamagnetic effect. Using lunar chemical and thermal models to determine plausible limits of magnetic permeability, we interpret measurements of the induced moment. The maximum likely radius of a lunar core is 580 km. Subsatellite and ALSEP measurements of the induced field are in disagreement. Resolving the differences is critical to determining whether a core could or does exist.