Benedict Eastaugh

Recursive counterexamples are classical mathematical theorems that are made false by restricting their quantifiers to computable objects. Historically, they have been important for analysing the mathematical limitations of foundational programs such as constructivism or predicativism. For example, the least upper bound principle is recursively false, and thus unprovable by constructivists. In this talk I will argue that while recursive counterexamples are valuable for analysing foundational positions from an external, set-theoretic point of view, the approach is limited in its applicability because the results themselves are not accessible to the foundational standpoints under consideration. This limitation can be overcome, to some extent, by internalising the recursive counterexamples within a theory acceptable to a proponent of a given foundation; this is, essentially, the method of reverse mathematics. I will examine to what extent the full import of reverse mathematical results can be appreciated from a given foundational standpoint, and propose an analysis based on an analogy with Brouwer’s weak and strong counterexamples. Finally, I will argue that, at least where the reverse mathematical analysis of foundations is concerned, the epistemic considerations above show that reverse mathematics should be carried out within a weak base theory such as RCA0, rather than by studying ω-models from a set-theoretic perspective.

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