Archive for Mathematical Logic最新文献

A partial solution to Ono’s problem P54 is given. Here Ono’s problem P54 is whether Harrop disjunction property is equivalent to disjunction property or not in intermediate predicate logics. As an application of this result it is shown that some intermediate predicate logics satisfy Harrop disjunction property.

We show a transfer principle for the property that all types realised in a given elementary extension are definable. It can be written as follows: a Henselian valued field is stably embedded in an elementary extension if and only if its value group is stably embedded in its corresponding extension, its residue field is stably embedded in its corresponding extension, and the extension of valued fields satisfies a certain algebraic condition. We show for instance that all types over the Hahn field (mathbb {R}((mathbb {Z}))) are definable. Similarly, all types over the quotient field of the Witt ring (W(mathbb {F}_p^{text {alg}})) are definable. This extends a work of Cubides and Delon and of Cubides and Ye.

We work with weakly precomplete equivalence relations introduced by Badaev. The weak precompleteness is a natural notion inspired by various fixed point theorems in computability theory. Let E be an equivalence relation on the set of natural numbers (omega ), having at least two classes. A total function f is a diagonal function for E if for every x, the numbers x and f(x) are not E-equivalent. It is known that in the case of c.e. relations E, the weak precompleteness of E is equivalent to the lack of computable diagonal functions for E. Here we prove that this result fails already for (Delta ^0_2) equivalence relations, starting with the (Pi ^{-1}_2) level. We focus on the Turing degrees of possible diagonal functions. We prove that for any noncomputable c.e. degree ({textbf{d}}), there exists a weakly precomplete c.e. equivalence E admitting a ({textbf{d}})-computable diagonal function. We observe that a Turing degree ({textbf{d}}) can compute a diagonal function for every (Delta ^0_2) equivalence relation E if and only if ({textbf{d}}) computes ({textbf{0}}'). On the other hand, every PA degree can compute a diagonal function for an arbitrary c.e. equivalence E. In addition, if ({textbf{d}}) computes diagonal functions for all c.e. E, then ({textbf{d}}) must be a DNC degree.

In Cieśliński (J Philos Logic 39:325–337, 2010), Cieśliński asked whether compositional truth theory with the additional axiom that all propositional tautologies are true is conservative over Peano Arithmetic. We provide a partial answer to this question, showing that if we additionally assume that truth predicate agrees with arithmetical truth on quantifier-free sentences, the resulting theory is as strong as (Delta _0)-induction for the compositional truth predicate, hence non-conservative. On the other hand, it can be shown with a routine argument that the principle of quantifier-free correctness is itself conservative.

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.