Archive for Mathematical Logic最新文献

When the Canonical Ramsey’s Theorem by Erdős and Rado is applied to regressive functions, one obtains the Regressive Ramsey’s Theorem by Kanamori and McAloon. Taylor proved a “canonical” version of Hindman’s Theorem, analogous to the Canonical Ramsey’s Theorem. We introduce the restriction of Taylor’s Canonical Hindman’s Theorem to a subclass of the regressive functions, the (lambda )-regressive functions, relative to an adequate version of min-homogeneity and prove some results about the Reverse Mathematics of this Regressive Hindman’s Theorem and of natural restrictions of it. In particular we prove that the first non-trivial restriction of the principle is equivalent to Arithmetical Comprehension. We furthermore prove that the well-ordering-preservation principle for base-(omega ) exponentiation is reducible to this same principle by a uniform computable reduction.

We present a cut-free sequent calculus for a class of first-order theories in negation normal form which include coherent and co-coherent theories alike. All structural rules, including cut, are admissible.

Inspired by a framework of multi lingual sequent calculus, we introduce a formal logical system called locally continuous sequent calculus to represent L-domains. By considering the logic states defined on locally continuous sequent calculi, we show that the collection of all logic states of a locally continuous sequent calculus with respect to set inclusion forms an L-domain, and every L-domain can be obtained in this way. Moreover, we define conjunctive consequence relations as morphisms between our sequent calculi, and prove that the category of locally continuous sequent calculi and conjunctive consequence relations is equivalent to that of L-domains and Scott-continuous functions. This result extends Abramsky’s “Domain theory in logical form” to a continuous setting.

Akama et al. [1] introduced a hierarchical classification of first-order formulas for a hierarchical prenex normal form theorem in semi-classical arithmetic. In this paper, we give a justification for the hierarchical classification in a general context of first-order theories. To this end, we first formalize the standard transformation procedure for prenex normalization. Then we show that the classes (textrm{E}_k) and (textrm{U}_k) introduced in [1] are exactly the classes induced by (Sigma _k) and (Pi _k) respectively via the transformation procedure in any first-order theory.

We show that we can interpret concatenation theories in arithmetical theories without coding sequences by identifying binary strings with (2times 2) matrices with determinant 1.

In this paper, we study some new examples of ideals on (omega ) with maximal Tukey type (that is, maximal among partial orders of size continuum). This discussion segues into an examination of a refinement of the Tukey order—known as the weak Rudin–Keisler order—and its structure when restricted to these ideals of maximal Tukey type. Mirroring a result of Fremlin (Note Mat 11:177–214, 1991) on the Tukey order, we also show that there is an analytic P-ideal above all other analytic P-ideals in the weak Rudin–Keisler order.

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.