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CONSERVATIVITY OF ULTRAFILTERS OVER SUBSYSTEMS OF SECOND ORDER ARITHMETIC
2018
Journal of Symbolic Logic (JSL)
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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
2012
Journal of Mathematical Logic
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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
2008
Annals of Pure and Applied Logic
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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
2006
Fundamenta Mathematicae
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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
2015
Proceedings of the American Mathematical Society
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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
2018
Annual Conference for Computer Science Logic
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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]
2015
arXiv
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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]
2018
arXiv
pre-print
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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
2012
Journal of Logic and Analysis
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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
1984
Journal of Symbolic Logic (JSL)
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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
1983
Journal of Symbolic Logic (JSL)
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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
2011
Bulletin of Symbolic Logic
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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
2009
Archive for Mathematical Logic
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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]
2019
arXiv
pre-print
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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]
2019
arXiv
pre-print
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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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