Annals of Pure and Applied Logic最新文献

The Gödel-McKinsey-Tarski embedding allows to view intuitionistic logic through the lenses of modal logic. In this work, an extension of the modal embedding to infinitary intuitionistic logic is introduced. First, a neighborhood semantics for a family of axiomatically presented infinitary modal logics is given and soundness and completeness are proved via the method of canonical models. The semantics is then exploited to obtain a labelled sequent calculus with good structural properties. Next, soundness and faithfulness of the embedding are established by transfinite induction on the height of derivations: the proof is obtained directly without resorting to non-constructive principles. Finally, the modal embedding is employed in order to relate classical, intuitionistic and modal derivability in infinitary logic extended with axioms.

We provide a general preservation theorem for preserving selective independent families along countable support iterations. The theorem gives a general framework for a number of results in the literature concerning models in which the independence number $i$ is strictly below $c$, including iterations of Sacks forcing, Miller partition forcing, h-perfect tree forcings, coding with perfect trees. Moreover, applying the theorem, we show that $i={\aleph }_1$ in the Miller Lite model. An important aspect of the preservation theorem is the notion of “Cohen preservation”, which we discuss in detail.

We generically construct a model in which the ${\Pi }_3^1$-reduction property is true and the ${\Pi }_3^1$-uniformization property is false, thus producing a model which separates these two principles for the first time.

The overall purpose of the present work is to lay the foundations for a new approach to bridge logic and probabilistic computation. To this aim we introduce extensions of classical and intuitionistic propositional logic with counting quantifiers, that is, quantifiers that measure to which extent a formula is true. The resulting systems, called $\mathrm{cCPL}$ and $\mathrm{iCPL}$, respectively, admit a natural semantics, based on the Borel σ-algebra of the Cantor space, together with a sound and complete proof system. Our main results consist in relating $\mathrm{cCPL}$ and $\mathrm{iCPL}$ with some central concepts in the study of probabilistic computation. On the one hand, the validity of $\mathrm{cCPL}$-formulae in prenex form characterizes the corresponding level of Wagner's hierarchy of counting complexity classes, closely related to probabilistic complexity. On the other hand, proofs in $\mathrm{iCPL}$ correspond, in the sense of Curry and Howard, to typing derivations for a randomized extension of the λ-calculus, so that counting quantifiers reveal the probability of termination of the underlying probabilistic programs.

We extend the main result of [1] to the first-order intuitionistic logic (with and without equality), showing that it is a maximal (with respect to expressive power) abstract logic satisfying a certain form of compactness, the Tarski union property and preservation under asimulations. A similar result is also shown for the intuitionistic logic of constant domains.

An ${\omega }_1$-compact space is a space in which every closed discrete subspace is countable. We give various general conditions under which a locally compact, ${\omega }_1$-compact space is σ-countably compact, i.e., the union of countably many countably compact spaces. These conditions involve very elementary properties.

Many results shown here are independent of the usual (ZFC) axioms of set theory, and the consistency of some may involve large cardinals. For example, it is independent of the ZFC axioms whether every locally compact, ${\omega }_1$-compact space of cardinality ${\aleph }_1$ is σ-countably compact. Whether ${\aleph }_1$ can be replaced with ${\aleph }_2$ is a difficult unsolved problem. Modulo large cardinals, it is also ZFC-independent whether every hereditarily normal, or every monotonically normal, locally compact, ${\omega }_1$-compact space is σ-countably compact.

As a result, it is also ZFC-independent whether there is a locally compact, ${\omega }_1$-compact Dowker space of cardinality ${\aleph }_1$, or one that does not contain both an uncountable closed discrete subspace and a copy of the ordinal space ${\omega }_1$.

Set theoretic tools used for the consistency results include the existence of a Souslin tree, the Proper Forcing Axiom (PFA), and models generically referred to as “MM(S)[S]”. Most of the work is done by the P-Ideal Dichotomy (PID) axiom, which holds in the latter two cases, and which requires no large cardinal axioms when directly applied to topological spaces of cardinality ${\aleph }_1$, as it is in several theorems.

Kunen refuted the existence of an elementary embedding from the universe of sets to itself assuming the Axiom of Choice. This paper concerns the ramifications of this hypothesis when the Axiom of Choice is not assumed. For example, the existence of such an embedding implies that there is a proper class of cardinals λ such that ${\lambda }^{+}$ is measurable.

We find a strong separation between two natural families of simple rank one theories in Keisler's order: the theories $T_m$ reflecting graph sequences, which witness that Keisler's order has the maximum number of classes, and the theories $T_{n,k}$, which are the higher-order analogues of the triangle-free random graph. The proof involves building Boolean algebras and ultrafilters “by hand” to satisfy certain model theoretically meaningful chain conditions. This may be seen as advancing a line of work going back through Kunen's construction of good ultrafilters in ZFC using families of independent functions. We conclude with a theorem on flexible ultrafilters, and open questions.

Variations on the splitting number $s$ are examined by localizing the splitting property to finite sets. To be more precise, rather than considering families of subsets of the integers that have the property that every infinite set is split into two infinite sets by some member of the family a stronger property is considered: Whenever an subset of the integers is represented as the disjoint union of a family of finite sets one can ask that each of the finite sets is split into two non-empty pieces by some member of the family. It will be shown that restricting the size of the finite sets can result in distinguishable properties. In §2 some inequalities will be established, while in §3 the main consistency result will be proved.