Neo-Logicism and Gödel’s incompleteness theorems

Thesis

There is a long-standing gap in the literature as to whether Gödelian incompleteness constitutes a challenge for Neo-Logicism, and if so how serious it is. In this thesis, I articulate and address the challenge in detail. The Neo-Logicist project is to demonstrate the analyticity of arithmetic by deriving all its truths from logical principles and suitable definitions. The specific concern raised by Gödel’s first incompleteness theorem is that no single sound system of logic syntactically implies all arithmetical truths. I set out some responses that initially seem appealing and explain why they are not compelling. The upshot is that NeoLogicism either offers an epistemic route only to some truths of arithmetic; or that it has to move from a syntactic to a semantic notion of logical consequence, which risks undermining its epistemic goals. I discuss Crispin Wright’s recent attempt to address Gödelian incompleteness, which I argue is not satisfactory. Instead, I explore alternative responses for the Neo-Logicist such as adopting a semantic consequence relationship and invoking reflection principles.

I also discusses whether Frege had a proto-concept of completeness. Existing literature denies this. Yet, Frege’s project is usually understood as finding a formal system from which all arithmetical truths could be proven. Furthermore, Frege is credited with devising the first calculus complete for first-order logic. How are we to reconcile the first claim with the latter two? I argue that Frege did not just stumble upon this complete calculus, but in fact had an early conception of both theory- and calculus-completeness.

Authors: Pregel, F

• DOI: 10.5287/ora-ng74va9kz
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English