Archive for Mathematical Logic最新文献

We prove that a countable direct sum of chains has one, countably many or else continuum many isomorphism classes of siblings. This proves Thomassé’s conjecture for such structures. Further, we show that a direct sum of chains of any cardinality has one or infinitely many siblings, up to isomorphism.

We consider cardinal invariants determined by Hausdorff measures. We separate many cardinal invariants of Hausdorff measure 0 ideals using two models that separate many cardinal invariants of Yorioka ideals at once from earlier work. Also, we show the uniformity numbers of s-dimensional Hausdorff measure 0 ideals for (0< s < 1) and of the Lebesgue null ideal can be separated using the Mathias forcing.

There is no infinite sequence of (Pi ^1_1)-sound extensions of (textsf{ACA}_0) each of which proves (Pi ^1_1)-reflection of the next. This engenders a well-founded “reflection ranking” of (Pi ^1_1)-sound extensions of (textsf{ACA}_0). For any (Pi ^1_1)-sound theory T extending (textsf{ACA}^+_0), the reflection rank of T equals the proof-theoretic ordinal of T. This provides an alternative characterization of the notion of “proof-theoretic ordinal,” which is one of the central concepts of proof theory. We provide an alternative proof of this theorem using cut-elimination for infinitary derivations.

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.