Archive for Mathematical Logic最新文献

It is known that a semicomputable continuum S in a computable topological space can be approximated by a computable subcontinuum by any given precision under condition that S is chainable and decomposable. In this paper we show that decomposability can be replaced by the assumption that S is chainable from a to b, where a is a computable point.

By a result of L.D. Beklemishev, the hierarchy of nested applications of the (Sigma _1)-collection rule over any (Pi _2)-axiomatizable base theory extending Elementary Arithmetic collapses to its first level. We prove that this result cannot in general be extended to base theories of arbitrary quantifier complexity. In fact, given any recursively enumerable set of true (Pi _2)-sentences, S, we construct a sound ((Sigma _2 ! vee ! Pi _2))-axiomatized theory T extending S such that the hierarchy of nested applications of the (Sigma _1)-collection rule over T is proper. Our construction uses some results on subrecursive degree theory obtained by L. Kristiansen.

We prove that if (mathcal {A}) is an infinite Boolean algebra in the ground model V and (mathbb {P}) is a notion of forcing adding any of the following reals: a Cohen real, an unsplit real, or a random real, then, in any (mathbb {P})-generic extension V[G], (mathcal {A}) has neither the Nikodym property nor the Grothendieck property. A similar result is also proved for a dominating real and the Nikodym property.

We prove a Mathias-type criterion for the Magidor iteration of Prikry forcings.

The (varepsilon )-elimination method of Hilbert’s (varepsilon )-calculus yields the up-to-date most direct algorithm for computing the Herbrand disjunction of an extensional formula. A central advantage is that the upper bound on the Herbrand complexity obtained is independent of the propositional structure of the proof. Prior (modern) work on Hilbert’s (varepsilon )-calculus focused mainly on the pure calculus, without equality. We clarify that this independence also holds for first-order logic with equality. Further, we provide upper bounds analyses of the extended first (varepsilon )-theorem, even if the formalisation incorporates so-called (varepsilon )-equality axioms.

This paper presents a novel proof of the conservativity of the intuitionistic theory of strictly positive fixpoints, (widehat{{textrm{ID}}}{}_{1}^{{textrm{i}}}{}), over Heyting arithmetic (({textrm{HA}})), originally proved in full generality by Arai (Ann Pure Appl Log 162:807–815, 2011. https://doi.org/10.1016/j.apal.2011.03.002). The proof embeds (widehat{{textrm{ID}}}{}_{1}^{{textrm{i}}}{}) into the corresponding theory over Beeson’s logic of partial terms and then uses two consecutive interpretations, a realizability interpretation of this theory into the subtheory generated by almost negative fixpoints, and a direct interpretation into Heyting arithmetic with partial terms using a hierarchy of satisfaction predicates for almost negative formulae. It concludes by applying van den Berg and van Slooten’s result (Indag Math 29:260–275, 2018. https://doi.org/10.1016/j.indag.2017.07.009) that Heyting arithmetic with partial terms plus the schema of self realizability for arithmetic formulae is conservative over ({textrm{HA}}).

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 degrees. In particular, we show that every (Delta ^0_2)-degree contains an NCR element.

We provide a general criterion for Fraenkel–Mostowski models of ({textsf{ZFA}}) (i.e. Zermelo–Fraenkel set theory weakened to permit the existence of atoms) which implies “every linearly ordered set can be well ordered” (({textsf{LW}})), and look at six models for ({textsf{ZFA}}) which satisfy this criterion (and thus ({textsf{LW}}) is true in these models) and “every Dedekind finite set is finite” (({textsf{DF}}={textsf{F}})) is true, and also consider various forms of choice for well-ordered families of well orderable sets in these models. In Model 1, the axiom of multiple choice for countably infinite families of countably infinite sets (({textsf{MC}}_{aleph _{0}}^{aleph _{0}})) is false. It was the open question of whether or not such a model exists (from Howard and Tachtsis “On metrizability and compactness of certain products without the Axiom of Choice”) that provided the motivation for this paper. In Model 2, which is constructed by first choosing an uncountable regular cardinal in the ground model, a strong form of Dependent choice is true, while the axiom of choice for well-ordered families of finite sets (({textsf{AC}}^{{textsf{WO}}}_{{textsf{fin}}})) is false. Also in this model the axiom of multiple choice for well-ordered families of well orderable sets fails. Model 3 is similar to Model 2 except for the status of ({textsf{AC}}^{{textsf{WO}}}_{{textsf{fin}}}) which is unknown. Models 4 and 5 are variations of Model 3. In Model 4 ({textsf{AC}}_{textrm{fin}}^{{textsf{WO}}}) is true. The construction of Model 5 begins by choosing a regular successor cardinal in the ground model. Model 6 is the only one in which (2{mathfrak {m}} = {mathfrak {m}}) for every infinite cardinal number ({mathfrak {m}}). We show that the union of a well-ordered family of well orderable sets is well orderable in Model 6 and that the axiom of multiple countable choice is false.

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.