Philosophy of Mathematics seminar

Wesley Wrigley, London School of Economics

Absolutely Undecidable Arithmetical Propositions  

In an unpublished draft written in the 1930s, Gödel argued that the incompleteness theorems give us no reason to believe that some arithmetical propositions are undecidable in an absolute sense. In this talk, I’ll argue that his position presupposes that we possess, under an appropriate idealisation, an ability which I call the recursive-ordinal recognition ability. I’ll show that we have this ability if, and only if, there are no absolutely undecidable arithmetical propositions. I’ll finish by outlining an argument against our possession of this ability, and indicate the broad kind of propositions to which the absolutely undecidable ones must belong.