Can We Follow the Omega Rule? Infinitary Inference and Human Reasoning

In his forthcoming Philosophical Quarterly article, Brett Topey challenges the assumption that finite beings cannot employ the ω‑rule—an infinitary rule of inference. Responding especially to Jared Warren, Topey argues that apparent uses of the ω‑rule can be more plausibly explained via finitary reasoning. This inquiry raises fundamental questions about arithmetical determinacy, mathematical language, and the limits of human inference.

This podcast delves into a philosophical debate regarding whether finite beings can follow infinitary rules of inference, specifically the omega rule. The omega rule posits that if one accepts an infinite number of premises asserting a property for each natural number, one can then conclude that all natural numbers possess that property. The source outlines Brett Topey's challenge to Jared Warren's argument, which claims we do follow this rule, primarily through the thought experiment of a "supertask computer" checking Goldbach's Conjecture. Topey counters with a finitary reconstruction of our reasoning, the "pause button" variant, and a reinterpretation of Warren's "Martian mathematicians" to demonstrate that our inferences are based on generalizations to instances, not the reverse. Ultimately, the text argues that if finite minds cannot follow the omega rule, then the determinacy of arithmetical truth cannot be explained by its application, suggesting that "some other route" is required.

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Pleae note that this NotebookLM podcast may contain errors and it is therefore important to refer to the source documents.

Brett Topey, Can we follow the omega rule?, The Philosophical Quarterly, 2025;, pqaf059, https://doi.org/10.1093/pq/pqaf059

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