Gödel initiated the program of finding and justifying axioms that effect a significant reduction in incompleteness and he drew a fundamental distinction between intrinsic and extrinsic justifications. Reflection principles are the most promising candidates for new axioms that are intrinsically justified. Taking as our starting point Tait’s work on general reflection principles, we prove a series of limitative results concerning this approach. These results collectively show that general reflection principles are either weak ) or inconsistent. The philosophical significance of these (...) results is discussed. (shrink)

We force a property of cardinals first proved relatively consistent by Sargsyan, that of being supercompact but not equation image-supercompact, starting from a model of set theory which does not satisfy equation image and that contains supercompact cardinals.