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CONSERVATIVITY OF ULTRAFILTERS OVER SUBSYSTEMS OF SECOND ORDER ARITHMETIC

ANTONIO MONTALBÁN, RICHARD A. SHORE
2018 Journal of Symbolic Logic (JSL)  
AbstractWe extend the usual language of second order arithmetic to one in which we can discuss an ultrafilter over of the sets of a given model.  ...  We prove that adding these axioms to IHT produce conservative extensions of ACA0 +IHT, ${\rm{ACA}}_{\rm{0}}^ +$, ATR0, ${\rm{\Pi }}_2^1$-CA0, and ${\rm{\Pi }}_2^1$-CA0 for all sentences of second order  ...  In the standard way, this then shows that R is conservative over T for all sentences of second order arithmetic.  ... 
doi:10.1017/jsl.2017.76 fatcat:chbnjayr5zaa7ilu4p7p3pxdfa

NON-PRINCIPAL ULTRAFILTERS, PROGRAM EXTRACTION AND HIGHER-ORDER REVERSE MATHEMATICS

ALEXANDER P. KREUZER
2012 Journal of Mathematical Logic  
We investigate the strength of the existence of a non-principal ultrafilter over fragments of higher order arithmetic.  ...  We show that ACA_0^ω+U is Π^1_2-conservative over ACA_0^ω and thus that ACA_0^ω+ is conservative over PA.  ...  All of these systems are conservative over their second-order counterparts, where the second-order part is given by functions instead of sets.  ... 
doi:10.1142/s021906131250002x fatcat:qf7nhx25tvg37kyaqesvoxrbgq

A standard model of Peano arithmetic with no conservative elementary extension

Ali Enayat
2008 Annals of Pure and Applied Logic  
numbers such that the expansion N A := (N, A) A∈A of the standard model N := (ω, +, ×) of Peano arithmetic has no conservative elementary extension, i.e., for any elementary extension there is a subset  ...  Schmerl, The Structure of Models of Peano Arithmetic, Oxford University Press, 2006, Question 7] by showing that there exists an uncountable arithmetically closed family A of subsets of the set ω of natural  ...  the proof of Theorem A.  ... 
doi:10.1016/j.apal.2008.07.005 fatcat:b6qddclth5amjhwbwcepulh3ti

Automorphisms of models of bounded arithmetic

Ali Enayat
2006 Fundamenta Mathematicae  
We establish the following model theoretic characterization of the fragment I∆ 0 +Exp+BΣ 1 of Peano arithmetic in terms of fixed points of automorphisms of models of bounded arithmetic (the fragment I∆  ...  Moreover, if j is fixed point free, then the fixed point set of j is isomorphic to M. Here Aut(X) is the group of automorphisms of the structure X, and Q is the ordered set of rationals.  ...  A REFINEMENT OF THEOREM A The author has recently established a refinement of Theorem A by characterizing the subsystem W KL * 0 of second order arithmetic in terms of automorphisms of models of I∆ 0 .  ... 
doi:10.4064/fm192-1-3 fatcat:45m34gnz2bafbp4s5xgbdgmuj4

Measure theory and higher order arithmetic

Alexander P. Kreuzer
2015 Proceedings of the American Mathematical Society  
We obtain that over ACA 0 ω + (μ) the existence of the Lebesgue measure is Π 1 2 -conservative over ACA 0 ω and with this conservative over PA.  ...  For this reason ACA 0 ω + (μ) is the weakest fragment of higher order arithmetic where σ-additive measures are directly definable.  ...  These systems are conservative over their second order counterparts, where the second order part is given by functions instead of sets.  ... 
doi:10.1090/proc/12671 fatcat:ff3tab76nnffzmmfrqsxedjv7u

MacNeille Completion and Buchholz' Omega Rule for Parameter-Free Second Order Logics

Kazushige Terui, Michael Wagner
2018 Annual Conference for Computer Science Logic  
Buchholz' Ω-rule is a way to give a syntactic, possibly ordinal-free proof of cut elimination for various subsystems of second order arithmetic.  ...  In this paper, we consider a family of sequent calculi LIP = n≥−1 LIP n for the parameterfree fragments of second order intuitionistic logic, that corresponds to the family ID <ω = n<ω ID n of arithmetical  ...  Introduction This paper is concerned with cut elimination for subsystems of second order logics.  ... 
doi:10.4230/lipics.csl.2018.37 dblp:conf/csl/Terui18 fatcat:am7snl5n2zcetginjmqfkgea4i

Measure theory and higher order arithmetic [article]

Alexander P. Kreuzer
2015 arXiv   pre-print
We obtain that over ACA_0^ω + (μ) the existence of the Lebesgue measure is Π^1_2-conservative over ACA_0^ω and with this conservative over PA.  ...  For this reasons ACA_0^ω + (μ) is the weakest fragment of higher order arithmetic where σ-additive measures are directly definable.  ...  These systems are conservative over their second order counterparts, where the second order part is given by functions instead of sets.  ... 
arXiv:1312.1531v2 fatcat:tctxumkoyvb6rapk3hdhc7553e

Proof Theory of Constructive Systems: Inductive Types and Univalence [article]

Michael Rathjen
2018 arXiv   pre-print
A further goal is to investigate the strength of intuitionistic theories of generalized inductive definitions in the framework of intuitionistic explicit mathematics that lie beyond the reach of Martin-Lof  ...  Some of the reductions are known only through ordinal-theoretic characterizations. The paper also addresses the strength of Voevodsky's univalence axiom.  ...  (ix) The classical subsystem of second order arithmetic (Σ 1 2 -AC) + BI (same as (∆ 1 2 -CA) + BI). (x ) ) The intuitionistic system IARI of second order arithmetic.  ... 
arXiv:1610.02191v2 fatcat:poqgfttkgjfrxo5yvpohajq7zq

A logical analysis of the generalized Banach contractions principle

Alexander Kreuzer
2012 Journal of Logic and Analysis  
We show that this principle is a consequence of Ramsey's theorem for pairs over, roughly, RCA 0 + Σ 0 2 -IA.  ...  The author would like to thank the anonymous referees for comments and suggestions that have helped to improve the presentation of this paper.  ...  The system RCA ω 0 is conservative over its second-order counterpart, where the second-order part is given by functions instead of sets. This second-order system can then be interpreted in RCA 0 .  ... 
doi:10.4115/jla.2012.4.17 fatcat:rl4gnjb7fjg5jd6uj66oc72i4y

Annual Meeting of the Association for Symbolic Logic

George Boolos, Sy Friedman
1984 Journal of Symbolic Logic (JSL)  
The next result shows a parallel between second order arithmetic and propositional modal logics with quantifiers over the propositional quantifiers.  ...  Tj + RDC in finite types is conservative over Tj w.r.t. arithmetic sentences. THEOREM 6.  ...  In the enriched logic, the property of being a natural number can be proved to be a I-lI property, using the permutability of the U quantifier with a second-order V quantifier to obtain also a 27 characterization  ... 
doi:10.2307/2274306 fatcat:kkksnqcog5g6zls66ku5abamd4

Annual Meeting of the Association for Symbolic Logic, Philadelphia 1981

Simon Kochen, Hugues Leblanc, Charles D. Parsons
1983 Journal of Symbolic Logic (JSL)  
The next result shows a parallel between second order arithmetic and propositional modal logics with quantifiers over the propositional quantifiers.  ...  Tj + RDC in finite types is conservative over Tj w.r.t. arithmetic sentences. THEOREM 6.  ...  In the enriched logic, the property of being a natural number can be proved to be a I-lI property, using the permutability of the U quantifier with a second-order V quantifier to obtain also a 27 characterization  ... 
doi:10.2307/2273495 fatcat:qwtcwtawv5hnjhb3ut4wgqogwq

Open Questions in Reverse Mathematics

Antonio Montalbán
2011 Bulletin of Symbolic Logic  
We also mention some of the areas of reverse mathematics that are starting to be developed and where interesting open question may be found.  ...  We present a list of open questions in reverse mathematics, including some relevant background information for each question.  ...  See [AS06] for a study of the fundamental aspects of the theory of metric, Hilbert, and Banach spaces in the context of subsystems of second-order arithmetic.  ... 
doi:10.2178/bsl/1309952320 fatcat:maim2j6vovfcpjjkrzm7xeicqi

$${\Pi^1_2}$$ -comprehension and the property of Ramsey

Christoph Heinatsch
2009 Archive for Mathematical Logic  
For example, take the subsystem ACA 0 (which is an abbreviation for arithmetical comprehension axiom) which is a conservative extension of Peano arithmetic.  ...  When we prove our main theorem, the situation will be similar: We will have a proof which uses a non principal ultrafilter on ω and formalize it in a subsystem of second order arithmetic by exploiting  ... 
doi:10.1007/s00153-009-0124-8 fatcat:b2e6rj5zcfhsvaqybn2o254xuq

Where Pigeonhole Principles meet König Lemmas [article]

David Belanger, Chitat Chong, Wei Wang, Tin Lok Wong, Yue Yang
2019 arXiv   pre-print
We show that the latter implies the existence of 2-random reals, and is conservative over the former.  ...  We study the pigeonhole principle for Σ_2-definable injections with domain twice as large as the codomain, and the weak König lemma for Δ^0_2-definable trees in which every level has at least half of the  ...  Acknowledgements We thank Keita Yokoyama for numerous fruitful discussions which led to a simplification of the proof of Theorem 5.10 and to the conception of Proposition 6.5.  ... 
arXiv:1912.03487v1 fatcat:pwhogaq6c5htbcjsnxwzd5s2s4

MacNeille completion and Buchholz' Omega rule for parameter-free second order logics [article]

Kazushige Terui
2019 arXiv   pre-print
Buchholz' Omega-rule is a way to give a syntactic, possibly ordinal-free proof of cut elimination for various subsystems of second order arithmetic.  ...  In this paper, we consider the parameter-free fragments LIP0, LIP1, LIP2, ... of the second order intuitionistic logic, that correspond to the arithmetical theories ID0, ID1, ID2, ... of iterated inductive  ...  Introduction This paper is concerned with cut elimination for subsystems of second order logics.  ... 
arXiv:1804.11066v2 fatcat:lcgdc3pltrdl5ouus7makbrouu
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