A note on Grzegorczyk's logic

Grzegorczyk's modal logic (Grz) corresponds to the class of upwards well‐founded partially ordered Kripke frames, however all known proofs of this fact utilize some form of the Axiom of Choice; G. Boolos asked in [1], whether it is provable in plain ZF. We answer his question negatively: Grz corresponds (in ZF) to a class of frames, which does not provably coincide with upwards well‐founded posets in ZF alone. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)