Induction rules in bounded arithmetic

Jeřábek, Emil

doi:10.1007/s00153-019-00702-w

Cited by 3 publications

(1 citation statement)

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“…At first sight, the previous result for IΣ − 1 could seem to be a very particular property of the induction scheme in the formal arithmetic setting. However, and this was already observed by Kaye in [17], this property again corresponds to a very general purely logical fact for theories described in terms of conditional sentences, see theorem 1.4 in [17] (let us observe that similar results in the context of bounded arithmetic theories have been obtained by Jeřábek in [26]. )…”

Section: Finite Sets Of Conditional Sentencessupporting

confidence: 73%

On Conditional Axioms and Associated Inference Rules

Borrego-Díaz,

Cordón-Franco,

Lara-Martín

2024

Axioms

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In the present paper, we address the following general question in the framework of classical first-order logic. Assume that a certain mathematical principle can be formalized in a first-order language by a set E of conditional formulas of the form α(v)→β(v). Given a base theory T, we can use the set of conditional formulas E to extend the base theory in two natural ways. Either we add to T each formula in E as a new axiom (thus obtaining a theory denoted by T+E) or we extend T by using the formulas in E as instances of an inference rule (thus obtaining a theory denoted by T+E−Rule). The theory T+E will be stronger than T+E−Rule, but how much stronger can T+E be? More specifically, is T+E conservative over T+E−Rule for theorems of some fixed syntactical complexity Γ? Under very general assumptions on the set of conditional formulas E, we obtain two main conservation results in this regard. Firstly, if the formulas in E have low syntactical complexity with respect to some prescribed class of formulas Π and in the applications of E−Rule side formulas from the class Π and can be eliminated (in a certain precise sense), then T+E is ∀B(Π)-conservative over T+E−Rule. Secondly, if, in addition, E is a finite set with m conditional sentences, then nested applications of E−Rule of a depth at most of m suffice to obtain ∀B(Π) conservativity. These conservation results between axioms and inference rules extend well-known conservation theorems for fragments of first-order arithmetics to a general, purely logical framework.

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Section: Finite Sets Of Conditional Sentencessupporting

confidence: 73%