Archive for Mathematical Logic最新文献

We work in the Baire space (mathbb {Z}^omega ) equipped with the coordinate-wise addition (+). Consider a (sigma -)ideal (mathcal {I}) and a family (mathbb {T}) of some kind of perfect trees. We are interested in results of the form: for every (Ain mathcal {I}) and a tree (Tin mathbb {T}) there exists (T'in mathbb {T}, T'subseteq T) such that (A+underbrace{[T']+[T']+dots +[T']}_{text {n--times}}in mathcal {I}) for each (nin omega ). Explored tree types include perfect trees, uniformly perfect trees, Miller trees, Laver trees and (omega -)Silver trees. The latter kind of trees is an analogue of Silver trees from the Cantor space. Besides the standard (sigma )-ideal (mathcal {M}) of meager sets, we also analyze (mathcal {M}_-) and fake null sets (mathcal {N}). The latter two are born out of the characterizations of their respective analogues in the Cantor space. The key ingredient in proofs were combinatorial characterizations of these ideals in the Baire space.

Definitionally: strongly effectively immune sets are infinite and their c.e. subsets have maximums effectively bounded in their c.e. indices; whereas, for effectively immune sets, their c.e. subsets’ cardinalities are what’re effectively bounded. This definitional difference between these two kinds of sets is very nicely paralleled by the following difference between their complements. McLaughlin: strongly effectively immune sets cannot have immune complements; whereas, the main theorem herein: effectively immune sets cannot have hyperimmune complements. Ullian: effectively immune sets can have effectively immune complements. The main theorem improves Arslanov’s, effectively hyperimmune sets cannot have effectively hyperimmune complements: the improvement omits the second ‘effectively’. Two natural examples of strongly effectively immune sets are presented with new cases of the first proved herein. The first is the set of minimal-Blum-size programs for the partial computable functions; the second, the set of Kolmogorov-random strings. A proved, natural example is presented of an effectively dense simple, not strongly effectively simple set; its complement is a set of maximal run-times. Further motivations for this study are presented. Kleene recursion theorem proofs herein emphasize how to conceptualize them. Finally, is suggested, future, related work—illustrated by a first, natural, strongly effectively (Sigma _2^0)-immune set—included: solution of an open problem from Rogers’ book.

We provide a complete axiomatization of modal inclusion logic—team-based modal logic extended with inclusion atoms. We review and refine an expressive completeness and normal form theorem for the logic, define a natural deduction proof system, and use the normal form to prove completeness of the axiomatization. Complete axiomatizations are also provided for two other extensions of modal logic with the same expressive power as modal inclusion logic: one augmented with a might operator and the other with a single-world variant of the might operator.

We develop an algebraic study of W.S. Cooper’s three-valued propositional logic of ordinary discourse ((mathcal{O}mathcal{L})). This logic displays a number of unusual features: (mathcal{O}mathcal{L}) is not weaker but incomparable with classical logic, it is connexive, paraconsistent and contradictory. As a non-structural logic, (mathcal{O}mathcal{L}) cannot be algebraized by the standard methods. However, we show that (mathcal{O}mathcal{L}) has an algebraizable structural companion, and determine its equivalent semantics, which turns out to be a finitely-generated discriminator variety. We provide an equational and a twist presentation for this class of algebras, which allow us to compare it with other well-known algebras of non-classical logics. In this way we establish that (mathcal{O}mathcal{L}) is definitionally equivalent to an expansion of the three-valued logic ({mathcal {J}}3) of D’Ottaviano and da Costa, itself a schematic extension of paraconsistent Nelson logic.

We investigate the property of elimination of imaginaries for some special cases of ordered abelian groups. We show that certain Hahn products of ordered abelian groups do not eliminate imaginaries in the pure language of ordered groups. Moreover, we prove that, adding finitely many constants to the language of ordered abelian groups, the theories of the finite lexicographic products (mathbb {Z}^n) and (mathbb {Z}^n times mathbb {Q}) have definable Skolem functions.

For a free filter F on (omega ), let (N_F=omega cup {p_F}), where (p_Fnot in omega ), be equipped with the following topology: every element of (omega ) is isolated whereas all open neighborhoods of (p_F) are of the form (Acup {p_F}) for (Ain F). The aim of this paper is to study spaces of the form (N_F) in the context of the Nikodym property of Boolean algebras. By (mathcal{A}mathcal{N}) we denote the class of all those ideals (mathcal {I}) on (omega ) such that for the dual filter (mathcal {I}^*) the space (N_{mathcal {I}^*}) carries a sequence (langle mu _n:nin omega rangle ) of finitely supported signed measures such that (Vert mu _nVert rightarrow infty ) and (mu _n(A)rightarrow 0) for every clopen subset (Asubseteq N_{mathcal {I}^*}). We prove that (mathcal {I}in mathcal{A}mathcal{N}) if and only if there exists a density submeasure (varphi ) on (omega ) such that (varphi (omega )=infty ) and (mathcal {I}) is contained in the exhaustive ideal (text{ Exh }(varphi )). Consequently, we get that if (mathcal {I}subseteq text{ Exh }(varphi )) for some density submeasure (varphi ) on (omega ) such that (varphi (omega )=infty ) and (N_{mathcal {I}^*}) is homeomorphic to a subspace of the Stone space (St(mathcal {A})) of a given Boolean algebra (mathcal {A}), then (mathcal {A}) does not have the Nikodym property. We observe that each (mathcal {I}in mathcal{A}mathcal{N}) is Katětov below the asymptotic density zero ideal (mathcal {Z}), and prove that the class (mathcal{A}mathcal{N}) has a subset of size (mathfrak {d}) which is dominating with respect to the Katětov order (le _K), but (mathcal{A}mathcal{N}) has no (le _K)-maximal element. We show that, when (mathcal {I}) is a density ideal, (mathcal {I}not in mathcal{A}mathcal{N}) holds if and only if (mathcal {I}) is totally bounded if and only if the Boolean algebra (mathcal {P}(omega )/mathcal {I}) contains a countable splitting family. Our results shed some new light on differences between the Nikodym property and the Grothendieck property of Boolean algebras.

In a previous paper of the author it was shown that the question whether systems of exponential diophantine equations are solvable in ({mathbb {Q}}) is undecidable. Now we show that the solvability of a conjunction of exponential diophantine equations in ({mathbb {Q}}) is equivalent to the solvability of just one such equation. It follows that the problem whether an exponential diophantine equation has solutions in ({mathbb {Q}}) is undecidable. We also show that two particular forms of exponential diophantine equations are undecidable.

A complete theory T has the Schröder–Bernstein property or simply the SB-property if any pair of elementarily bi-embeddable models are isomorphic. This property has been studied in the discrete first-order setting and can be seen as a first step towards classification theory. This paper deals with the SB-property on continuous theories. Examples of complete continuous theories that have this property include Hilbert spaces and any completion of the theory of probability algebras. We also study a weaker notion, the SB-property up to perturbations. This property holds if any two elementarily bi-embeddable models are isomorphic up to perturbations. We prove that the theory of Hilbert spaces expanded with a bounded self-adjoint operator has the SB-property up to perturbations of the operator and that the theory of atomless probability algebras with a generic automorphism have the SB-property up to perturbations of the automorphism. We also study how the SB-property behaves with respect to randomizations. Finally we prove, in the continuous setting, that if T is a strictly stable theory then T does not have the SB-property.

This paper studies categories and functors in the context of reverse and computable mathematics. In ordinary reverse mathematics, we only focuses on categories whose objects and morphisms can be represented by natural numbers. We first consider morphism sets of categories and prove several associated theorems equivalent to (mathrm ACA_{0}) over the base system (mathrm RCA_{0}). The Yoneda Lemma is a basic result in category theory and homological algebra. We then develop an effective version of the Yoneda Lemma in (mathrm RCA_{0}); as an application, we formalize an effective version of the Yoneda Embedding in (mathrm RCA_{0}). Products and coproducts are basic notions for defining special categories like semi-additive categories and additive categories. We study properties of products and coproducts of a sequence of objects of categories and provide effective characterizations of semi-additive categories and additive categories in terms of products and coproducts. Finally, we further consider the strength of theorems of category theory that are studied in this paper by methods of higher-order reverse mathematics

We discuss the externally definable Ramsey property, a weakening of the Ramsey property for relational structures, where the only colourings considered are those that are externally definable: that is, definable with parameters in an elementary extension. We show a number of basic results analogous to the classical Ramsey theory, and show that, for an ultrahomogeneous structure M with countable age, the externally definable Ramsey property is equivalent to the dynamical statement that, for all (n in mathbb {N} ), every subflow of the space (S_n(M)) of n-types has a fixed point. We discuss a range of examples, including results regarding the lexicographic product of structures.