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John T. Baldwin
2011 Proceedings of the 11th Asian Logic Conference  
It reflects discussions with set theorists during my stay at Mittag-Leffler and discussion with the Infinity project members in Barcelona.  ...  In particular, we illustrate the role of higher dimensional amalgamations and sketch the role of a weak extension of ZFC in the proof.  ...  So this is also arithmetic. ii) By i), the set of p such that 'p is in S at (A)' is arithmetic (a fortiori Σ 1 1 ) in A, so by Lemma 5.4.ii, each such p is hyperarithmetic in A.  ... 
doi:10.1142/9789814360548_0002 fatcat:doszxnnu5reqnh6gz2amz3v6tq

Partial evaluation of the discrete solution of elliptic boundary value problems

Wolfgang Hackbusch, Florian Drechsler
2012 Computing and Visualization in Science  
The latter approach is in particular suited for problems with oscillatory coefficients, where one is not interested in all details of the solution.  ...  This reduces the storage amount and still enables a partial evaluation of the solution (restricted to the skeletons of the remaining subdomains).  ...  The low rank blocks are selected from a hierarchy of partitions organised in a so-called cluster tree that provide hierarchies of partitionings.  ... 
doi:10.1007/s00791-013-0211-6 fatcat:wxylvou2tbeeda5h2jejxewdr4

A Robust Unscented Transformation for Uncertain Moments

Hugo T.M. Kussaba, João Y. Ishihara, Leonardo R.A.X. Menezes
2019 Journal of the Franklin Institute  
In this scenario, the moment matching equations are reformulated as a system of polynomial equations and inequalities, and it is proposed to use the Chebychev center of the solution set as a robust UT.  ...  This paper proposes a robust version of the unscented transform (UT) for one-dimensional random variables. It is assumed that the moments are not exactly known, but are known to lie in intervals.  ...  A naive choice of sigma points would be the use of the arithmetic mean of the lower and upper bounds for the moments in (19).  ... 
doi:10.1016/j.jfranklin.2019.02.018 fatcat:4mnosd6kwbh3fm4ctow6tzrkqi

Completely determined Borel sets and measurability [article]

Linda Westrick
2021 arXiv   pre-print
We consider the reverse math strength of the statement 𝖢-𝖣𝖬:"Every completely determined Borel set is measurable."  ...  Whenever M⊆ 2^ω is the second-order part of an ω-model of 𝖢-𝖣𝖬, then for every Z ∈ M, there is a R ∈ M such that R is Δ^1_1-random relative to Z.  ...  Thus there is an ordinal b ∈ O such that H R0 b computes either a jump hierarchy on or a descending sequence in a * . But recognizing a jump hierarchy or a descending sequence is arithmetic.  ... 
arXiv:2001.01881v2 fatcat:nyyuzn6qqbeqnep5jbxwdcabyq

Complete L_ω_1,ω-Sentences with Maximal Models in Multiple Cardinalities [article]

John Baldwin, Ioannis Souldatos
2018 arXiv   pre-print
In [BKS15] examples of incomplete sentences are given with maximal models in more than one cardinality. The question was raised whether one can find similar examples of complete sentences.  ...  From (homogeneous) characterizability of κ we construct sentences with maximal models in κ and in one of κ^+, κ^ω, 2^κ and more.  ...  All the examples in this paper have maximal models in some cardinalities and using settheory we can identify the maximality cardinals in the ℵ-hierarchy.  ... 
arXiv:1508.06620v3 fatcat:sa6zy2querbq7mvscv7g3bbh3a

Well Quasiorders and Hierarchy Theory [article]

Victor Selivanov
2018 arXiv   pre-print
While the classical hierarchies of sets usually degenerate to structures very close to ordinals, the extension of them to functions requires more complicated WQOs, and the same applies to reducibilities  ...  We discuss some applications of WQOs to several fields were hierarchies and reducibilities are the principal classification tools, notably to Descriptive Set Theory, Computability theory and Automata Theory  ...  I am grateful to Leibniz-Zentrum für Informatik for accepting and taking care of Dagstuhl Seminars 08271, 11411, 15392, 16031 which were important for promoting the topic of this paper.  ... 
arXiv:1809.02941v1 fatcat:7ztqokcr7ret3fiusnzszegyxa

Bi-interpretation in weak set theories [article]

Alfredo Roque Freire, Joel David Hamkins
2020 arXiv   pre-print
Similarly, using a construction of Mathias, we prove that every model of ZF is bi-interpretable with a model of Zermelo set theory in which the replacement axiom fails.  ...  In contrast to the robust mutual interpretability phenomenon in set theory, Ali Enayat proved that bi-interpretation is absent: distinct theories extending ZF are never bi-interpretable and models of ZF  ...  Was the use of the V α hierarchy significant? Can one prove the analogues of the theorems for ZFC − , that is, for set theory without the power set axiom?  ... 
arXiv:2001.05262v2 fatcat:nbt6d73iwrffvofconvds2si24

HyperLTL Satisfiability is Σ_1^1-complete, HyperCTL* Satisfiability is Σ_1^2-complete [article]

Marie Fortin, Louwe B. Kuijer, Patrick Totzke, Martin Zimmermann
2021 arXiv   pre-print
Finally, we show that the membership problem for every level of the HyperLTL quantifier alternation hierarchy is Π_1^1-complete.  ...  In this paper we settle the exact complexity of these problems, showing that both are in fact highly undecidable: we prove that HyperLTL satisfiability is Σ_1^1-complete and HyperCTL* satisfiability is  ...  So, Σ 0 1 is part of the first level of the arithmetic hierarchy, Σ 1 1 is part of the first level of the analytical hierarchy, while Σ 2 1 is not even analytical.  ... 
arXiv:2105.04176v1 fatcat:of66ktmjtrhcrja43lgrqyl4wi

Can logic be combined with probability? Probably

Colin Howson
2009 Journal of Applied Logic  
In the nineteen sixties seminal work was done by Gaifman and then Scott and Krauss in adapting the concepts, tools and procedures of the model theory of modern logic to provide a corresponding model theory  ...  in which notions of probabilistic consistency and consequence are defined analogously to the way they are defined (semantically) in the deductive case.  ...  various classes of probability functions definable in L(N) appear in the arithmetical hierarchy.  ... 
doi:10.1016/j.jal.2007.11.003 fatcat:logxhdrc3vcxlkv6fbn7ffaxq4

Fixpoints, games and the difference hierarchy

Julian C. Bradfield
2003 RAIRO - Theoretical Informatics and Applications  
This both provides a simple characterization of the fixpoint hierarchy, and refines existing results on the power of the game quantifier in descriptive set theory.  ...  Drawing on an analogy with temporal fixpoint logic, we relate the arithmetic fixpoint definable sets to the winning positions of certain games, namely games whose winning conditions lie in the difference  ...  The following "well-known" result on the transfinite difference hierarchy over Kleene pointclasses is the motivation for extending Theorem 8: Fact 12. ∂ <ω1 (Σ 0 n ) = ∆ 0 n+1 and in particular, Σ ∂ <ω1  ... 
doi:10.1051/ita:2003011 fatcat:4hwj5rljynabbm7vc7af2y4kim

"Mathematics is the Logic of the Infinite": Zermelo's Project of Infinitary Logic

Jerzy Pogonowski
2021 Studies in Logic, Grammar and Rhetoric  
I also stress the fact that the second axiomatization of set theory provided by Zermelo in 1930 involved the use of extremal axioms of a very specific sort.1  ...  In this paper I discuss Ernst Zermelo's ideas concerning the possibility of developing a system of infinitary logic that, in his opinion, should be suitable for mathematical inferences.  ...  The standard model of Peano arithmetic can be characterized in Lω1 ω ; the same concerns the class of all finite sets. 2.  ... 
doi:10.2478/slgr-2021-0042 fatcat:nnowhogl3bhklm234xa7dion7u

Noncommutative polynomial optimization under symmetry [article]

Marie Ioannou, Denis Rosset
2021 arXiv   pre-print
We also apply our technique to the Sliwa inequalities in the Bell scenario with three parties with binary measurements settings/outcomes.  ...  Symmetry reduction is key to scale the applications of the NPA relaxation, and our formalism encompasses and generalizes the approaches found in the literature.  ...  First of all, it reduces the problem size so that it can be handled in exact arithmetic.  ... 
arXiv:2112.10803v1 fatcat:ksue5itlq5dclfjob4lio6ymv4


2014 Bulletin of Symbolic Logic  
Moreover, these methods (e.g., the stability hierarchy) provide an organization for much mathematics which gives specific content to dreams of Bourbaki about the architecture of mathematics.  ...  The stability hierarchy extends this virtue to other complete theories.  ...  with tools of the arithmetic algebraic geometry.  ... 
doi:10.1017/bsl.2014.3 fatcat:mgy3v22mz5b6fhk6h7s5je47d4

The determined property of Baire in reverse math [article]

Eric P. Astor, Damir Dzhafarov, Antonio Montalbán, Reed Solomon, Linda Brown Westrick
2018 arXiv   pre-print
We define the notion of a determined Borel code in reverse math, and consider the principle DPB, which states that every determined Borel set has the property of Baire.  ...  We show that whenever M⊆ 2^ω is the second-order part of an ω-model of DPB, then for every Z ∈ M, there is a G ∈ M such that G is Δ^1_1-generic relative to Z.  ...  This is because, in general, it might take a jump hierarchy the height of the rank of T in order to produce an evaluation map.  ... 
arXiv:1809.03940v2 fatcat:gcdbwftcwvffxcxrj6haz3xypa

Incomparable ω_1-like models of set theory [article]

Gunter Fuchs and Victoria Gitman and Joel David Hamkins
2015 arXiv   pre-print
We show that the analogues of the Hamkins embedding theorems, proved for the countable models of set theory, do not hold when extended to the uncountable realm of ω_1-like models of set theory.  ...  sets is not universal for the ω_1-like models of set theory.  ...  The research of the third author has been supported in part by NSF grant DMS-0800762, PSC-CUNY grant 64732-00-42, CUNY Collaborative Incentive Award 80209-06 20 and Simons Foundation grant 209252.  ... 
arXiv:1501.01022v1 fatcat:ixsqgzst7fb3dj6h3twoeiwlfi
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