Archive for Mathematical Logic

ISSNs: 0933-5846, 1432-0665

34 found

View year:

  1.  19
    Games characterizing certain families of functions.Marek Balcerzak, Tomasz Natkaniec & Piotr Szuca - 2024 - Archive for Mathematical Logic 63 (7):759-772.
    We obtain several game characterizations of Baire 1 functions between Polish spaces _X_, _Y_ which extends the recent result of V. Kiss. Then we propose similar characterizations for equi-Bare 1 families of functions. Also, using related ideas, we give game characterizations of Baire measurable and Lebesgue measurable functions.
    No categories
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark  
  2.  13
    Spectral MV-algebras and equispectrality.Giuseppina Gerarda Barbieri, Antonio Di Nola & Giacomo Lenzi - 2024 - Archive for Mathematical Logic 63 (7):893-919.
    In this paper we study the set of MV-algebras with given prime spectrum and we introduce the class of spectral MV-algebras. An MV-algebra is spectral if it is generated by the union of all its prime ideals (or proper ideals, or principal ideals, or maximal ideals). Among spectral MV-algebras, special attention is devoted to bipartite MV-algebras. An MV-algebra is bipartite if it admits an homomorphism onto the MV-algebra of two elements. We prove that both bipartite MV-algebras and spectral MV-algebras can (...)
    No categories
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark  
  3.  13
    Katětov order between Hindman, Ramsey and summable ideals.Rafał Filipów, Krzysztof Kowitz & Adam Kwela - 2024 - Archive for Mathematical Logic 63 (7):859-876.
    A family $$\mathcal {I}$$ I of subsets of a set X is an ideal on X if it is closed under taking subsets and finite unions of its elements. An ideal $$\mathcal {I}$$ I on X is below an ideal $$\mathcal {J}$$ J on Y in the Katětov order if there is a function $$f{: }Y\rightarrow X$$ f : Y → X such that $$f^{-1}[A]\in \mathcal {J}$$ f - 1 [ A ] ∈ J for every $$A\in \mathcal {I}$$ A (...)
    No categories
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark  
  4.  15
    Quantifier-free induction for lists.Stefan Hetzl & Jannik Vierling - 2024 - Archive for Mathematical Logic 63 (7):813-835.
    We investigate quantifier-free induction for Lisp-like lists constructed inductively from the empty list $$ nil $$ nil and the operation $${\textit{cons}}$$ cons, that adds an element to the front of a list. First we show that, for $$m \ge 1$$ m ≥ 1, quantifier-free $$m$$ m -step induction does not simulate quantifier-free $$(m + 1)$$ ( m + 1 ) -step induction. Secondly, we show that for all $$m \ge 1$$ m ≥ 1, quantifier-free $$m$$ m -step induction does not (...)
    No categories
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark  
  5.  11
    Separablilty of metric measure spaces and choice axioms.Paul Howard - 2024 - Archive for Mathematical Logic 63 (7):987-1003.
    In set theory without the Axiom of Choice we prove that the assertion “For every metric space (_X_, _d_) with a Borel measure \(\mu \) such that the measure of every open ball is positive and finite, (_X_, _d_) is separable.’ is implied by the axiom of choice for countable collections of sets and implies the axiom of choice for countable collections of finite sets. We also show that neither implication is reversible in Zermelo–Fraenkel set theory weakend to permit the (...)
    No categories
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark  
  6.  10
    On undecidability of the propositional logic of an associative binary modality.Michael Kaminski - 2024 - Archive for Mathematical Logic 63 (7):837-857.
    It is shown that both classical and intuitionistic propositional logics of an associative binary modality are undecidable. The proof is based on the deduction theorem for these logics.
    No categories
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark  
  7.  7
    Positive indiscernibles.Mark Kamsma - 2024 - Archive for Mathematical Logic 63 (7):921-940.
    We generalise various theorems for finding indiscernible trees and arrays to positive logic: based on an existing modelling theorem for s-trees, we prove modelling theorems for str-trees, str$$_0$$ 0 -trees (the reduct of str-trees that forgets the length comparison relation) and arrays. In doing so, we prove stronger versions for basing—rather than locally basing or EM-basing—str-trees on s-trees and str$$_0$$ 0 -trees on str-trees. As an application we show that a thick positive theory has k-$$\mathsf {TP_2}$$ TP 2 iff it (...)
    No categories
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark  
  8.  9
    Glivenko–Cantelli classes and NIP formulas.Karim Khanaki - 2024 - Archive for Mathematical Logic 63 (7):1005-1031.
    We give several new equivalences of NIP for formulas and new proofs of known results using Talagrand (Ann Probab 15:837–870, 1987) and Haydon et al. (in: Functional Analysis Proceedings, The University of Texas at Austin 1987–1989, Lecture Notes in Mathematics, Springer, New York, 1991). We emphasize that Keisler measures are more complicated than types (even in the NIP context), in an analytic sense. Among other things, we show that for a first order theory T and a formula $$\phi (x,y)$$, the (...)
    No categories
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark  
  9.  10
    Fragments of IOpen.Konstantin Kovalyov - 2024 - Archive for Mathematical Logic 63 (7):969-986.
    In this paper we consider some fragments of $$\textsf{IOpen}$$ (Robinson arithmetic $$\mathsf Q$$ with induction for quantifier-free formulas) proposed by Harvey Friedman and answer some questions he asked about these theories. We prove that $$\mathsf {I(lit)}$$ is equivalent to $$\textsf{IOpen}$$ and is not finitely axiomatizable over $$\mathsf Q$$, establish some inclusion relations between $$\mathsf {I(=)}, \mathsf {I(\ne )}, \mathsf {I(\leqslant )}$$ and $$\textsf{I} (\nleqslant )$$. We also prove that the set of diophantine equations solvable in models of $$\mathsf I (=)$$ (...)
    No categories
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark  
  10.  24
    The Josefson–Nissenzweig theorem and filters on $$\omega $$.Witold Marciszewski & Damian Sobota - 2024 - Archive for Mathematical Logic 63 (7):773-812.
    For a free filter F on $$\omega $$ ω, endow the space $$N_F=\omega \cup \{p_F\}$$ N F = ω ∪ { p F }, where $$p_F\not \in \omega $$ p F ∉ ω, with the topology in which every element of $$\omega $$ ω is isolated whereas all open neighborhoods of $$p_F$$ p F are of the form $$A\cup \{p_F\}$$ A ∪ { p F } for $$A\in F$$ A ∈ F. Spaces of the form $$N_F$$ N F constitute the (...)
    No categories
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark  
  11.  9
    Pathology of submeasures and $$F_{\sigma }$$ ideals.Jorge Martínez, David Meza-Alcántara & Carlos Uzcátegui - 2024 - Archive for Mathematical Logic 63 (7):941-967.
    We address some phenomena about the interaction between lower semicontinuous submeasures on $${\mathbb {N}}$$ N and $$F_{\sigma }$$ F σ ideals. We analyze the pathology degree of a submeasure and present a method to construct pathological $$F_{\sigma }$$ F σ ideals. We give a partial answers to the question of whether every nonpathological tall $$F_{\sigma }$$ F σ ideal is Katětov above the random ideal or at least has a Borel selector. Finally, we show a representation of nonpathological $$F_{\sigma }$$ (...)
    No categories
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark  
  12.  8
    On two consequences of CH established by Sierpiński.R. Pol & P. Zakrzewski - 2024 - Archive for Mathematical Logic 63 (7):877-891.
    We study the relations between two consequences of the Continuum Hypothesis discovered by Wacław Sierpiński, concerning uniform continuity of continuous functions and uniform convergence of sequences of real-valued functions, defined on subsets of the real line of cardinality continuum.
    No categories
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark  
  13.  24
    Indiscernibles and satisfaction classes in arithmetic.Ali Enayat - 2024 - Archive for Mathematical Logic 63 (5):655-677.
    We investigate the theory Peano Arithmetic with Indiscernibles ( \(\textrm{PAI}\) ). Models of \(\textrm{PAI}\) are of the form \(({\mathcal {M}},I)\), where \({\mathcal {M}}\) is a model of \(\textrm{PA}\), _I_ is an unbounded set of order indiscernibles over \({\mathcal {M}}\), and \(({\mathcal {M}},I)\) satisfies the extended induction scheme for formulae mentioning _I_. Our main results are Theorems A and B following. _Theorem A._ _Let_ \({\mathcal {M}}\) _be a nonstandard model of_ \(\textrm{PA}\) _ of any cardinality_. \(\mathcal {M }\) _has an expansion (...)
    No categories
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark  
  14.  15
    On computable numberings of families of Turing degrees.Marat Faizrahmanov - 2024 - Archive for Mathematical Logic 63 (5):609-622.
    In this work, we study computable families of Turing degrees introduced and first studied by Arslanov and their numberings. We show that there exist finite families of Turing c.e. degrees both those with and without computable principal numberings and that every computable principal numbering of a family of Turing degrees is complete with respect to any element of the family. We also show that every computable family of Turing degrees has a complete with respect to each of its elements computable (...)
    No categories
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark  
  15.  15
    Herbrandized modified realizability.Gilda Ferreira & Paulo Firmino - 2024 - Archive for Mathematical Logic 63 (5):703-721.
    Realizability notions in mathematical logic have a long history, which can be traced back to the work of Stephen Kleene in the 1940s, aimed at exploring the foundations of intuitionistic logic. Kleene’s initial realizability laid the ground for more sophisticated notions such as Kreisel’s modified realizability and various modern approaches. In this context, our work aligns with the lineage of realizability strategies that emphasize the accumulation, rather than the propagation of precise witnesses. In this paper, we introduce a new notion (...)
    No categories
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark  
  16.  23
    Errata: on the role of the continuum hypothesis in forcing principles for subcomplete forcing.Gunter Fuchs - 2024 - Archive for Mathematical Logic 63 (5):509-521.
    In this note, I will list instances where in the literature on subcomplete forcing and its forcing principles (mostly in articles of my own), the assumption of the continuum hypothesis, or that we are working above the continuum, was omitted. I state the correct statements and provide or point to correct proofs. There are also some new results, most of which revolve around showing the necessity of the extra assumption.
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark  
  17.  18
    Around accumulation points and maximal sequences of indiscernibles.Moti Gitik - 2024 - Archive for Mathematical Logic 63 (5):591-608.
    Answering a question of Mitchell (Trans Am Math Soc 329(2):507–530, 1992) we show that a limit of accumulation points can be singular in $${\mathcal {K}}$$ K. Some additional constructions are presented.
    No categories
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark  
  18.  28
    Varieties of truth definitions.Piotr Gruza & Mateusz Łełyk - 2024 - Archive for Mathematical Logic 63 (5):563-589.
    We study the structure of the partial order induced by the definability relation on definitions of truth for the language of arithmetic. Formally, a definition of truth is any sentence $$\alpha $$ which extends a weak arithmetical theory (which we take to be $${{\,\mathrm{I\Delta _{0}+\exp }\,}}$$ ) such that for some formula $$\Theta $$ and any arithmetical sentence $$\varphi $$, $$\Theta (\ulcorner \varphi \urcorner )\equiv \varphi $$ is provable in $$\alpha $$. We say that a sentence $$\beta $$ is definable (...)
    No categories
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark  
  19.  21
    Cohesive powers of structures.Valentina Harizanov & Keshav Srinivasan - 2024 - Archive for Mathematical Logic 63 (5):679-702.
    A cohesive power of a structure is an effective analog of the classical ultrapower of a structure. We start with a computable structure, and consider its effective power over a cohesive set of natural numbers. A cohesive set is an infinite set of natural numbers that is indecomposable with respect to computably enumerable sets. It plays the role of an ultrafilter, and the elements of a cohesive power are the equivalence classes of certain partial computable functions determined by the cohesive (...)
    No categories
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark  
  20.  23
    On the extendability to $$\mathbf {\Pi }_3^0$$ ideals and Katětov order.Jialiang He, Jintao Luo & Shuguo Zhang - 2024 - Archive for Mathematical Logic 63 (5):523-528.
    We show that there is a $$ \varvec{\Sigma }_4^0$$ ideal such that it’s neither extendable to any $$ \varvec{\Pi }_3^0$$ ideal nor above the ideal $$ \textrm{Fin}\times \textrm{Fin} $$ in the sense of Katětov order, answering a question from M. Hrušák.
    No categories
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark  
  21.  16
    Square compactness and Lindelöf trees.Pedro E. Marun - 2024 - Archive for Mathematical Logic 63 (5):741-757.
    We prove that every weakly square compact cardinal is a strong limit cardinal, and therefore weakly compact. We also study Aronszajn trees with no uncountable finitely splitting subtrees, characterizing them in terms of being Lindelöf with respect to a particular topology. We prove that the class of such trees is consistently non-empty and lies between the classes of Suslin and Aronszajn trees.
    No categories
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark  
  22.  20
    Pcf without choice Sh835.Saharon Shelah - 2024 - Archive for Mathematical Logic 63 (5):623-654.
    We mainly investigate models of set theory with restricted choice, e.g., ZF + DC + the family of countable subsets of $$\lambda $$ is well ordered for every $$\lambda $$ (really local version for a given $$\lambda $$ ). We think that in this frame much of pcf theory, (and combinatorial set theory in general) can be generalized. We prove here, in particular, that there is a proper class of regular cardinals, every large enough successor of singular is not measurable (...)
    No categories
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark  
  23.  21
    The extent of saturation of induced ideals.Kenta Tsukuura - 2024 - Archive for Mathematical Logic 63 (5):723-739.
    We construct a model with a saturated ideal _I_ over \({\mathcal {P}}_{\kappa }\lambda \) and study the extent of saturation of _I_.
    No categories
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark  
  24.  24
    Essential hereditary undecidability.Albert Visser - 2024 - Archive for Mathematical Logic 63 (5):529-562.
    In this paper we study essential hereditary undecidability. Theories with this property are a convenient tool to prove undecidability of other theories. The paper develops the basic facts concerning essentially hereditary undecidability and provides salient examples, like a construction of essentially hereditarily undecidable theories due to Hanf and an example of a rather natural essentially hereditarily undecidable theory strictly below. We discuss the (non-)interaction of essential hereditary undecidability with recursive boolean isomorphism. We develop a reduction relation essential tolerance, or, in (...)
    No categories
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  25.  20
    Indestructibility and the linearity of the Mitchell ordering.Arthur W. Apter - 2024 - Archive for Mathematical Logic 63 (3):473-482.
    Suppose that \(\kappa \) is indestructibly supercompact and there is a measurable cardinal \(\lambda > \kappa \). It then follows that \(A_0 = \{\delta is a measurable cardinal and the Mitchell ordering of normal measures over \(\delta \) is nonlinear \(\}\) is unbounded in \(\kappa \). If the Mitchell ordering of normal measures over \(\lambda \) is also linear, then by reflection (and without any use of indestructibility), \(A_1= \{\delta is a measurable cardinal and the Mitchell ordering of normal measures (...)
    No categories
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark  
  26.  25
    Vector spaces with a union of independent subspaces.Alessandro Berarducci, Marcello Mamino & Rosario Mennuni - 2024 - Archive for Mathematical Logic 63 (3):499-507.
    We study the theory of K-vector spaces with a predicate for the union X of an infinite family of independent subspaces. We show that if K is infinite then the theory is complete and admits quantifier elimination in the language of K-vector spaces with predicates for the n-fold sums of X with itself. If K is finite this is no longer true, but we still have that a natural completion is near-model-complete.
    No categories
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark  
  27.  38
    Regressive versions of Hindman’s theorem.Lorenzo Carlucci & Leonardo Mainardi - 2024 - Archive for Mathematical Logic 63 (3):447-472.
    When the Canonical Ramsey’s Theorem by Erdős and Rado is applied to regressive functions, one obtains the Regressive Ramsey’s Theorem by Kanamori and McAloon. Taylor proved a “canonical” version of Hindman’s Theorem, analogous to the Canonical Ramsey’s Theorem. We introduce the restriction of Taylor’s Canonical Hindman’s Theorem to a subclass of the regressive functions, the $$\lambda $$ λ -regressive functions, relative to an adequate version of min-homogeneity and prove some results about the Reverse Mathematics of this Regressive Hindman’s Theorem and (...)
    No categories
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark  
  28.  25
    Cut elimination for coherent theories in negation normal form.Paolo Maffezioli - 2024 - Archive for Mathematical Logic 63 (3):427-445.
    We present a cut-free sequent calculus for a class of first-order theories in negation normal form which include coherent and co-coherent theories alike. All structural rules, including cut, are admissible.
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark  
  29.  22
    Weak essentially undecidable theories of concatenation, part II.Juvenal Murwanashyaka - 2024 - Archive for Mathematical Logic 63 (3):353-390.
    We show that we can interpret concatenation theories in arithmetical theories without coding sequences by identifying binary strings with \(2\times 2\) matrices with determinant 1.
    No categories
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark  
  30.  35
    Nondefinability results with entire functions of finite order in polynomially bounded o-minimal structures.Hassan Sfouli - 2024 - Archive for Mathematical Logic 63 (3):491-498.
    Let \({\mathcal {R}}\) be a polynomially bounded o-minimal expansion of the real field. Let _f_(_z_) be a transcendental entire function of finite order \(\rho \) and type \(\sigma \in [0,\infty ]\). The main purpose of this paper is to show that if ( \(\rho ) or ( \(\rho =1\) and \(\sigma =0\) ), the restriction of _f_(_z_) to the real axis is not definable in \({\mathcal {R}}\). Furthermore, we give a generalization of this result for any \(\rho \in [0,\infty )\).
    No categories
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark  
  31.  28
    The second-order version of Morley’s theorem on the number of countable models does not require large cardinals.Franklin D. Tall & Jing Zhang - 2024 - Archive for Mathematical Logic 63 (3):483-490.
    The consistency of a second-order version of Morley’s Theorem on the number of countable models was proved in [EHMT23] with the aid of large cardinals. We here dispense with them.
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark  
  32.  22
    L-domains as locally continuous sequent calculi.Longchun Wang & Qingguo Li - 2024 - Archive for Mathematical Logic 63 (3):405-425.
    Inspired by a framework of multi lingual sequent calculus, we introduce a formal logical system called locally continuous sequent calculus to represent _L_-domains. By considering the logic states defined on locally continuous sequent calculi, we show that the collection of all logic states of a locally continuous sequent calculus with respect to set inclusion forms an _L_-domain, and every _L_-domain can be obtained in this way. Moreover, we define conjunctive consequence relations as morphisms between our sequent calculi, and prove that (...)
    No categories
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark  
  33.  21
    The fixed point and the Craig interpolation properties for sublogics of $$\textbf{IL}$$.Sohei Iwata, Taishi Kurahashi & Yuya Okawa - 2024 - Archive for Mathematical Logic 63 (1):1-37.
    We study the fixed point property and the Craig interpolation property for sublogics of the interpretability logic \(\textbf{IL}\). We provide a complete description of these sublogics concerning the uniqueness of fixed points, the fixed point property and the Craig interpolation property.
    No categories
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark  
  34.  17
    Turing degrees and randomness for continuous measures.Mingyang Li & Jan Reimann - 2024 - Archive for Mathematical Logic 63 (1):39-59.
    We study degree-theoretic properties of reals that are not random with respect to any continuous probability measure (NCR). To this end, we introduce a family of generalized Hausdorff measures based on the iterates of the “dissipation” function of a continuous measure and study the effective nullsets given by the corresponding Solovay tests. We introduce two constructions that preserve non-randomness with respect to a given continuous measure. This enables us to prove the existence of NCR reals in a number of Turing (...)
    No categories
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark  
 Previous issues
  
Next issues