Abstract: Abstract We consider several game versions of the cardinal invariants \({\mathfrak {t}}\) , \({\mathfrak {u}}\) and \({\mathfrak {a}}\) . We show that the standard proof that parametrized diamond principles prove that the cardinal invariants are small actually shows that their game counterparts are small. On the other hand we show that \({\mathfrak {t}}<{\mathfrak {t}}_{Builder}\) and \({\mathfrak {u}}<{\mathfrak {u}}_{Builder}\) are both relatively consistent with ZFC, where \({\mathfrak {t}}_{Builder}\) and \({\mathfrak {u}}_{Builder}\) are the principal game versions of \({\mathfrak {t}}\) and \({\mathfrak {u}}\) , respectively. The corresponding question for \({\mathfrak {a}}\) remains open. PubDate: 2019-03-22
Abstract: Abstract For each continuous t-norm &, a class of fuzzy topological spaces, called &-topological spaces, is introduced. The motivation stems from the idea that to each many-valued logic there may correspond a theory of many-valued topology, in particular, each continuous t-norm may lead to a theory of fuzzy topology. It is shown that for each continuous t-norm &, the subcategory consisting of &-topological spaces is simultaneously reflective and coreflective in the category of fuzzy topological spaces, hence gives rise to an autonomous theory of fuzzy topology. Topologizing a fuzzy pre-ordered set with the fuzzy Scott topology yields a functor from the category of fuzzy pre-ordered sets and maps that preserve suprema of flat ideals to the category of &-topological spaces. It is proved that this functor is a full one if and only if the t-norm & is Archimedean. PubDate: 2019-03-16
Abstract: Abstract We study two ideals which are naturally associated to independent families. The first of them, denoted \(\mathcal {J}_\mathcal {A}\) , is characterized by a diagonalization property which allows along a cofinal sequence (the order type of which of uncountable cofinality) of stages along a finite support iteration to adjoin a maximal independent family. The second ideal, denoted \(\mathrm {id}(\mathcal {A})\) , originates in Shelah’s proof of \(\mathfrak {i}<\mathfrak {u}\) in Shelah (Arch Math Log 31(6), 433–443, 1992). We show that for every independent family \(\mathcal {A}\) , \(\mathrm {id}(\mathcal {A})\subseteq \mathcal {J}_\mathcal {A}\) and define a class of maximal independent families, to which we refer as densely maximal, for which the two ideals coincide. Building upon the techniques of Shelah (1992) we characterize Sacks indestructibility for such families in terms of properties of \(\mathrm {id}(\mathcal {A})\) and devise a countably closed poset which adjoins a Sacks indestructible densely maximal independent family. PubDate: 2019-03-15
Abstract: Abstract Tarski algebras, also known as implication algebras or semi-boolean algebras, are the \(\left\{ \rightarrow \right\} \) -subreducts of Boolean algebras. In this paper we shall introduce and study the complete and atomic Tarski algebras. We shall prove a duality between the complete and atomic Tarski algebras and the class of covering Tarski sets, i.e., structures \(\left<X,{\mathcal {K}}\right>\) , where X is a non-empty set and \({\mathcal {K}}\) is non-empty family of subsets of X such that \(\bigcup {\mathcal {K}}=X\) . This duality is a generalization of the known duality between sets and complete and atomic Boolean algebras. We shall also analize the case of complete and atomic Tarski algebras endowed with a complete modal operator, and we will prove a duality for these algebras. PubDate: 2019-03-02
Abstract: Abstract We examine conditions under which, in a computable topological space, a semicomputable set is computable. It is known that in a computable metric space a semicomputable set S is computable if S is a continuum chainable from a to b, where a and b are computable points, or S is a circularly chainable continuum which is not chainable. We prove that this result holds in any computable topological space. PubDate: 2019-03-01
Abstract: Abstract Fontanella (J Symb Logic 79(1):193–207, 2014) showed that if \(\langle \kappa _n:n<\omega \rangle \) is an increasing sequence of supercompacts and \(\nu =\sup _n\kappa _n\) , then the strong tree property holds at \(\nu ^+\) . Building on a proof by Neeman (J Math Log 9:139–157, 2010), we show that the strong tree property at \(\kappa ^+\) is consistent with \(\lnot SCH_\kappa \) , where \(\kappa \) is singular strong limit of countable cofinality. PubDate: 2019-02-25
Abstract: Abstract We study the set of possible sizes of maximal independent families to which we refer as spectrum of independence and denote \(\hbox {Spec}(mif)\) . Here mif abbreviates maximal independent family. We show that: whenever \(\kappa _1<\cdots <\kappa _n\) are finitely many regular uncountable cardinals, it is consistent that \(\{\kappa _i\}_{i=1}^n\subseteq \hbox {Spec}(mif)\) ; whenever \(\kappa \) has uncountable cofinality, it is consistent that \(\hbox {Spec}(mif)=\{\aleph _1,\kappa =\mathfrak {c}\}\) . Assuming large cardinals, in addition to (1) above, we can provide that $$\begin{aligned} (\kappa _i,\kappa _{i+1})\cap \hbox {Spec}(mif)=\emptyset \end{aligned}$$ for each i, \(1\le i<n\) . PubDate: 2019-02-25
Abstract: Abstract We show that \(\mathsf {RT} (2,4)\) cannot be proved with one typical application of \(\mathsf {RT} (2,2)\) in an intuitionistic extension of \({\mathsf {RCA}}_{0}\) to higher types, but that this does not remain true when the law of the excluded middle is added. The argument uses Kohlenbach’s axiomatization of higher order reverse mathematics, results related to modified reducibility, and a formalization of Weihrauch reducibility. PubDate: 2019-02-22
Abstract: Abstract We define a generic Vopěnka cardinal to be an inaccessible cardinal \(\kappa \) such that for every first-order language \({\mathcal {L}}\) of cardinality less than \(\kappa \) and every set \({\mathscr {B}}\) of \({\mathcal {L}}\) -structures, if \( {\mathscr {B}} = \kappa \) and every structure in \({\mathscr {B}}\) has cardinality less than \(\kappa \) , then an elementary embedding between two structures in \({\mathscr {B}}\) exists in some generic extension of V. We investigate connections between generic Vopěnka cardinals in models of ZFC and the number and complexity of \(\aleph _1\) -Suslin sets of reals in models of ZF. In particular, we show that ZFC + (there is a generic Vopěnka cardinal) is equiconsistent with ZF + \((2^{\aleph _1} \not \le S_{\aleph _1} )\) where \(S_{\aleph _1}\) is the pointclass of all \(\aleph _1\) -Suslin sets of reals, and also with ZF + \((S_{\aleph _1} = {{\varvec{\Sigma }}}^1_2)\) + \((\varTheta = \aleph _2)\) where \(\varTheta \) is the least ordinal that is not a surjective image of the reals. PubDate: 2019-02-20
Abstract: Abstract We study the algebraic implications of the non-independence property and variants thereof (dp-minimality) on infinite fields, motivated by the conjecture that all such fields which are neither real closed nor separably closed admit a (definable) henselian valuation. Our results mainly focus on Hahn fields and build up on Will Johnson’s “The canonical topology on dp-minimal fields” (J Math Log 18(2):1850007, 2018). PubDate: 2019-02-09
Abstract: Abstract In this paper, we answer a question asked in Koepke et al. (J Symb Logic 78:85–100, 2013) regarding a Mathias criteria for Tree-Prikry forcing. Also we will investigate Prikry forcing using various filters. For completeness and self inclusion reasons, we will give proofs of many known theorems. PubDate: 2019-02-07
Abstract: Abstract By a well-known result of Kotlarski et al. (1981), first-order Peano arithmetic \({{\mathsf {P}}}{{\mathsf {A}}}\) can be conservatively extended to the theory \({{\mathsf {C}}}{{\mathsf {T}}}^{-}\mathsf {[PA]}\) of a truth predicate satisfying compositional axioms, i.e., axioms stating that the truth predicate is correct on atomic formulae and commutes with all the propositional connectives and quantifiers. This result motivates the general question of determining natural axioms concerning the truth predicate that can be added to \({{\mathsf {C}}}{{\mathsf {T}}}^{-}\mathsf {[PA]}\) while maintaining conservativity over \( {{\mathsf {P}}}{{\mathsf {A}}}\) . Our main result shows that conservativity fails even for the extension of \({{\mathsf {C}}}{{\mathsf {T}}}^{-}\mathsf {[PA]}\) obtained by the seemingly weak axiom of disjunctive correctness \({{\mathsf {D}}}{{\mathsf {C}}}\) that asserts that the truth predicate commutes with disjunctions of arbitrary finite size. In particular, \({{\mathsf {C}}}{\mathsf {T}}^{-}\mathsf {[PA]}+\mathsf {DC}\) implies \(\mathsf {Con}(\mathsf {PA})\) . Our main result states that the theory \({\mathsf {C}}{\mathsf {T}}^{-}\mathsf {[PA]}+\mathsf {DC}\) coincides with the theory \({\mathsf {C}}{\mathsf {T}}_{0}\mathsf {[PA]}\) obtained by adding \( \Delta _{0}\) -induction in the language with the truth predicate. This result strengthens earlier work by Kotlarski (1986) and Cieśliński (2010). For our proof we develop a new general form of Visser’s theorem on non-existence of infinite descending chains of truth definitions and prove it by reduction to (Löb’s version of) Gödel’s second incompleteness theorem, rather than by using the Visser–Yablo paradox, as in Visser’s original proof (1989). PubDate: 2019-02-04
Abstract: Abstract The mantle is the intersection of all ground models of V. We show that if there exists an extendible cardinal then the mantle is the smallest ground model of V. PubDate: 2019-02-01
Abstract: Abstract We show constructively that every quasi-convex, uniformly continuous function \(f:C \rightarrow \mathbb {R}\) with at most one minimum point has a minimum point, where C is a convex compact subset of a finite dimensional normed space. Applications include a result on strictly quasi-convex functions, a supporting hyperplane theorem, and a short proof of the constructive fundamental theorem of approximation theory. PubDate: 2019-02-01
Abstract: Abstract We study classes of atomic models \(\mathbf{At}_T\) of a countable, complete first-order theory T. We prove that if \(\mathbf{At}_T\) is not \(\mathrm{pcl}\) -small, i.e., there is an atomic model N that realizes uncountably many types over \(\mathrm{pcl}_N(\bar{a})\) for some finite \(\bar{a}\) from N, then there are \(2^{\aleph _1}\) non-isomorphic atomic models of T, each of size \(\aleph _1\) . PubDate: 2019-02-01
Authors:Victoria Gitman; Joel David Hamkins Abstract: Abstract The generic Vopěnka principle, we prove, is relatively consistent with the ordinals being non-Mahlo. Similarly, the generic Vopěnka scheme is relatively consistent with the ordinals being definably non-Mahlo. Indeed, the generic Vopěnka scheme is relatively consistent with the existence of a \(\Delta _2\) -definable class containing no regular cardinals. In such a model, there can be no \(\Sigma _2\) -reflecting cardinals and hence also no remarkable cardinals. This latter fact answers negatively a question of Bagaria, Gitman and Schindler. PubDate: 2018-05-14 DOI: 10.1007/s00153-018-0632-5
Authors:Luciano J. González Abstract: Abstract We define when a ternary term m of an algebraic language \(\mathcal {L}\) is called a distributive nearlattice term ( \(\mathrm {DN}\) -term) of a sentential logic \(\mathcal {S}\) . Distributive nearlattices are ternary algebras generalising Tarski algebras and distributive lattices. We characterise the selfextensional logics with a \(\mathrm {DN}\) -term through the interpretation of the DN-term in the algebras of the algebraic counterpart of the logics. We prove that the canonical class of algebras (under the point of view of Abstract Algebraic Logic) associated with a selfextensional logic with a \(\mathrm {DN}\) -term is a variety, and we obtain that the logic is in fact fully selfextensional. PubDate: 2018-05-11 DOI: 10.1007/s00153-018-0628-1
Authors:Stefano Baratella Abstract: Abstract We study a modal extension of the Continuous First-Order Logic of Ben Yaacov and Pedersen (J Symb Logic 75(1):168–190, 2010). We provide a set of axioms for such an extension. Deduction rules are just Modus Ponens and Necessitation. We prove that our system is sound with respect to a Kripke semantics and, building on Ben Yaacov and Pedersen (2010), that it satisfies a number of properties similar to those of first-order predicate logic. Then, by means of a canonical model construction, we get that every consistent set of formulas is satisfiable. From the latter result we derive an Approximated Strong Completeness Theorem, in the vein of Continuous Logic, and a Compactness Theorem. PubDate: 2018-05-10 DOI: 10.1007/s00153-018-0630-7
Authors:Marija Boričić Abstract: Abstract Gentzen’s approach to deductive systems, and Carnap’s and Popper’s treatment of probability in logic were two fruitful ideas that appeared in logic of the mid-twentieth century. By combining these two concepts, the notion of sentence probability, and the deduction relation formalized in the sequent calculus, we introduce the notion of ’probabilized sequent’ \(\Gamma \vdash _a^b\Delta \) with the intended meaning that “the probability of truthfulness of \(\Gamma \vdash \Delta \) belongs to the interval [a, b]”. This method makes it possible to define a system of derivations based on ’axioms’ of the form \(\Gamma _i\vdash _{a_i}^{b_i}\Delta _i\) , obtained as a result of empirical research, and then infer conclusions of the form \(\Gamma \vdash _a^b\Delta \) . We discuss the consistency, define the models, and prove the soundness and completeness for the defined probabilized sequent calculus. PubDate: 2018-05-07 DOI: 10.1007/s00153-018-0626-3
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