CONSTRUCTIVE ARITHMETICAL IMPOSSIBILITIES AND THEIR RELATION TO PARADOXES

Abstract This study focuses on certain combinations of rules or conditions involving a would-be ‘provability’ or ‘truth’ predicate that would render a system of arithmetic containing them either straightforwardly inconsistent (if those predicates were assumed to be definable) or logico-semantically paradoxical (if those predicates were taken as primitive and governed by the rules in question). These two negative properties are not to be conflated; we conjecture, however, that they are complementary. Logico-semantic paradoxicality, we contend, admits of proof-theoretic analysis: the ‘disproofs’ involved do not reveal straightforward inconsistency. This is because, unlike the disproofs involved in establishing straightforward inconsistencies, these paradox-revealing ‘disproofs’ cannot be brought into normal form. The border between metamathematical proofs of certain (constructive) impossibility results and the non-normalizable (and always constructive) disproofs engendered by semantic paradoxicality is not fully understood. The respective strategies of reasoning on each side—genuine proofs of inconsistency versus whatever kind of ‘disproof’ uncovers semantic paradoxicality—seem somehow similar. They seem to involve the same ‘lines of reasoning’. But there is an important and principled difference between them. This difference will be emphasized throughout our discussion of certain arithmetical impossibility results, and closely related semantic paradoxes. The proof-theoretic criterion for paradoxicality is that in the case of paradoxes (as opposed to genuine inconsistencies) the apparent ‘disproofs’ that use the rules stipulated for the primitive predicates in question cannot be brought into normal form. In proof-theoretic terminology: their reduction sequences do not terminate. This means that cut fails for languages generating paradox. But cut holds for the language of arithmetic. It follows that the paradox-generating primitive predicates of a semantically closed language cannot be defined in arithmetical terms. For, if they could be, then they could be replaced by their definitions within the paradoxical disproofs, and the resulting disproofs would be normalizable.