Archive for Mathematical Logic最新文献

We adjust the notion of finitary filter pair, which was coined for creating and analyzing finitary logics, in such a way that we can treat logics of cardinality (kappa ), where (kappa ) is a regular cardinal. The corresponding new notion is called (kappa )-filter pair. A filter pair can be seen as a presentation of a logic, and we ask what different (kappa )-filter pairs give rise to a fixed logic of cardinality (kappa ). To make the question well-defined we restrict to a subcollection of filter pairs and establish a bijection from that collection to the set of natural extensions of that logic by a set of variables of cardinality (kappa ). Along the way we use (kappa )-filter pairs to construct natural extensions for a given logic, work out the relationships between this construction and several others proposed in the literature, and show that the collection of natural extensions forms a complete lattice. In an optional section we introduce and motivate the concept of a general filter pair.

We investigate several ideal versions of the pseudointersection number (mathfrak {p}), ideal slalom numbers, and associated topological spaces with the focus on selection principles. However, it turns out that well-known pseudointersection invariant (mathtt {cov}^*({mathcal I})) has a crucial influence on the studied notions. For an invariant (mathfrak {p}_mathrm {K}({mathcal J})) introduced by Borodulin-Nadzieja and Farkas (Arch. Math. Logic 51:187–202, 2012), and an invariant (mathfrak {p}_mathrm {K}({mathcal I},{mathcal J})) introduced by Repický (Real Anal. Exchange 46:367–394, 2021), we have

respectively. In addition to the first inequality, for a slalom invariant (mathfrak {sl_e}({mathcal I},{mathcal J})) introduced in  Šupina (J. Math. Anal. Appl. 434:477–491, 2016), we show that

Finally, we obtain a consistency that ideal versions of the Fréchet–Urysohn property and the strictly Fréchet–Urysohn property are distinguished.

We introduce a logic for a class of probabilistic Kripke structures that we call type structures, as they are inspired by Harsanyi type spaces. The latter structures are used in theoretical economics and game theory. A strong completeness theorem for an associated infinitary propositional logic with probabilistic operators was proved by Meier. By simplifying Meier’s proof, we prove that our logic is strongly complete with respect to the class of type structures. In order to do that, we define a canonical model (in the sense of modal logics), which turns out to be a terminal object in a suitable category. Furthermore, we extend some standard model-theoretic constructions to type structures and we prove analogues of first-order results for those constructions.

We prove that every quasi-Hopfian finitely presented structure A has a d-(Sigma _2) Scott sentence, and that if in addition A is computable and Aut(A) satisfies a natural computable condition, then A has a computable d-(Sigma _2) Scott sentence. This unifies several known results on Scott sentences of finitely presented structures and it is used to prove that other not previously considered algebraic structures of interest have computable d-(Sigma _2) Scott sentences. In particular, we show that every right-angled Coxeter group of finite rank has a computable d-(Sigma _2) Scott sentence, as well as any strongly rigid Coxeter group of finite rank. Finally, we show that the free projective plane of rank 4 has a computable d-(Sigma _2) Scott sentence, thus exhibiting a natural example where the assumption of quasi-Hopfianity is used (since this structure is not Hopfian).

This is an introductory paper to a series of results linking generic absoluteness results for second and third order number theory to the model theoretic notion of model companionship. Specifically we develop here a general framework linking Woodin’s generic absoluteness results for second order number theory and the theory of universally Baire sets to model companionship and show that (with the required care in details) a (Pi _2)-property formalized in an appropriate language for second order number theory is forcible from some (Tsupseteq mathsf {ZFC}+)large cardinals if and only if it is consistent with the universal fragment of T if and only if it is realized in the model companion of T. In particular we show that the first order theory of (H_{omega _1}) is the model companion of the first order theory of the universe of sets assuming the existence of class many Woodin cardinals, and working in a signature with predicates for (Delta _0)-properties and for all universally Baire sets of reals. We will extend these results also to the theory of (H_{aleph _2}) in a follow up of this paper.

In this paper, we mainly investigate the existence of states based on the Glivenko theorem in bounded semihoops, which are building blocks for the algebraic semantics for relevant fuzzy logics. First, we extend algebraic formulations of the Glivenko theorem to bounded semihoops and give some characterizations of Glivenko semihoops and regular semihoops. The category of regular semihoops is a reflective subcategory of the category of Glivenko semihoops. Moreover, by means of the negative translation term, we characterize the Glivenko variety. Then we show that the regular semihoop of regular elements of a free algebra in the variety of Glivenko semihoops is free in the corresponding variety of regular semihoops. Similar results are derived for the semihoop of dense elements of free Glivenko semihoops. Finally, we give a purely algebraic method to check the existence of states on Glivenko semihoops. In particular, we prove that a bounded semihoop has Bosbach states if and only if it has a divisible filter, and a bounded semihoop has Riečan states if and only if it has a semi-divisible filter.

We give recursion-theoretic characterizations of the counting class (textsf {#P} ), the class of those functions which count the number of accepting computations of non-deterministic Turing machines working in polynomial time. Moreover, we characterize in a recursion-theoretic manner all the levels ({textsf {#P} _k}_{kin {mathbb {N}}}) of the counting hierarchy of functions (textsf {FCH} ), which result from allowing queries to functions of the previous level, and (textsf {FCH} ) itself as a whole. This is done in the style of Bellantoni and Cook’s safe recursion, and it places (textsf {#P} ) in the context of implicit computational complexity. Namely, it relates (textsf {#P} ) with the implicit characterizations of (textsf {FPTIME} ) (Bellantoni and Cook, Comput Complex 2:97–110, 1992) and (textsf {FPSPACE} ) (Oitavem, Math Log Q 54(3):317–323, 2008), by exploiting the features of the tree-recursion scheme of (textsf {FPSPACE} ).

We systematically study the interrelations between all possible variations of (Delta ^0_1) variants of the law of excluded middle and related principles in the context of intuitionistic arithmetic and analysis.

We construct a 2-equivalence (mathfrak {CohTheory}^{op }simeq mathfrak {TypeSpaceFunc}). Here (mathfrak {CohTheory}) is the 2-category of positive theories and (mathfrak {TypeSpaceFunc}) is the 2-category of type space functors. We give a precise definition of interpretations for positive logic, which will be the 1-cells in (mathfrak {CohTheory}). The 2-cells are definable homomorphisms. The 2-equivalence restricts to a duality of categories, making precise the philosophy that a theory is ‘the same’ as the collection of its type spaces (i.e. its type space functor). In characterising those functors that arise as type space functors, we find that they are specific instances of (coherent) hyperdoctrines. This connects two different schools of thought on the logical structure of a theory. The key ingredient, the Deligne completeness theorem, arises from topos theory, where positive theories have been studied under the name of coherent theories.

A Kaufmann model is an (omega _1)-like, recursively saturated, rather classless model of ({{mathsf {P}}}{{mathsf {A}}}) (or ({{mathsf {Z}}}{{mathsf {F}}} )). Such models were constructed by Kaufmann under the combinatorial principle (diamondsuit _{omega _1}) and Shelah showed they exist in (mathsf {ZFC}) by an absoluteness argument. Kaufmann models are an important witness to the incompactness of (omega _1) similar to Aronszajn trees. In this paper we look at some set theoretic issues related to this motivated by the seemingly naïve question of whether such a model can be “killed” by forcing without collapsing (omega _1). We show that the answer to this question is independent of (mathsf {ZFC}) and closely related to similar questions about Aronszajn trees. As an application of these methods we also show that it is independent of (mathsf {ZFC}) whether or not Kaufmann models can be axiomatized in the logic (L_{omega _1, omega } (Q)) where Q is the quantifier “there exists uncountably many”.