Archive for Mathematical Logic最新文献

We expand our effective framework for weak convergence of measures on the real line by showing that effective convergence in the Prokhorov metric is equivalent to effective weak convergence. In addition, we establish a framework for the study of the effective theory of vague convergence of measures. We introduce a uniform notion and a non-uniform notion of vague convergence, and we show that both these notions are equivalent. However, limits under effective vague convergence may not be computable even when they are finite. We give an example of a finite incomputable effective vague limit measure, and we provide a necessary and sufficient condition so that effective vague convergence produces a computable limit. Finally, we determine a sufficient condition for which effective weak and vague convergence of measures coincide. As a corollary, we obtain an effective version of the equivalence between classical weak and vague convergence of sequences of probability measures.

We provide a non-Gentzen, though fully syntactical, cut-elimination algorithm for classical propositional logic. The designed procedure is implemented on (textsf{GS4}), the one-sided version of Kleene’s sequent system (textsf{G4}). The algorithm here proposed proves to be more ‘dexterous’ than other, more traditional, Gentzen-style techniques as the size of proofs decreases at each step of reduction. As a corollary result, we show that analyticity always guarantees minimality of the size of (textsf{GS4})-proofs.

Let T be a complete geometric theory and let (T_P) be the theory of dense pairs of models of T. We show that if T is superrosy with -rank 1 then (T_P) is superrosy with -rank at most (omega ).

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.