On parameter free induction schemas (original) (raw)

The Nonαxiomαtizαbility of 1.(0^) by Finitely Many Schemata

2003

Under set-theoretic hypotheses, it is proved by Magidor and Malitz that logic with the Magidor-Malitz quantifier in the Ki -interpretation is recursively axiomatizable. It is shown here, under no additional settheoretic hypotheses, that this logic cannot be axiomatized by finitely many schemata. Magidor and Malitz [2] introduced the ^-variable-binding quantifiers Q. The language L(Q") is formed by adding Q to first-order predicate logic. For an infinite cardinal K, QnX\X2.. .xnφ may be assigned the so-called /c-interpretation in a structure ΐPίί, wherein QX\.. .xnφ is satisfied if there exists a n A c ΐPίί of power K that is homogeneous for φ, i.e., for any au... ,an E A, φ(ctι,... 9an) holds in 9K. Among many other results Magidor and Malitz establish, under the set-theoretic axiom 0K l, a completeness theorem for L(Q") in the Kt-interpretation (hereafter LίQ^)). Unfortunately, the complete axiom system for L(Qκj) exhibited in [2] lacks the simplicity of, e.g., Keisler&#3...

A Note on Induction Schemas in Bounded Arithmetic

Arxiv preprint cs/0210011, 2002

Abstract: As is well known, Buss' theory of bounded arithmetic $ S^{1} _ {2} $ proves Sigma0b(Sigma1b)−LIND\ Sigma_ {0}^{b}(\ Sigma_ {1}^{b})-LIND Sigma0b(Sigma1b)−LIND; however, we show that Allen's $ D_ {2}^{1} $ does not prove Sigma0b(Sigma1b)−LLIND\ Sigma_ {0}^{b}(\ Sigma_ {1}^{b})-LLIND Sigma0b(Sigma1b)−LLIND unless $ P= NC .Wealsogivesomeinterestingalternativeaxiomatisationsof. We also give some interesting alternative axiomatisations of .Wealsogivesomeinterestingalternativeaxiomatisationsof S^{1} _ {2} $.

Axiomatizations of Compositional Inductive-Recursive Definitions

2018

xi Acknowledgement xiii 0 Introduction 1 0.1 Martin-Löf Type Theory and Inductive Types . . . . . . . . . . . . . . . 2 0.2 Basic Examples of Induction-Recursion . . . . . . . . . . . . . . . . . . . 3 0.3 Axiomatizations of Induction-Recursion . . . . . . . . . . . . . . . . . . . 5 0.3.1 DS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 0.3.2 DS′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 0.4 Compositionality (Outline of the Thesis) . . . . . . . . . . . . . . . . . . 7 0.4.1 Why Composing Codes? . . . . . . . . . . . . . . . . . . . . . . . 8 0.4.2 Uniform Codes for Induction-Recursion (UF) . . . . . . . . . . . . 8 0.4.3 Polynomial Codes for Induction-Recursion (PN) . . . . . . . . . . 9 0.5 Formalization in Agda, and Notation . . . . . . . . . . . . . . . . . . . . 10 0.6 Structure of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 0.6.1 Publications and Contributions . . . . . . . . . . . . . . . . ....