Tarski's theorem and liar-like paradoxes
Ming Hsiung · Logic Journal of IGPL · 2013
Tarski's theorem essentially says that the Liar paradox is paradoxical in the minimal reflexive frame. We generalize this result to the Liar-like paradox λα for all ordinal α ≥ 1. The main result is that for any positive integer n = 2i(2j +1), the paradox λn is paradoxical in a frame iff this frame contains at least a cycle the depth of which is not divisible by 2i+1; and for any ordinal α ≥ ω, the paradox λα is paradoxical in a frame iff this frame contains at least an infinite walk that has an arbitrarily large depth. We thus get that λn has a degree of paradoxicality no more than λm iff the multiplicity of 2 in the (unique) prime factorization of n is no more than that in the prime factorization of m; and all tranfinite λα has the same degree of paradoxcality but has a higher degree of paradoxicality than any λn.