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A characterisation of Π₁-conservativity over IΣ₁

Conference contribution
Authors Rasmus Blanck
Published in Journées sur les Arithmétiques Faibles 35, 6/6-7/6 2016, Lisbon, Portugal
Publication year 2016
Published at Department of Philosophy, Linguistics and Theory of Science
Language en
Keywords arithmetised metamathematics, partial conservativity, fragments of arithmetic
Subject categories Mathematical logic, Logic


By putting together a number of classic results due to Orey, Hájek, Guaspari and Lindström we get the well known characterisation of Π₁-conservativity over extensions T of Peano arithmetic PA. In short, the following are equivalent for a sentence φ: 1. T + φ is Π₁-conservative over T, 2 T + φ is interpretable in T, 3. for each n ∈ ω, T ⊢ Con(T|n + φ) 4. every model of T can be end-extended to a model of T + φ, 5. every countable model of T can be end-extended to a model of T + φ, 6. for every model M of T, T + Th-Σ₁(M) + φ is consistent. If we instead consider extensions T of IΣ₁, the characterisation breaks down. In this case, neither of 1 or 2 implies the other; we can never have 3 if T is finitely axiomatised; and regarding 4, it is not even known if every model of IΣ₁ has a proper end-extension to a model of IΣ₁. In this talk, which reports on joint work with Ali Enayat, we show that it is possible to salvage parts of this characterisation for extensions of IΣ₁. The equivalence of 1, 5 and 6 can still be shown to hold, and we also present another equivalent condition, which is similar to 3, but phrased in terms of bounded provability.

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