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  1.  1
    Universal Coding and Prediction on Ergodic Random Points.Łukasz Dębowski & Tomasz Steifer - 2022 - Bulletin of Symbolic Logic 28 (3):387-412.
    Suppose that we have a method which estimates the conditional probabilities of some unknown stochastic source and we use it to guess which of the outcomes will happen. We want to make a correct guess as often as it is possible. What estimators are good for this? In this work, we consider estimators given by a familiar notion of universal coding for stationary ergodic measures, while working in the framework of algorithmic randomness, i.e., we are particularly interested in prediction of (...)
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  2.  5
    An Axiomatic Approach to Forcing in a General Setting.Rodrigo A. Freire & Peter Holy - 2022 - Bulletin of Symbolic Logic 28 (3):427-450.
    The technique of forcing is almost ubiquitous in set theory, and it seems to be based on technicalities like the concepts of genericity, forcing names and their evaluations, and on the recursively defined forcing predicates, the definition of which is particularly intricate for the basic case of atomic first order formulas. In his [3], the first author has provided an axiomatic framework for set forcing over models of $\mathrm {ZFC}$ that is a collection of guiding principles for extensions over which (...)
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  3.  3
    A Note on Fragments of Uniform Reflection in Second Order Arithmetic.Emanuele Frittaion - 2022 - Bulletin of Symbolic Logic 28 (3):451-465.
    We consider fragments of uniform reflection for formulas in the analytic hierarchy over theories of second order arithmetic. The main result is that for any second order arithmetic theory $T_0$ extending $\mathsf {RCA}_0$ and axiomatizable by a $\Pi ^1_{k+2}$ sentence, and for any $n\geq k+1$, $$\begin{align*}T_0+ \mathrm{RFN}_{\varPi^1_{n+2}} \ = \ T_0 + \mathrm{TI}_{\varPi^1_n}, \end{align*}$$ $$\begin{align*}T_0+ \mathrm{RFN}_{\varSigma^1_{n+1}} \ = \ T_0+ \mathrm{TI}_{\varPi^1_n}^{-}, \end{align*}$$ where T is $T_0$ augmented with full induction, and $\mathrm {TI}_{\varPi ^1_n}^{-}$ denotes the schema of transfinite induction up (...)
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  4.  29
    The Collapse of the Hilbert Program: A Variation on the Gödelian Theme.Saul A. Kripke - 2022 - Bulletin of Symbolic Logic 28 (3):413-426.
    The Hilbert program was actually a specific approach for proving consistency, a kind of constructive model theory. Quantifiers were supposed to be replaced by ε-terms. εxA(x) was supposed to denote a witness to ∃xA(x), or something arbitrary if there is none. The Hilbertians claimed that in any proof in a number-theoretic system S, each ε-term can be replaced by a numeral, making each line provable and true. This implies that S must not only be consistent, but also 1-consistent. Here we (...)
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  5.  6
    Affine Logic for Constructive Mathematics.Michael Shulman - 2022 - Bulletin of Symbolic Logic 28 (3):327-386.
    We show that numerous distinctive concepts of constructive mathematics arise automatically from an “antithesis” translation of affine logic into intuitionistic logic via a Chu/Dialectica construction. This includes apartness relations, complemented subsets, anti-subgroups and anti-ideals, strict and non-strict order pairs, cut-valued metrics, and apartness spaces. We also explain the constructive bifurcation of some classical concepts using the choice between multiplicative and additive affine connectives. Affine logic and the antithesis construction thus systematically “constructivize” classical definitions, handling the resulting bookkeeping automatically.
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  6.  1
    Uniform Properties of Ideals in Rings of Restricted Power Series.Madeline G. Barnicle - 2022 - Bulletin of Symbolic Logic 28 (2):258-258.
    When is an ideal of a ring radical or prime? By examining its generators, one may in many cases definably and uniformly test the ideal’s properties. We seek to establish such definable formulas in rings of p-adic power series, such as $\mathbb Q_{p}\langle X\rangle $, $\mathbb Z_{p}\langle X\rangle $, and related rings of power series over more general valuation rings and their fraction fields. We obtain a definable, uniform test for radicality, and, in the one-dimensional case, for primality. This builds (...)
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  7.  1
    Constructing Wadge Classes.Raphaël Carroy, Andrea Medini & Sandra Müller - 2022 - Bulletin of Symbolic Logic 28 (2):207-257.
    We show that, assuming the Axiom of Determinacy, every non-selfdual Wadge class can be constructed by starting with those of level $\omega _1$ and iteratively applying the operations of expansion and separated differences. The proof is essentially due to Louveau, and it yields at the same time a new proof of a theorem of Van Wesep. The exposition is self-contained, except for facts from classical descriptive set theory.
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  8.  5
    The Jacobson Radical of a Propositional Theory.Giulio Fellin, Peter Schuster & Daniel Wessel - 2022 - Bulletin of Symbolic Logic 28 (2):163-181.
    Alongside the analogy between maximal ideals and complete theories, the Jacobson radical carries over from ideals of commutative rings to theories of propositional calculi. This prompts a variant of Lindenbaum’s Lemma that relates classical validity and intuitionistic provability, and the syntactical counterpart of which is Glivenko’s Theorem. The Jacobson radical in fact turns out to coincide with the classical deductive closure. As a by-product we obtain a possible interpretation in logic of the axioms-as-rules conservation criterion for a multi-conclusion Scott-style entailment (...)
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  9.  8
    Order Types of Models of Fragments of Peano Arithmetic.Lorenzo Galeotti & Benedikt Löwe - 2022 - Bulletin of Symbolic Logic 28 (2):182-206.
    The complete characterisation of order types of non-standard models of Peano arithmetic and its extensions is a famous open problem. In this paper, we consider subtheories of Peano arithmetic, in particular, theories formulated in proper fragments of the full language of arithmetic. We study the order types of their non-standard models and separate all considered theories via their possible order types. We compare the theories with and without induction and observe that the theories without induction tend to have an algebraic (...)
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  10.  6
    Formal Theories of Occurrences and Substitutions.René Gazzari - 2022 - Bulletin of Symbolic Logic 28 (2):261-263.
  11.  4
    Non-Deterministic Matrices: Theory and Applications to Algebraic Semantics.Ana Claudia de Jesus Golzio - 2022 - Bulletin of Symbolic Logic 28 (2):260-261.
  12.  2
    P-Points, MAD Families and Cardinal Invariants.Osvaldo Guzmán González - 2022 - Bulletin of Symbolic Logic 28 (2):258-260.
    The main topics of this thesis are cardinal invariants, P -points and MAD families. Cardinal invariants of the continuum are cardinal numbers that are bigger than $\aleph _{0}$ and smaller or equal than $\mathfrak {c}.$ Of course, they are only interesting when they have some combinatorial or topological definition. An almost disjoint family is a family of infinite subsets of $\omega $ such that the intersection of any two of its elements is finite. A MAD family is a maximal almost (...)
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  13.  4
    On Logics and Semantics for Interpretability.Luka Mikec - 2022 - Bulletin of Symbolic Logic 28 (2):265-265.
  14.  2
    The Combinatorics and Absoluteness of Definable Sets of Real Numbers.Zach Norwood - 2022 - Bulletin of Symbolic Logic 28 (2):263-264.
    This thesis divides naturally into two parts, each concerned with the extent to which the theory of $L$ can be changed by forcing.The first part focuses primarily on applying generic-absoluteness principles to how that definable sets of reals enjoy regularity properties. The work in Part I is joint with Itay Neeman and is adapted from our paper Happy and mad families in $L$, JSL, 2018. The project was motivated by questions about mad families, maximal families of infinite subsets of $\omega (...)
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  15.  7
    Paraconsistent Logic Programming in Three and Four-Valued Logics.Kleidson Êglicio Carvalho da Silva Oliveira - 2022 - Bulletin of Symbolic Logic 28 (2):260-260.
  16.  1
    Proof Mining with the Bounded Functional Interpretation.Pedro Pinto - 2022 - Bulletin of Symbolic Logic 28 (2):265-266.
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  17.  3
    Semantics Modulo Satisfiability with Applications: Function Representation, Probabilities and Game Theory.Sandro Márcio da Silva Preto - 2022 - Bulletin of Symbolic Logic 28 (2):264-265.
    In the context of propositional logics, we apply semantics modulo satisfiability—a restricted semantics which comprehends only valuations that satisfy some specific set of formulas—with the aim to efficiently solve some computational tasks. Three possible such applications are developed.We begin by studying the possibility of implicitly representing rational McNaughton functions in Łukasiewicz Infinitely-valued Logic through semantics modulo satisfiability. We theoretically investigate some approaches to such representation concept, called representation modulo satisfiability, and describe a polynomial algorithm that builds representations in the newly (...)
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  18.  6
    A Journey Through Computability, Topology and Analysis.Manlio Valenti - 2022 - Bulletin of Symbolic Logic 28 (2):266-267.
    This thesis is devoted to the exploration of the complexity of some mathematical problems using the framework of computable analysis and descriptive set theory. We will especially focus on Weihrauch reducibility as a means to compare the uniform computational strength of problems. After a short introduction of the relevant background notions, we investigate the uniform computational content of problems arising from theorems that lie at the higher levels of the reverse mathematics hierarchy.We first analyze the strength of the open and (...)
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  19.  2
    Theorems of Hyperarithmetic Analysis and Almost Theorems of Hyperarithmetic Analysis.James S. Barnes, Jun le Goh & Richard A. Shore - 2022 - Bulletin of Symbolic Logic 28 (1):133-149.
    Theorems of hyperarithmetic analysis occupy an unusual neighborhood in the realms of reverse mathematics and recursion-theoretic complexity. They lie above all the fixed iterations of the Turing jump but below ATR $_{0}$. There is a long history of proof-theoretic principles which are THAs. Until the papers reported on in this communication, there was only one mathematical example. Barnes, Goh, and Shore [1] analyze an array of ubiquity theorems in graph theory descended from Halin’s [9] work on rays in graphs. They (...)
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  20.  26
    Level Theory, Part 3: A Boolean Algebra of Sets Arranged in Well-Ordered Levels.Tim Button - 2022 - Bulletin of Symbolic Logic 28 (1):1-26.
    On a very natural conception of sets, every set has an absolute complement. The ordinary cumulative hierarchy dismisses this idea outright. But we can rectify this, whilst retaining classical logic. Indeed, we can develop a boolean algebra of sets arranged in well-ordered levels. I show this by presenting Boolean Level Theory, which fuses ordinary Level Theory with ideas due to Thomas Forster, Alonzo Church, and Urs Oswald. BLT neatly implement Conway’s games and surreal numbers; and a natural extension of BLT (...)
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  21.  1
    An Introduction to the Scott Complexity of Countable Structures and a Survey of Recent Results.Matthew Harrison-Trainor - 2022 - Bulletin of Symbolic Logic 28 (1):71-103.
    Every countable structure has a sentence of the infinitary logic $\mathcal {L}_{\omega _1 \omega }$ which characterizes that structure up to isomorphism among countable structures. Such a sentence is called a Scott sentence, and can be thought of as a description of the structure. The least complexity of a Scott sentence for a structure can be thought of as a measurement of the complexity of describing the structure. We begin with an introduction to the area, with short and simple proofs (...)
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  22.  2
    Degrees of Randomized Computability.Rupert Hölzl & Christopher P. Porter - 2022 - Bulletin of Symbolic Logic 28 (1):27-70.
    In this survey we discuss work of Levin and V’yugin on collections of sequences that are non-negligible in the sense that they can be computed by a probabilistic algorithm with positive probability. More precisely, Levin and V’yugin introduced an ordering on collections of sequences that are closed under Turing equivalence. Roughly speaking, given two such collections $\mathcal {A}$ and $\mathcal {B}$, $\mathcal {A}$ is below $\mathcal {B}$ in this ordering if $\mathcal {A}\setminus \mathcal {B}$ is negligible. The degree structure associated (...)
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  23.  4
    In Memoriam: Gerald E. Sacks, 1933–2019.Manuel Lerman & Theodore A. Slaman - 2022 - Bulletin of Symbolic Logic 28 (1):150-155.
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  24.  25
    Invariance Criteria as Meta-Constraints.Gil Sagi - 2022 - Bulletin of Symbolic Logic 28 (1):104-132.
    Invariance criteria are widely accepted as a means to demarcate the logical vocabulary of a language. In previous work, I proposed a framework of “semantic constraints” for model theoretic consequence which does not rely on a strict distinction between logical and nonlogical terms, but rather on a range of constraints on models restricting the interpretations of terms in the language in different ways. In this paper I show how invariance criteria can be generalized so as to apply to semantic constraints (...)
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