Abstract: Publication date: Available online 15 October 2018Source: Annals of Pure and Applied LogicAuthor(s): Peter Holy, Philipp Lücke, Ana Njegomir We show that many large cardinal notions can be characterized in terms of the existence of certain elementary embeddings between transitive set-sized structures, that map their critical point to the large cardinal in question. As an application, we use such embeddings to provide new proofs of results of Christoph Weiß on the consistency strength of certain generalized tree properties. These new proofs eliminate problems contained in the original proofs provided by Weiß.

Abstract: Publication date: Available online 21 September 2018Source: Annals of Pure and Applied LogicAuthor(s): Sohei Iwata, Taishi Kurahashi In this paper, we establish a stronger version of Artemov's arithmetical completeness theorem of the Logic of Proofs LP0. Moreover, we prove a version of the uniform arithmetical completeness theorem of LP0.

Abstract: Publication date: Available online 21 September 2018Source: Annals of Pure and Applied LogicAuthor(s): Yatir Halevi We give a geometric description of the pair (V,p), where V is an algebraic variety over a non-trivially valued algebraically closed field K with valuation ring OK and p is a Zariski dense generically stable type concentrated on V, by defining a fully faithful functor to the category of schemes over OK with residual dominant morphisms over OK.Under this functor, the pair (an algebraic group, a generically stable generic type of a subgroup) gets sent to a group scheme over OK. This returns a geometric description of the subgroup as the set of OK-points of the group scheme, generalizing a previous result in the affine case.We also study a maximum modulus principle on schemes over OK and show that the schemes obtained by this functor enjoy it.

Abstract: Publication date: Available online 10 September 2018Source: Annals of Pure and Applied LogicAuthor(s): Christian Espíndola We present a unified categorical treatment of completeness theorems for several classical and intuitionistic infinitary logics with a proposed axiomatization. This provides new completeness theorems and subsumes previous ones by Gödel, Kripke, Beth, Karp and Joyal. As an application we prove, using large cardinals assumptions, the disjunction and existence properties for infinitary intuitionistic first-order logics.

Abstract: Publication date: Available online 6 September 2018Source: Annals of Pure and Applied LogicAuthor(s): Åsa Hirvonen, Tapani Hyttinen Trying to interpret B. Zilber's project on model theory of quantum mechanics we study a way of building limit models from finite-dimensional approximations. Our point of view is that of metric model theory, and we develop a method of taking ultraproducts of unbounded operators. We first calculate the Feynman propagator for the free particle as defined by physicists as an inner product 〈x0 Kt x1〉 of the eigenvector x0〉 of the position operator with eigenvalue x0 and Kt( x1〉), where Kt is the time evolution operator. However, due to a discretising effect, the eigenvector method does not work as expected, and straightforward calculations give the wrong value. We look at this phenomenon, and then complement this by showing how to instead correctly calculate the kernel of the time evolution operator (for both the free particle and the harmonic oscillator) in the limit model. We believe that our method of calculating these is new.

ADR
+Θ
+is+measurable&rft.title=Annals+of+Pure+and+Applied+Logic&rft.issn=0168-0072&rft.date=&rft.volume=">Hod up to
AD
R
+
Θ
is measurable

Abstract: Publication date: Available online 31 August 2018Source: Annals of Pure and Applied LogicAuthor(s): Rachid Atmai, Grigor Sargsyan Suppose M is a transitive class size model of ADR+“Θ is regular”. M is a minimal model of ADR+“Θ is measurable” if (i) R,Ord⊆M (ii) there is μ∈M such that M⊨“μ is a normal R-complete measure on Θ” and (iii) for any transitive class size N⊊M such that R⊆N, N⊨“there is no R-complete measure on Θ”. Continuing Trang's work in [8], we compute HOD of a minimal model of ADR+“Θ is measurable”.

Abstract: Publication date: Available online 30 August 2018Source: Annals of Pure and Applied LogicAuthor(s): Roman Kuznets Proof-theoretic method has been successfully used almost from the inception of interpolation properties to provide efficient constructive proofs thereof. Until recently, the method was limited to sequent calculi (and their notational variants), despite the richness of generalizations of sequent structures developed in structural proof theory in the meantime. In this paper, we provide a systematic and uniform account of the recent extension of this proof-theoretic method to hypersequents, nested sequents, and labelled sequents for normal modal logic. The method is presented in terms and notation easily adaptable to other similar formalisms, and interpolant transformations are stated for typical rule types rather than for individual rules.

Abstract: Publication date: Available online 23 August 2018Source: Annals of Pure and Applied LogicAuthor(s): Adam Brandenburger, H. Jerome Keisler The hidden-variable question is whether or not various properties — randomness or correlation, for example — that are observed in the outcomes of an experiment can be explained via introduction of extra (hidden) variables which are unobserved by the experimenter. The question can be asked in both the classical and quantum domains. In the latter, it is fundamental to the interpretation of the quantum formalism (Bell [2], Kochen and Specker [10], and others). In building a suitable mathematical model of an experiment, the physical set-up will guide us on how to model the observable variables — i.e., the measurement and outcome spaces. But, by definition, we cannot know what structure to put on the hidden-variable space. Nevertheless, we show that, under a measure-theoretic condition, the hidden-variable question can be put into a canonical form. The condition is that the σ-algebras on the measurement and outcome spaces are countably generated. An argument using a classical result on isomorphisms of measure algebras then shows that the hidden-variable space can always be taken to be the unit interval equipped with the Lebesgue measure on the Borel sets.

Abstract: Publication date: Available online 17 August 2018Source: Annals of Pure and Applied LogicAuthor(s): Vera Koponen We describe all binary simple homogeneous structures M in terms of ∅-definable equivalence relations on M, which “coordinatize” M and control dividing, and extension properties that respect these equivalence relations.

Abstract: Publication date: Available online 16 August 2018Source: Annals of Pure and Applied LogicAuthor(s): Barbara F. Csima, Jonathan Stephenson We first give an example of a rigid structure of computable dimension 2 such that the unique isomorphism between two non-computably isomorphic computable copies has Turing degree strictly below 0″, and not above 0′. This gives a first example of a computable structure with a degree of categoricity that does not belong to an interval of the form [0(α),0(α+1)] for any computable ordinal α. We then extend the technique to produce a rigid structure of computable dimension 3 such that if d0, d1, and d2 are the degrees of isomorphisms between distinct representatives of the three computable equivalence classes, then each di

Abstract: Publication date: Available online 15 August 2018Source: Annals of Pure and Applied LogicAuthor(s): Toshiyasu Arai In this paper we calibrate the strength of the soundness of a set theory KPω+(Π1-Collection) with the assumption that ‘there exists an uncountable regular ordinal’ in terms of the existence of ordinals.

Abstract: Publication date: Available online 14 August 2018Source: Annals of Pure and Applied LogicAuthor(s): Matthias Baaz, Norbert Preining We characterize the recursively enumerable first order Gödel logics with △ with respect to validity and non-satisfiability. The finitely valued and four infinitely valued Gödel logics with △ are recursively enumerable, not-satisfiability is recursively enumerable if validity is recursively enumerable. This is in contrast to first order Gödel logics without △, where validity is recursively enumerable for finitely valued and two infinitely valued Gödel logics, not-satisfiability is recursively enumerable if validity is recursively enumerable or 0 isolated in the truth value set.

Abstract: Publication date: Available online 13 August 2018Source: Annals of Pure and Applied LogicAuthor(s): Gilda Ferreira We present a purely proof-theoretic proof of the existence property for the full intuitionistic first-order predicate calculus, via natural deduction, in which commuting conversions are not needed. Such proof illustrates the potential of an atomic polymorphic system with only three generators of formulas – conditional and first and second-order universal quantifiers – as a tool for proof-theoretical studies.

Abstract: Publication date: Available online 10 August 2018Source: Annals of Pure and Applied LogicAuthor(s): Steve Awodey We construct an algebraic weak factorization system (L,R) on the category of cartesian cubical sets, in which the canonical path object factorization A→AI→A×A, induced by the 1-cube I, is an (L,R)-factorization for any R-object A.

Abstract: Publication date: Available online 8 August 2018Source: Annals of Pure and Applied LogicAuthor(s): Richard Pettigrew In this paper, I describe and motivate a new species of mathematical structuralism, which I call Instrumental Nominalism about Set-Theoretic Structuralism. As the name suggests, this approach takes standard Set-Theoretic Structuralism of the sort championed by Bourbaki, and removes its ontological commitments by taking an instrumental nominalist approach to that ontology of the sort described by Joseph Melia and Gideon Rosen. I argue that this avoids all of the problems that plague other versions of structuralism.

Abstract: Publication date: Available online 8 August 2018Source: Annals of Pure and Applied LogicAuthor(s): Danielle Macbeth For over a century we have been reading Frege's Begriffsschrift notation as a variant of standard notation. But Frege's notation can also be read differently, in a way enabling us to understand how reasoning in Begriffsschrift is at once continuous with and a significant advance beyond earlier mathematical practices of reasoning within systems of signs. It is this second reading that I outline here, beginning with two preliminary claims. First, I show that one does not reason in specially devised systems of signs of mathematics as one reasons in natural language; the signs are not abbreviations of words. Then I argue that even given a system of signs within which to reason in mathematics, there are two ways one can read expressions involving those signs, either mathematically or mechanically. These two lessons are then applied to a reading of Frege's proof of Theorem 133 in Part III of his 1879 logic, a proof that Frege claims is at once strictly deductive and ampliative, a real extension of our knowledge. In closing, I clarify what this might mean, and how it might be possible.

Abstract: Publication date: Available online 8 August 2018Source: Annals of Pure and Applied LogicAuthor(s): Victoria Gitman, Ralf Schindler We introduce the concept of virtual large cardinals and apply it to obtain a hierarchy of new large cardinal notions between ineffable cardinals and 0#. Given a large cardinal notion A characterized by the existence of elementary embeddings j:Vα→Vβ satisfying some list of properties, we say that a cardinal is virtually A if the embeddings j:VαV→VβV exist in the generic multiverse of V. Unlike their ideological cousins generic large cardinals, virtual large cardinals are actual large cardinals that are compatible with V=L. We study virtual versions of extendible, n-huge, and rank-into-rank cardinals and determine where they fit into the large cardinal hierarchy.

Abstract: Publication date: Available online 30 July 2018Source: Annals of Pure and Applied LogicAuthor(s): Damian Sobota We present a general method of constructing Boolean algebras with the Nikodym property and of some given cardinalities. The construction is dependent on the values of some classical cardinal characteristics of the continuum. As a result we obtain a consistent example of an infinite Boolean algebra with the Nikodym property and of cardinality strictly less than the continuum c. It follows that the existence of such an algebra is undecidable by the usual axioms of set theory. Besides, our results shed some new light on the Efimov problem and cofinalities of Boolean algebras.

Abstract: Publication date: Available online 20 July 2018Source: Annals of Pure and Applied LogicAuthor(s): Amitayu Banerjee, Mohamed Khaled Let α≥2 be any ordinal. We consider the class Drsα of relativized diagonal free set algebras of dimension α. With same technique, we prove several important results concerning this class. Among these results, we prove that almost all free algebras of Drsα are atomless, and none of these free algebras contains zero-dimensional elements other than zero and top element. The class Drsα corresponds to first order logic, without equality symbol, with α-many variables and on relativized semantics. Hence, in this variation of first order logic, there is no finitely axiomatizable, complete and consistent theory.

Abstract: Publication date: Available online 5 July 2018Source: Annals of Pure and Applied LogicAuthor(s): Paul Potgieter We consider some random series parametrised by Martin-Löf random sequences. The simplest case is that of Rademacher series, independent of a time parameter. This is then extended to the case of Fourier series on the circle with Rademacher coefficients. Finally, a specific Fourier series which has coefficients determined by a computable function is shown to converge to an algorithmically random Brownian motion.

Abstract: Publication date: Available online 21 June 2018Source: Annals of Pure and Applied LogicAuthor(s): A. Alibek, B.S. Baizhanov, B.Sh. Kulpeshov, T.S. Zambarnaya We study Vaught's problem for weakly o-minimal theories of convexity rank 1. We investigate such theories having less than 2ω countable models and prove their binarity. The main result of the paper is a description of the countable spectrum of weakly o-minimal theories of convexity rank 1.

Abstract: Publication date: Available online 19 June 2018Source: Annals of Pure and Applied LogicAuthor(s): Steven Givant, Hajnal Andréka A relation algebra is called measurable when its identity is the sum of measurable atoms, where an atom is called measurable if its square is the sum of functional elements.In this paper we show that atomic measurable relation algebras have rather strong structural properties: they are constructed from systems of groups, coordinated systems of isomorphisms between quotients of the groups, and systems of cosets that are used to “shift” the operation of relative multiplication. An atomic and complete measurable relation algebra is completely representable if and only if there is a stronger coordination between these isomorphisms induced by a scaffold (the shifting cosets are not needed in this case). We also prove that a measurable relation algebra in which the associated groups are all finite is atomic.