Undecidability and the Reference of Formal Systems

Abstract

This paper attempts to provide an account of the reference of formal systems. I assume (on grounds that I cannot lay out fully) that formal systems can be considered to be referential, that is, capable of formulating truths in the correspondence sense, on two conditions: 1. that they are consistent and 2. that they contain true but unprovable formulas. The first of these conditions is self-evident; the second, by contrast, cannot be assumed without begging the question, without presupposing truth before accounting for its possibility. I argue, however, that Kurt Gödel’s proof of the inevitability of undecidable formulas in any formal system provides a ground for assuming the existence of true but unprovable sentences without presupposing objective truth. For this, however, we need to develop a different sense of ‘true’ from what is usually assigned to the undecidable formula. Using insights from Jacques Derrida, I argue that we can legitimately conceptualize the truth of the undecidable formula as referring not to some objective reality but to the formal system itself.