Annals of Pure and Applied Logic最新文献

Taking inspiration from the monadicity of complete atomic Boolean algebras, we prove that profinite modal algebras are monadic over Set. While analyzing the monadic functor, we recover the universal model construction - a construction widely used in the modal logic literature for describing finitely generated free modal algebras and the essentially finite generated subframes of their canonical models.

In this paper we aim to compare Kurepa trees and Aronszajn trees. Moreover, we analyze the effect of large cardinal assumptions on this comparison. Using the method of walks on ordinals, we will show it is consistent with ZFC that there is a Kurepa tree and every Kurepa tree contains an Aronszajn subtree, if there is an inaccessible cardinal. This is stronger than Komjath's theorem in [5], where he proves the same consistency from two inaccessible cardinals. Moreover, we prove it is consistent with ZFC that there is a Kurepa tree T such that if $U\subset T$ is a Kurepa tree with the inherited order from T, then U has an Aronszajn subtree. This theorem uses no large cardinal assumption. Our last theorem immediately implies the following: If ${\mathrm{MA}}_{{\omega }_2}$ holds and ${\omega }_2$ is not a Mahlo cardinal in

In the affine fragment of continuous logic, type spaces are compact convex sets. I study some model theoretic properties of extreme types. It is proved that every complete theory T has an extremal model, i.e. a model which realizes only extreme types. Extremal models form an elementary class in the full continuous logic sense if and only if the set of extreme n-types is closed in $S_n\left(T\right)$ for each n. Also, some applications are given in the special cases where the theory has a compact or first order model.

In this paper we study admissible extensions of several theories T of reverse mathematics. The idea is that in such an extension the structure $M=(N,S,\in )$ of the natural numbers and collection of sets of natural numbers $S$ has to obey the axioms of T while simultaneously one also has a set-theoretic world with transfinite levels erected on top of $M$ governed by the axioms of Kripke-Platek set theory, $\mathrm{KP}$.

In some respects, the admissible extension of T can be viewed as a proof-theoretic analog of Barwise's admissible cover of an arbitrary model of set theory; see [2]. However, by contrast, the admissible extension of T is usually not a conservative extension of T. Owing to the interplay of T and $\mathrm{KP}$, either theory's axioms may force new sets of naturals to exist which in turn may engender yet new sets of naturals on account of the axioms of the other.

The paper discerns a general pattern though. It turns out that for many familiar theories T, the second order part of the admissible cover of T equates to T augmented by transfinite induction over all initial segments of the Bachmann-Howard ordinal. Technically, the paper uses a novel type of ordinal analysis, expanding that for $\mathrm{KP}$ to the higher set-theoretic universe while at the same time treating the world of subsets of $N$ as an unanalyzed class-sized urelement structure.

Among the systems of reverse mathematics, for which we determine the admissible extension, are ${\Pi }_1^1\text{-}{\mathrm{CA}}_0$ and ${\mathrm{ATR}}_0$ as well as the theory of bar induction, $\mathrm{BI}$.

Semiconic idempotent logic is a common generalization of intuitionistic logic, relevance logic with mingle, and semilinear idempotent logic. It is an algebraizable logic and it admits a cut-free hypersequent calculus. We give a structural decomposition of its characteristic algebraic semantics, conic idempotent residuated lattices, showing that each of these is an ordinal sum of simpler partially ordered structures. This ordinal sum is indexed by a totally ordered residuated lattice, which serves as its skeleton and is both a subalgebra and nuclear image. We equationally characterize the totally ordered residuated lattices appearing as such skeletons. Further, we describe both congruence and subalgebra generation in conic idempotent residuated lattices, proving that every variety generated by these enjoys the congruence extension property. In particular, this holds for semilinear idempotent residuated lattices. Based on this analysis, we obtain a local deduction theorem for semiconic idempotent logic, which also specializes to semilinear idempotent logic.

We study expansions of a vector space V over a field $F$, possibly with extra structure, with a generic submodule over a subring of $F$. We construct a natural expansion by existentially defined functions so that the expansion in the extended language satisfies quantifier elimination. We show that this expansion preserves tame model theoretic properties such as stability, NIP, NTP₁, NTP₂ and NSOP₁. We also study induced independence relations in the expansion.

We prove that if A is a computable Hopfian finitely presented structure, then A has a computable d-${\Sigma }_2$ Scott sentence if and only if the weak Whitehead problem for A is decidable. We use this to infer that every hyperbolic group as well as any polycyclic-by-finite group has a computable d-${\Sigma }_2$ Scott sentence, thus covering two main classes of finitely presented groups. Our proof also implies that every weakly Hopfian finitely presented group is strongly defined by its ${\exists }^{+}$-types, a question which arose in a different context.

We explore different generalizations of the classical concept of independent families on ω following the study initiated by Kunen, Fischer, Eskew and Montoya. We show that under ${\left(D\ell \right)}_{\kappa }^{⁎}$ we can get strongly κ-independent families of size $2^{\kappa }$ and present an equivalence of $\mathrm{GCH}$ in terms of strongly independent families. We merge the two natural ways of generalizing independent families through a filter or an ideal and we focus on the $C$-independent families, where $C$ is the club filter. Also we show a relationship between the existence of $J$-independent families and the saturation of the ideal $J$.

We develop a transfer principle of structural Ramsey theory from finite structures to ultraproducts. We show that under certain mild conditions, when a class of finite structures has finite small Ramsey degrees, under the (Generalized) Continuum Hypothesis the ultraproduct has finite big Ramsey degrees for internal colorings. The necessity of restricting to internal colorings is demonstrated by the example of the ultraproduct of finite linear orders. Under CH, this ultraproduct $L^{⁎}$ has, as a spine, ${\eta }_1$, an uncountable analogue of the order type of rationals η. Finite big Ramsey degrees for η were exactly calculated by Devlin in [5]. It is immediate from [39] that ${\eta }_1$ fails to have finite big Ramsey degrees. Moreover, we extend Devlin's coloring to ${\eta }_1$ to show that it witnesses big Ramsey degrees of finite tuples in η on every copy of η in ${\eta }_1$, and consequently in $L^{⁎}$. This work gives additional confirmation that ultraproducts are a suitable environment for studying Ramsey properties of finite and infinite structures.

In this paper, we establish an explicit higher reciprocity law for the polynomial ring over a nonprincipal ultraproduct of finite fields. Such an ultraproduct can be taken over the same finite field, which allows to recover the classical higher reciprocity law for the polynomial ring $F_q\left[t\right]$ over a finite field $F_q$ that is due to Dedekind, Kühne, Artin, and Schmidt. On the other hand, when the ultraproduct is taken over finite fields of unbounded cardinalities, we obtain an explicit higher reciprocity law for the polynomial ring over an infinite field in both characteristics 0 and $p>0$ for some prime p. We then use the higher reciprocity law to prove an analogue of the Grunwald–Wang theorem for such a polynomial ring in both characteristics 0 and $p>0$ for some prime p.