Annals of Pure and Applied Logic最新文献

Here we give a complete list of the groups definable in Presburger arithmetic up to a finite index subgroup.

Given a weakly compact cardinal κ, we give an axiomatization of intuitionistic first-order logic over $L_{{\kappa }^{+},\kappa }$ and prove it is sound and complete with respect to Kripke models. As a consequence we get the disjunction and existence properties for that logic. This generalizes the work of Nadel in [8] for intuitionistic logic over $L_{{\omega }_1,\omega }$. When κ is a regular cardinal such that ${\kappa }^{<\kappa }=\kappa$, we deduce, by an easy modification of the proof, a complete axiomatization of intuitionistic first-order logic over $L_{{\kappa }^{+},\kappa ,\kappa }$, the language with disjunctions of at most κ formulas, conjunctions of less than κ formulas and quantification on less than κ many variables. In particular, this applies to any regular cardinal under the Generalized Continuum Hypothesis.

Let $\mu <\kappa <\lambda$ be three infinite cardinals, the first two being regular. We show that if there is no inner model with large cardinals, $u(\kappa ,\lambda )$ is regular, where $u(\kappa ,\lambda )$ denotes the least size of a cofinal subset in $\left(P_{\kappa }\right(\lambda ),\subseteq )$, and $\mathrm{cf}\left(\lambda \right)\ne \mu$, then (a) the μ-club filters on $P_{\kappa }\left(\lambda \right)$ and $P_{\kappa }\left(u\right(\kappa ,\lambda \left)\right)$ are isomorphic, and (b) the ideal dual to the μ-club filter on $P_{\kappa }\left(\lambda \right)$ (and hence the restriction of the nonstationary ideal on $P_{\kappa }\left(\lambda \right)$ to sets of uniform cofinality μ) is not $I_{\kappa ,\lambda }$-$b_{u(\kappa ,\lambda )}$-saturated.

Semi-algebraic proof systems such as sum-of-squares ($\mathrm{SoS}$) have attracted a lot of attention due to their relation to approximation algorithms: constant degree semi-algebraic proofs lead to conjecturally optimal polynomial-time approximation algorithms for important $\mathrm{NP}$-hard optimization problems. Motivated by the need to allow a more streamlined and uniform framework for working with $\mathrm{SoS}$ proofs than the restrictive propositional level, we initiate a systematic first-order logical investigation into the kinds of reasoning possible in algebraic and semi-algebraic proof systems. Specifically, we develop first-order theories that capture in a precise manner constant degree algebraic and semi-algebraic proof systems: every statement of a certain form that is provable in our theories translates into a family of constant degree polynomial calculus or $\mathrm{SoS}$ refutations, respectively; and using a reflection principle, the converse also holds.

This places algebraic and semi-algebraic proof systems in the established framework of bounded arithmetic, while providing theories corresponding to systems that vary quite substantially from the usual propositional-logic ones.

We give examples of how our semi-algebraic theory proves statements such as the pigeonhole principle, we provide a separation between algebraic and semi-algebraic theories, and we describe initial attempts to go beyond these theories by introducing extensions that use the inequality symbol, identifying along the way which extensions lead outside the scope of constant degree $\mathrm{SoS}$. Moreover, we prove new results for propositional proofs, and specifically extend Berkholz's dynamic-by-static simulation of polynomial calculus (PC) by $\mathrm{SoS}$ to PC with the radical rule.

Inspired by Owings's problem, we investigate whether, for a given an Abelian group G and cardinal numbers $\kappa ,\theta$, every colouring $c:G⟶\theta$ yields a subset $X\subseteq G$ with $\left|X\right|=\kappa$ such that $X+X$ is monochromatic. (Owings's problem asks this for $G=Z$, $\theta =2$ and $\kappa ={\aleph }_0$; this is known to be false for the same G and κ but $\theta =3$.) We completely settle the question for κ and θ both finite (by obtaining sufficient and necessary conditions for a positive answer) and for κ and θ both infinite (with a negative answer). Also, in the case where θ is infinite but κ is finite, we obtain some sufficient conditions for a negative answer as well as an example with a positive answer.

We prove that the standard computable presentation of the space $C[0,1]$ of continuous real-valued functions on the unit interval is computably and punctually (primitively recursively) universal. From the perspective of modern computability theory, this settles a problem raised by Sierpiński in the 1940s.

We prove that the original Urysohn's construction of the universal separable Polish space $U$ is punctually universal. We also show that effectively compact, punctual Stone spaces are punctually homeomorphically embeddable into Cantor space $2^{\omega }$; note that we do not require effective compactness be primitive recursive. We also prove that effective compactness cannot be dropped from the premises by constructing a counterexample.

As consequence of the VC theorem, any pseudo-finite measure over an NIP ultraproduct is generically stable. We demonstrate a converse of this theorem and prove that any finitely approximable measure over an ultraproduct is itself pseudo-finite (even without the NIP assumption). We also analyze the connection between the Morley product and the pseudo-finite product. In particular, we show that if μ is definable and both μ and ν are pseudo-finite, then the Morley product of μ and ν agrees with the pseudo-finite product of μ and ν. Using this observation, we construct generically stable idempotent measures on pseudo-finite NIP groups.

Let $T^{⁎}$ be the theory of lattice-ordered, convex subrings of von Neumann regular real closed rings that are divisible-projectable, sc-regular ([12]) and have no minimal (non zero) idempotents. In this paper, we introduce and study a local divisibility binary relation that, added to the language for lattice-ordered rings, together with the (usual) divisibility relation and the radical relation associated to the minimal prime spectrum ([19]) yields quantifier elimination for $T^{⁎}$.

We introduce a model-complete theory which completely axiomatizes the structure $Z_{\alpha }=〈Z,+,0,1,f〉$ where $f:x\mapsto \lfloor \alpha x\rfloor$ is a unary function with α a fixed transcendental number. Moreover, we show that decidability of $Z_{\alpha }$ is equivalent to computability of α. This result fits into the more general theme of adding traces of multiplication to integers without losing decidability.

In this paper we continue the study in [11] of compactness and incompactness principles at double successors, focusing here on the case of double successors of singulars of countable cofinality. We obtain models which satisfy the tree property and club stationary reflection at these double successors. Moreover, we can additionally obtain either approachability or its failure. We also show how to obtain our results on ${\aleph }_{\omega +2}$ by incorporating collapses; particularly relevant for these circumstances is a new indestructibility theorem of ours showing that posets satisfying certain linked assumptions preserve club stationary reflection.