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9Semi-honest subrecursive degrees and the collection rule in arithmeticArchive for Mathematical Logic 63 (1): 163-180. 2023.By a result of L.D. Beklemishev, the hierarchy of nested applications of the $$\Sigma _1$$ -collection rule over any $$\Pi _2$$ -axiomatizable base theory extending Elementary Arithmetic collapses to its first level. We prove that this result cannot in general be extended to base theories of arbitrary quantifier complexity. In fact, given any recursively enumerable set of true $$\Pi _2$$ -sentences, S, we construct a sound $$(\Sigma _2 \! \vee \! \Pi _2)$$ -axiomatized theory T extending S such …Read more
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31Some Results on LΔ — n+1Mathematical Logic Quarterly 47 (4): 503-512. 2001.We study the quantifier complexity and the relative strength of some fragments of arithmetic axiomatized by induction and minimization schemes for Δn+1 formulas
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28Verifying the bridge between simplicial topology and algebra: the Eilenberg-Zilber algorithmLogic Journal of the IGPL 22 (1): 39-65. 2014.
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14Lipschitz and Wadge binary games in second order arithmeticAnnals of Pure and Applied Logic 174 (9): 103301. 2023.
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33A note on parameter free Π1 -induction and restricted exponentiationMathematical Logic Quarterly 57 (5): 444-455. 2011.We characterize the sets of all Π2 and all equation image theorems of IΠ−1 in terms of restricted exponentiation, and use these characterizations to prove that both sets are not deductively equivalent. We also discuss how these results generalize to n > 0. As an application, we prove that a conservation theorem of Beklemishev stating that IΠ−n + 1 is conservative over IΣ−n with respect to equation image sentences cannot be extended to Πn + 2 sentences. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, We…Read more
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24Envelopes, indicators and conservativenessMathematical Logic Quarterly 52 (1): 51-70. 2006.A well known theorem proved by J. Paris and H. Friedman states that BΣn +1 is a Πn +2-conservative extension of IΣn . In this paper, as a continuation of our previous work on collection schemes for Δn +1-formulas , we study a general version of this theorem and characterize theories T such that T + BΣn +1 is a Πn +2-conservative extension of T . We prove that this conservativeness property is equivalent to a model-theoretic property relating Πn-envelopes and Πn-indicators for T . The analysis of…Read more
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33Fragments of Arithmetic and true sentencesMathematical Logic Quarterly 51 (3): 313-328. 2005.By a theorem of R. Kaye, J. Paris and C. Dimitracopoulos, the class of the Πn+1-sentences true in the standard model is the only consistent Πn+1-theory which extends the scheme of induction for parameter free Πn+1-formulas. Motivated by this result, we present a systematic study of extensions of bounded quantifier complexity of fragments of first-order Peano Arithmetic. Here, we improve that result and show that this property describes a general phenomenon valid for parameter free schemes. As a …Read more
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28Existentially closed models in the framework of arithmeticJournal of Symbolic Logic 81 (2): 774-788. 2016.
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8On the Optimality of Conservation Results for Local Reflection in ArithmeticJournal of Symbolic Logic 78 (4): 1025-1035. 2013.
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7Local induction and provably total computable functionsAnnals of Pure and Applied Logic 165 (9): 1429-1444. 2014.Let Iπ2 denote the fragment of Peano Arithmetic obtained by restricting the induction scheme to parameter free Π2Π2 formulas. Answering a question of R. Kaye, L. Beklemishev showed that the provably total computable functions of Iπ2 are, precisely, the primitive recursive ones. In this work we give a new proof of this fact through an analysis of certain local variants of induction principles closely related to Iπ2. In this way, we obtain a more direct answer to Kaye's question, avoiding the meta…Read more
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McDermott, J., B11 Milders, M., B23 Needham, A., 215 Newman, RS, B45 Niedeggen, M., B23Cognition 94 257. 2005.
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17On axiom schemes for T-provably $${\Delta_{1}}$$ Δ 1 formulasArchive for Mathematical Logic 53 (3-4): 327-349. 2014.This paper investigates the status of the fragments of Peano Arithmetic obtained by restricting induction, collection and least number axiom schemes to formulas which are Δ1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta_1}$$\end{document} provably in an arithmetic theory T. In particular, we determine the pr…Read more
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A note on parameter free N1-induction and restricted exponentiationMathematical Logic Quarterly 57 (5): 444-455. 2011.
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18Induction, minimization and collection for Δ n+1 (T)–formulasArchive for Mathematical Logic 43 (4): 505-541. 2004.For a theory T, we study relationships among IΔ n +1 (T), LΔ n+1 (T) and B * Δ n+1 (T). These theories are obtained restricting the schemes of induction, minimization and (a version of) collection to Δ n+1 (T) formulas. We obtain conditions on T (T is an extension of B * Δ n+1 (T) or Δ n+1 (T) is closed (in T) under bounded quantification) under which IΔ n+1 (T) and LΔ n+1 (T) are equivalent. These conditions depend on Th Πn +2 (T), the Π n+2 –consequences of T. The first condition is connected …Read more
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19On the quantifier complexity of Δ n+1 (T)– inductionArchive for Mathematical Logic 43 (3): 371-398. 2004.In this paper we continue the study of the theories IΔ n+1 (T), initiated in [7]. We focus on the quantifier complexity of these fragments and theirs (non)finite axiomatization. A characterization is obtained for the class of theories such that IΔ n+1 (T) is Π n+2 –axiomatizable. In particular, IΔ n+1 (IΔ n+1 ) gives an axiomatization of Th Π n+2 (IΔ n+1 ) and is not finitely axiomatizable. This fact relates the fragment IΔ n+1 (IΔ n+1 ) to induction rule for Δ n+1 –formulas. Our arguments, invo…Read more
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45Science as public sphere?Social Epistemology 21 (1). 2007.In this paper we argue that the best way to explain the normative framework of science is to adopt a model inspired in the democratic characterization of a public sphere. This model assumes and develops some deliberative democratic principles about the inclusiveness of the concerned, the parity of the reasons and the general interest of the subjects. In contrast to both bargaining models and to power-inspired models of the scientific activities, the model of scientific public sphere proposes to …Read more
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University of SydneyGraduate student
Areas of Interest
Aesthetics |
Philosophy of Social Science |