Archive for Mathematical Logic最新文献

In this paper we investigate two logics (and their fragments) from an algebraic point of view. The two logics are: (textsf{MALL}) (multiplicative-additive Linear Logic) and (textsf{LL}) (classical Linear Logic). Both logics turn out to be strongly algebraizable in the sense of Blok and Pigozzi and their equivalent algebraic semantics are, respectively, the variety of Girard algebras and the variety of girales. We show that any variety of girales has a TD-term and hence equationally definable principal congruences. Also we investigate the structure of the algebras in question, thus obtaining a representation theorem for Girard algebras and girales. We also prove that congruence lattices of girales are really congruence lattices of Heyting algebras, thus determining the simple and subdirectly irreducible girales. Finally we introduce a class of examples showing that the variety of girales contains infinitely many nonisomorphic finite simple algebras.

Non-archimedean models of the theory of the real ordered field with restricted analytic functions may not support a total exponential function, but they always have partial exponentials defined in certain convex subrings. On face of this, we study the first order theory of non-archimedean ordered valued fields with all restricted analytic functions and an exponential function defined in the valuation ring, which extends the restricted analytic exponential. We obtain model completeness and other desirable properties for this theory. In particular, any model embeds in a model where the partial exponential extends to a total one.

We develop the model theory of (omega )-stable K-loops and symmetric spaces. Continuing Poizat’s seminal work, we notably establish an appropriate version of the indecomposability theorem and we adapt Lascar’s analysis to this context.

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.