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Incoming links: Well-orders are not first-order axiomatizable
Past exam of the mathematics course of the University of Cambridge 2018 ii Paper 3 16G iv Solution Created 2026-09-24 Updated 2026-10-03
The class of well-orderings is not first-order axiomatizable. If axiomatized it, add constants and sentencesEvery finite subset has a finite well-order containing the required finite descending chain. By the compactness theorem, the whole set has a model; that model satisfies but contains an infinite descending chain, so it is not a well-order. This contradiction proves the claim, as summarized by well-orders are not first-order axiomatizable.