Archive For Mathematical Logic Research Articles

On the rigidity of Souslin trees and their generic branches

We show it is consistent that there is a Souslin tree S such that after forcing with S, S is Kurepa and for all clubs C subset omega _1, Supharpoonright C is rigid. This answers the questions in Fuchs (Arch Math Logic 52(1–2):47–66, 2013). Moreover, we show it is consistent with diamondsuit that for every Souslin tree T there is a dense X subseteq T which does not contain a copy of T. This is related to a question due to Baumgartner in Baumgartner (Ordered sets (Banff, Alta., 1981), volume 83 of NATO Adv. Study Inst. Ser. C: Math. Phys. Sci., Reidel, Dordrecht-Boston, pp 239–277, 1982).

Strong downward Löwenheim–Skolem theorems for stationary logics, II: reflection down to the continuum

Continuing (Fuchino et al. in Arch Math Log, 2020. https://doi.org/10.1007/s00153-020-00730-x), we study the Strong Downward Lowenheim–Skolem Theorems (SDLSs) of the stationary logic and their variations. In Fuchino et al. (2020) it has been shown that the SDLS for the ordinary stationary logic with weak second-order parameters $$\textsf {SDLS}({\mathcal {L}}^{\aleph _0}_{stat},{<}\,\aleph _2)$$ down to $${<}\,\aleph _2$$ is equivalent to the conjunction of CH and Cox’s Diagonal Reflection Principle for internally clubness. We show that the SDLS for the stationary logic without weak second-order parameters $$\textsf {SDLS}^-({\mathcal {L}}^{\aleph _0}_{stat},{<}\,2^{\aleph _0})$$ down to $${<}\,2^{\aleph _0}$$ implies that the size of the continuum is $$\aleph _2$$ . In contrast, an internal interpretation of the stationary logic can satisfy the SDLS down to $${<}\,2^{\aleph _0}$$ under the continuum being of size $$>\aleph _2$$ . This SDLS is shown to be equivalent to an internal version of the Diagonal Reflection Principle down to an internally stationary set of size $${<}\,2^{\aleph _0}$$ . We also consider a version of the stationary logic and show that the SDLS for this logic in internal interpretation $$\textsf {SDLS}^{int}_+({\mathcal {L}}^{PKL}_{stat},{<}\,2^{\aleph _0})$$ for reflection down to $${<}\,2^{\aleph _0}$$ is consistent under the assumption of the consistency of ZFC $$+$$ “the existence of a supercompact cardinal” and this SDLS implies that the continuum is (at least) weakly Mahlo. These three “axioms” in terms of SDLS are consequences of three instances of a strengthening of generic supercompactness which we call Laver-generic supercompactness. Existence of a Laver-generic supercompact cardinal in each of these three instances also fixes the cardinality of the continuum to be $$\aleph _1$$ or $$\aleph _2$$ or very large respectively. We also show that the existence of one of these generic large cardinals implies the “ $$++$$ ” version of the corresponding forcing axiom.

Cicho\u0144\u2019s diagram and localisation cardinals

We reimplement the creature forcing construction used by Fischer et al. (Arch Math Log 56(7–8):1045–1103, 2017. https://doi.org/10.1007/S00153-017-0553-8. arXiv:1402.0367 [math.LO]) to separate Cichoń’s diagram into five cardinals as a countable support product. Using the fact that it is of countable support, we augment our construction by adding uncountably many additional cardinal characteristics, sometimes referred to as localisation cardinals.

Chang’s Conjecture with $$\square _{\omega _1, 2}$$ from an $$\omega _1$$-Erdős cardinal

Answering a question of Sakai (Arch Math Logic 52(1–2):29–45, 2013), we show that the existence of an $$\omega _1$$ -Erdős cardinal suffices to obtain the consistency of Chang’s Conjecture with $$\square _{\omega _1, 2}$$ . By a result of Donder (In: Set theory and model theory (Bonn, 1979), volume 872 of lecture notes in mathematics. Springer, Berlin, pp 55–97, 1981) this is best possible. We also give an answer to another question of Sakai relating to the incompatibility of $$\square _{\lambda , 2}$$ and $$(\lambda ^+, \lambda ) \twoheadrightarrow (\kappa ^+, \kappa )$$ for uncountable $$\kappa $$ .

Scott sentences for equivalence structures

For a computable structure $${\mathcal {A}}$$, if there is a computable infinitary Scott sentence, then the complexity of this sentence gives an upper bound for the complexity of the index set $$I({\mathcal {A}})$$. If we can also show that $$I({\mathcal {A}})$$ is m-complete at that level, then there is a correspondence between the complexity of the index set and the complexity of a Scott sentence for the structure. There are results (Calvert et al. in Algebra Logic 45:306–325, 2006; Carson et al. in Trans Am Math Soc 364:5715–5728, 2012; Knight and Saraph in Scott sentences for certain groups, pre-print; McCoy and Wallbaum in Trans Am Math Soc 364:5729–5734, 2012) that suggest that these complexities will always match. However, it was shown in Knight and McCoy (Arch Math Logic 53:519–524, 2014) that there is a structure (a particular subgroup of $${\mathbb {Q}}$$) for which the index set is m-complete $$d-\varSigma ^0_2$$, though there is no computable $$d-\varSigma _2$$ Scott sentence. In the present paper, we give an example of a particular equivalence structure for which the index set is m-complete $$\varPi _3^0$$ but for which there is no computable $$\varPi _3$$ Scott sentence. There is, however, a computable $$\varPi _3$$ pseudo-Scott sentence for the structure, that is, a sentence that acts as a Scott sentence if we only consider computable structures.

Proof-theoretic strengths of the well-ordering principles

In this note the proof-theoretic ordinal of the well-ordering principle for the normal functions $$\mathsf {g}$$ on ordinals is shown to be equal to the least fixed point of $$\mathsf {g}$$ . Moreover corrections to the previous paper (Arai in Arch Math Log 57:649–664, 2017) are made.

Free sequences in $${\mathscr {P}}\left( \omega \right) /\text {fin}$$

We investigate maximal free sequences in the Boolean algebra $${\mathscr {P}}\left( \omega \right) {/}\text {fin}$$ , as defined by Monk (Comment Math Univ Carol 52(4):593–610, 2011). We provide some information on the general structure of these objects and we are particularly interested in the minimal cardinality of a free sequence, a cardinal characteristic of the continuum denoted $${\mathfrak {f}}$$ . Answering a question of Monk, we demonstrate the consistency of $$\omega _1 = {\mathfrak {i}} = {\mathfrak {f}} < {\mathfrak {u}} = \omega _2$$ . In fact, this consistency is demonstrated in the model of Shelah for $${\mathfrak {i}}< {\mathfrak {u}}$$ (Shelah in Arch Math Log 31(6):433–443, 1992). Our paper provides a streamlined and mostly self contained presentation of this construction.

Ideals of independence

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.

Evasion and prediction V: Unsymmetric game ideals, constant prediction, and strong porosity ideals

We study and clarify the close connection between ideals determined by unsymmetric games, the combinatorial notion of constant prediction, and the concept of strong porosity. We obtain a number of new results, mostly in ZFC, about cardinal invariants related to these notions. Our results answer questions of Newelski and Roslanowski (Proc Amer Math Soc 117(3):823–831, 1993, Problem 3.6), of Kamo (J Math Soc Japan 53(1):35–57, 2001, Question 2), of Brendle and Shelah (Arch Math Logic 42(4):349–360, 2003, Questions 3.11 and 3.12), and of Guzman et al. (J Log Anal 9(6), 2017, Questions 5.2, 5.3, 5.4, and 5.5).

Operations and structures derived from non-associative MV-algebras

The so-called non-associative MV-algebras were introduced recently by the first author and J. Kühr in order to have an appropriate tool for certain logics used in expert systems where associativity of the binary operation is excluded, see, e.g., Botur and Halaš (Arch Math Log 48:243–255, 2009). Since implication is an important logical connective in practically every propositional logic, in the present paper we investigate the implication reducts of non-associative MV-algebras. We also determine their structures based on the underlying posets. The natural question when a poset with the greatest element equipped with sectional switching involutions can be organized into an implication NMV-algebra is solved. Moreover, congruence properties of the variety of implication NMV-algebras with, respectively, without zero are investigated. Analogously to classical propositional logic, we introduce a certain kind of Sheffer operation and we obtain a one-to-one correspondence between NMV-algebras and certain algebras built up by a Sheffer-like operation together with a unary operation.

Proof Compression and NP Versus PSPACE

We show that arbitrary tautologies of Johansson’s minimal propositional logic are provable by “small” polynomial-size dag-like natural deductions in Prawitz’s system for minimal propositional logic. These “small” deductions arise from standard “large” tree-like inputs by horizontal dag-like compression that is obtained by merging distinct nodes labeled with identical formulas occurring in horizontal sections of deductions involved. The underlying geometric idea: if the height, $$h\left( \partial \right) $$ , and the total number of distinct formulas, $$\phi \left( \partial \right) $$ , of a given tree-like deduction $$\partial $$ of a minimal tautology $$\rho $$ are both polynomial in the length of $$\rho $$ , $$\left| \rho \right| $$ , then the size of the horizontal dag-like compression $$\partial ^{{\textsc {c}} }$$ is at most $$h\left( \partial \right) \times \phi \left( \partial \right) $$ , and hence polynomial in $$\left| \rho \right| $$ . That minimal tautologies $$ \rho $$ are provable by tree-like natural deductions $$\partial $$ with $$\left| \rho \right| $$ -polynomial $$h\left( \partial \right) $$ and $$\phi \left( \partial \right) $$ follows via embedding from Hudelmaier’s result that there are analogous sequent calculus deductions of sequent $$\Rightarrow \rho $$ . The notion of dag-like provability involved is more sophisticated than Prawitz’s tree-like one and its complexity is not clear yet. Our approach nevertheless provides a convergent sequence of NP lower approximations of PSPACE-complete validity of minimal logic (Savitch in J Comput Syst Sci 4(2):177–192, 1970); Statman in Theor Comput Sci 9(1):67–72, 1979; Svejdar in Arch Math Log 42(7):711–716, 2003).

Implicational (semilinear) logics III: completeness properties

This paper presents an abstract study of completeness properties of non-classical logics with respect to matricial semantics. Given a class of reduced matrix models we define three completeness properties of increasing strength and characterize them in several useful ways. Some of these characterizations hold in absolute generality and others are for logics with generalized implication or disjunction connectives, as considered in the previous papers. Finally, we consider completeness with respect to matrices with a linear dense order and characterize it in terms of an extension property and a syntactical metarule. This is the final part of the investigation started and developed in the papers (Cintula and Noguera in Arch Math Logic 49(4):417–446, 2010; Arch Math Logic 53(3):353–372, 2016).

On Katětov and Katětov–Blass orders on analytic P-ideals and Borel ideals

Minami–Sakai (Arch Math Logic 55(7–8):883–898, 2016) investigated the cofinal types of the Katětov and the Katětov–Blass orders on the family of all $$F_\sigma $$ ideals. In this paper we discuss these orders on analytic P-ideals and Borel ideals. We prove the following: In the course of the proof of the latter result, we also prove that for any analytic ideal $$\mathcal {I}$$ there is a Borel ideal $$\mathcal {J}$$ with $$\mathcal {I} \subseteq \mathcal {J}$$ .

Reduced axioms for the propositional logics induced by basic algebras

A certain logic induced by basic algebras was already studied by the first author in Chajda (Int J Theor Phys 54:4306–4312, 2015) and, for the particular case of the so-called commutative basic algebras, axiom system was established by Botur and Halas (Arch Math Logic 48:243–255, 2009). In Kolařik (Discuss Math Gen Algebra Appl 36:113–116, 2016) the just mentioned axiom system was essentially reduced. The aim of this paper is to reduce the original axiom system from Chajda (Int J Theor Phys 54:4306–4312, 2015) and to show that it is the best possible reduction in the sense that the remaining axioms are independent.

Tight Size-Degree Bounds for Sums-of-Squares Proofs

We exhibit families of 4-CNF formulas over n variables that have sums-of-squares (SOS) proofs of unsatisfiability of degree (a.k.a. rank) d but require SOS proofs of size $${n^{\Omega{(d)}}}$$ for values of d = d(n) from constant all the way up to $${n^{\delta}}$$ for some universal constant $${\delta}$$ . This shows that the $${{n^{{\rm O}{(d)}}}}$$ running time obtained by using the Lasserre semidefinite programming relaxations to find degree-d SOS proofs is optimal up to constant factors in the exponent. We establish this result by combining NP-reductions expressible as low-degree SOS derivations with the idea of relativizing CNF formulas in Krajicek (Arch Math Log 43(4):427–441, 2004) and Dantchev & Riis (Proceedings of the 17th international workshop on computer science logic (CSL ’03), 2003) and then applying a restriction argument as in Atserias et al. (J Symb Log 80(2):450–476, 2015; ACM Trans Comput Log 17:19:1–19:30, 2016). This yields a generic method of amplifying SOS degree lower bounds to size lower bounds and also generalizes the approach used in Atserias et al. (2016) to obtain size lower bounds for the proof systems resolution, polynomial calculus, and Sherali–Adams from lower bounds on width, degree, and rank, respectively.

An Easton like theorem in the presence of Shelah cardinals

We show that Shelah cardinals are preserved under the canonical $${{\mathrm{GCH}}}$$GCH forcing notion. We also show that if $${{\mathrm{GCH}}}$$GCH holds and $$F:{{\mathrm{REG}}}\rightarrow {{\mathrm{CARD}}}$$F:REGźCARD is an Easton function which satisfies some weak properties, then there exists a cofinality preserving generic extension of the universe which preserves Shelah cardinals and satisfies $$\forall \kappa \in {{\mathrm{REG}}},~ 2^{\kappa }=F(\kappa )$$źźźREG,2ź=F(ź). This gives a partial answer to a question asked by Cody (Arch Math Logic 52(5---6):569---591, 2013) and independently by Honzik (Acta Univ Carol 1:55---72, 2015). We also prove an indestructibility result for Shelah cardinals.

Generic Vopěnka’s Principle, remarkable cardinals, and the weak Proper Forcing Axiom

We introduce and study the first-order Generic Vopĕnka's Principle, which states that for every definable proper class of structures $$\mathcal {C}$$C of the same type, there exist $$B\ne A$$BźA in $$\mathcal {C}$$C such that B elementarily embeds into A in some set-forcing extension. We show that, for $$n\ge 1$$nź1, the Generic Vopĕnka's Principle fragment for $$\Pi _n$$źn-definable classes is equiconsistent with a proper class of n-remarkable cardinals. The n-remarkable cardinals hierarchy for $$n\in \omega $$nźź, which we introduce here, is a natural generic analogue for the $$C^{(n)}$$C(n)-extendible cardinals that Bagaria used to calibrate the strength of the first-order Vopĕnka's Principle in Bagaria (Arch Math Logic 51(3---4):213---240, 2012). Expanding on the theme of studying set theoretic properties which assert the existence of elementary embeddings in some set-forcing extension, we introduce and study the weak Proper Forcing Axiom, $$\mathrm{wPFA}$$wPFA. The axiom $$\mathrm{wPFA}$$wPFA states that for every transitive model $$\mathcal M$$M in the language of set theory with some $$\omega _1$$ź1-many additional relations, if it is forced by a proper forcing $$\mathbb P$$P that $$\mathcal M$$M satisfies some $$\Sigma _1$$Σ1-property, then V has a transitive model $$\bar{\mathcal M}$$M�, satisfying the same $$\Sigma _1$$Σ1-property, and in some set-forcing extension there is an elementary embedding from $$\bar{\mathcal M}$$M� into $$\mathcal M$$M. This is a weakening of a formulation of $$\mathrm{PFA}$$PFA due to Claverie and Schindler (J Symb Logic 77(2):475---498, 2012), which asserts that the embedding from $$\bar{\mathcal M}$$M� to $$\mathcal M$$M exists in V. We show that $$\mathrm{wPFA}$$wPFA is equiconsistent with a remarkable cardinal. Furthermore, the axiom $$\mathrm{wPFA}$$wPFA implies $$\mathrm{PFA}_{\aleph _2}$$PFAź2, the Proper Forcing Axiom for antichains of size at most $$\omega _2$$ź2, but it is consistent with $$\square _\kappa $$źź for all $$\kappa \ge \omega _2$$źźź2, and therefore does not imply $$\mathrm{PFA}_{\aleph _3}$$PFAź3.

Hybrid generalized Bosbach and Rie c̆ an states on non-commutative residuated lattices

Generalized Bosbach and Rie c̆ an states, which are useful for the development of an algebraic theory of probabilistic models for commutative or non-commutative fuzzy logics, have been investigated in the literature. In this paper, a new way arising from generalizing residuated lattice-based filters from commutative case to non-commutative one is applied to introduce new notions of generalized Bosbach and Rie c̆ an states, which are called hybrid ones, on non-commutative residuated lattices is provided, and the relationships between hybrid generalized states and those existing ones are studied, examples show that they are different. In particular, two problems from L.C. Ciungu, G. Georgescu, and C. Mure, “Generalized Bosbach States: Part I” (Archive for Mathematical Logic 52 (2013):335–376) are solved, and properties of hybrid generalized states, which are similar to those on commutative residuated lattices, are obtained without the condition “strong”.

Implicational (semilinear) logics II: additional connectives and characterizations of semilinearity

This is the continuation of the paper (Cintula and Noguera in Arch Math Log 49(4):417---446, 2010). We continue the abstract study of non-classical logics based on the kind of generalized implication connectives they possess and we focus on semilinear logics, i.e. those that are complete with respect to the class of models where the implication defines a linear order. We obtain general characterizations of semilinearity in terms of the intersection-prime extension property, the syntactical semilinearity metarule and the class of finitely subdirectly irreducible models. Moreover, we consider extensions of the language with lattice connectives and generalized disjunctions, study their interplay with implication and obtain axiomatizations and further descriptions of semilinear logics in terms of disjunctions and the proof by cases property.

Itay Neeman. Aronszajn trees and failure of the Singular Cardinal Hypothesis. Journal of Mathematical Logic, vol. 9, no. 1 (2009), pp. 139–157. - Dima Sinapova. The tree property at אּω+1. Journal of Symbolic Logic, vol. 77, no. 1 (2012), pp. 279–290. - Dima Sinapova. The tree property and the failure of SCH at uncountable cofinality. Archive for Mathematical Logic, vol.

Itay Neeman. Aronszajn trees and failure of the Singular Cardinal Hypothesis. Journal of Mathematical Logic, vol. 9, no. 1 (2009), pp. 139–157. - Dima Sinapova. The tree property at אּω+1. Journal of Symbolic Logic, vol. 77, no. 1 (2012), pp. 279–290. - Dima Sinapova. The tree property and the failure of SCH at uncountable cofinality. Archive for Mathematical Logic, vol. 51, no. 5-6 (2012), pp. 553–562. - Dima Sinapova. The tree property and the failure of the Singular Cardinal Hypothesis at אּω2. Journal of Symbolic Logic, vol. 77, no. 3 (2012), pp. 934–946. - Spencer Unger. Aronszajn trees and the successors of a singular cardinal. Archive for Mathematical Logic, vol. 52, no. 5-6 (2013), pp. 483–496. - Itay Neeman. The tree property up to אּω+1. Journal of Symbolic Logic. vol. 79, no. 2 (2014), pp. 429–459. - Volume 21 Issue 2