Annals of Pure and Applied Logic最新文献

A field K in a ring language $L$ is finitely undecidable if $\text{Cons}\left(\Sigma \right)$ is undecidable for every nonempty finite $\Sigma \subseteq \text{Th}(K;L)$. We adapt arguments originating with Cherlin-van den Dries-Macintyre/Ershov (for PAC fields) and Haran (for PRC fields) to prove all PAC and PRC fields are finitely undecidable. We describe the difficulties that arise in adapting the proof to PpC fields, and show no bounded PpC field is finitely axiomatisable. This work is drawn from the author's PhD thesis [44, Chapter 4].

A well-known version of Rowbottom's theorem for supercompactness ultrafilters leads naturally to notions of two-cardinal Ramseyness and corresponding normal ideals introduced herein. Generalizing results of Baumgartner, Feng and the first author, from the cardinal setting to the two-cardinal setting, we study hierarchies associated with a particular version of two-cardinal Ramseyness and a strong version of two-cardinal ineffability, as well as the relationships between these hierarchies and a natural notion of transfinite two-cardinal indescribability.

Hardy functions are defined by transfinite recursion and provide upper bounds for the growth rate of the provably total computable functions in various formal theories, making them an essential ingredient in many proofs of independence. Their definition is contingent on a choice of fundamental sequences, which approximate limits in a ‘canonical’ way. In order to ensure that these functions behave as expected, including the aforementioned unprovability results, these fundamental sequences must enjoy certain regularity properties.

In this article, we prove that Buchholz's system of fundamental sequences for the ϑ function enjoys such conditions, including the Bachmann property. We partially extend these results to variants of the ϑ function, including a version without addition for countable ordinals. We conclude that the Hardy functions based on these notation systems enjoy natural monotonicity properties and majorize all functions defined by primitive recursion along $\vartheta \left({\epsilon }_{\Omega +1}\right)$.

Forgetting is removing variables from a logical formula while preserving the constraints on the other variables. In spite of reducing information, it does not always decrease the size of the formula and may sometimes increase it. This article discusses the implications of such an increase and analyzes the computational properties of the phenomenon. Given a propositional Horn formula, a set of variables and a maximum allowed size, deciding whether forgetting the variables from the formula can be expressed in that size is $D^p$-hard in ${\Sigma }_2^p$. The same problem for unrestricted CNF propositional formulae is $D_2^p$-hard in ${\Sigma }_3^p$.

Combining creature forcing approaches from [16] and [8], we show that, under ch, there is a proper ${\omega }^{\omega }$-bounding poset with ℵ₂-cc that forces continuum many pairwise different cardinal characteristics, parametrised by reals, for each one of the following six types: uniformity and covering numbers of Yorioka ideals as well as both kinds of localisation and anti-localisation cardinals, respectively. This answers several open questions from [17].

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.