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CONSERVATIVITY OF ULTRAFILTERS OVER SUBSYSTEMS OF SECOND ORDER ARITHMETIC

2018
*
Journal of Symbolic Logic (JSL)
*

AbstractWe extend the usual language

doi:10.1017/jsl.2017.76
fatcat:chbnjayr5zaa7ilu4p7p3pxdfa
*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*. ...##
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NON-PRINCIPAL ULTRAFILTERS, PROGRAM EXTRACTION AND HIGHER-ORDER REVERSE MATHEMATICS

2012
*
Journal of Mathematical Logic
*

We investigate the strength

doi:10.1142/s021906131250002x
fatcat:qf7nhx25tvg37kyaqesvoxrbgq
*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. ...##
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A standard model of Peano arithmetic with no conservative elementary extension

2008
*
Annals of Pure and Applied Logic
*

numbers such that the expansion N A := (N, A) A∈A

doi:10.1016/j.apal.2008.07.005
fatcat:b6qddclth5amjhwbwcepulh3ti
*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. ...##
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Automorphisms of models of bounded arithmetic

2006
*
Fundamenta Mathematicae
*

We establish the following model theoretic characterization

doi:10.4064/fm192-1-3
fatcat:45m34gnz2bafbp4s5xgbdgmuj4
*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 . ...##
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Measure theory and higher order arithmetic

2015
*
Proceedings of the American Mathematical Society
*

We obtain that

doi:10.1090/proc/12671
fatcat:ff3tab76nnffzmmfrqsxedjv7u
*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. ...##
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MacNeille Completion and Buchholz' Omega Rule for Parameter-Free Second Order Logics

2018
*
Annual Conference for Computer Science Logic
*

Buchholz' Ω-rule is a way to give a syntactic, possibly ordinal-free proof

doi:10.4230/lipics.csl.2018.37
dblp:conf/csl/Terui18
fatcat:am7snl5n2zcetginjmqfkgea4i
*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. ...##
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Measure theory and higher order arithmetic
[article]

2015
*
arXiv
*
pre-print

We obtain that

arXiv:1312.1531v2
fatcat:tctxumkoyvb6rapk3hdhc7553e
*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. ...##
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Proof Theory of Constructive Systems: Inductive Types and Univalence
[article]

2018
*
arXiv
*
pre-print

A further goal is to investigate the strength

arXiv:1610.02191v2
fatcat:poqgfttkgjfrxo5yvpohajq7zq
*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*. ...##
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A logical analysis of the generalized Banach contractions principle

2012
*
Journal of Logic and Analysis
*

We show that this principle is a consequence

doi:10.4115/jla.2012.4.17
fatcat:rl4gnjb7fjg5jd6uj66oc72i4y
*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 . ...##
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Annual Meeting of the Association for Symbolic Logic

1984
*
Journal of Symbolic Logic (JSL)
*

The next result shows a parallel between

doi:10.2307/2274306
fatcat:kkksnqcog5g6zls66ku5abamd4
*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 ...##
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Annual Meeting of the Association for Symbolic Logic, Philadelphia 1981

1983
*
Journal of Symbolic Logic (JSL)
*

The next result shows a parallel between

doi:10.2307/2273495
fatcat:qwtcwtawv5hnjhb3ut4wgqogwq
*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 ...##
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Open Questions in Reverse Mathematics

2011
*
Bulletin of Symbolic Logic
*

We also mention some

doi:10.2178/bsl/1309952320
fatcat:maim2j6vovfcpjjkrzm7xeicqi
*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*. ...##
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$${\Pi^1_2}$$ -comprehension and the property of Ramsey

2009
*
Archive for Mathematical Logic
*

For example, take the

doi:10.1007/s00153-009-0124-8
fatcat:b2e6rj5zcfhsvaqybn2o254xuq
*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 ...##
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Where Pigeonhole Principles meet König Lemmas
[article]

2019
*
arXiv
*
pre-print

We show that the latter implies the existence

arXiv:1912.03487v1
fatcat:pwhogaq6c5htbcjsnxwzd5s2s4
*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. ...##
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MacNeille completion and Buchholz' Omega rule for parameter-free second order logics
[article]

2019
*
arXiv
*
pre-print

Buchholz' Omega-rule is a way to give a syntactic, possibly ordinal-free proof

arXiv:1804.11066v2
fatcat:lcgdc3pltrdl5ouus7makbrouu
*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. ...
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