THE PROBLEMATIC NATURE OF GÖDEL’S DISJUNCTIONS AND LUCAS-PENROSE’S THESES

We show that the name “Lucas-Penrose thesis” encompasses several differ-ent theses. All these theses refer to extremely vague concepts, and so are either practically meaningless, or obviously false. The arguments for the various theses, in turn, are based on confusions with regard to the meaning(s) of these vague notions, and on unjustified hidden assumptions concerning them. All these observations are true also for all interest-ing versions of the much weaker (and by far more widely accepted) thesis known as “Gö-del disjunction”. Our main conclusions are that pure mathematical theorems cannot decide alone any question which is not purely mathematical, and that an argument that cannot be fully formalized cannot be taken as a mathematical proof.