Annals of Pure and Applied Logic最新文献

In this paper, we investigate the fresh function spectrum of forcing notions, where a new function on an ordinal is called fresh if all its initial segments are in the ground model. We determine the fresh function spectrum of several forcing notions and discuss the difference between fresh functions and fresh subsets. Furthermore, we consider the question which sets are realizable as the fresh function spectrum of a homogeneous forcing. We show that under GCH all sets with a certain closure property are realizable, while consistently there are sets which are not realizable.

Let M be strongly minimal and constructed by a ‘Hrushovski construction’ with a single ternary relation. If the Hrushovski algebraization function μ is in a certain class $T$ (μ triples) we show that for independent I with $\left|I\right|>1$, ${\mathrm{dcl}}^{⁎}\left(I\right)=\varnothing$ (* means not in dcl of a proper subset). This implies the only definable truly n-ary functions f (f ‘depends’ on each argument), occur when $n=1$. We prove for Hrushovski's original construction and for the strongly minimal k-Steiner systems of Baldwin and Paolini that the symmetric definable closure, ${\mathrm{sdcl}}^{⁎}\left(I\right)=\varnothing$ (Definition 2.7). Thus, no such theory admits elimination of imaginaries. As, we show that in an arbitrary strongly minimal theory, elimination of imaginaries implies ${\mathrm{sdcl}}^{⁎}\left(I\right)\ne \varnothing$. In particular, such strongly minimal Steiner systems with line-length at least 4 do not interpret a quasigroup, even though they admit a coordinatization if $k=p^n$. The case structure depends on properties of the Hrushovski μ-function. The proofs depend on our introduction, for appropriate $G\subseteq \mathrm{aut}\left(M\right)$ (setwise or pointwise stabilizers of finite independent sets), the notion of a G-normal substructure $A$ of M and of a G-decomposition of any finite such $A$. These results lead to a finer classification of strongly minimal structures with flat geometry, according to what sorts of definable functions they admit.

We classify all ${\aleph }_0$-categorical and C-minimal C-sets up to elementary equivalence. As usual the Ryll-Nardzewski Theorem makes the classification of indiscernible ${\aleph }_0$-categorical C-minimal sets as a first step. We first define solvable good trees, via a finite induction. The trees involved in initial and induction steps have a set of nodes, either consisting of a singleton, or having dense branches without endpoints and the same number of branches at each node. The class of colored good trees is the elementary class of solvable good trees. We show that a pure C-set M is indiscernible, finite or ${\aleph }_0$-categorical and C-minimal iff its canonical tree $T\left(M\right)$ is a colored good tree. The classification of general ${\aleph }_0$-categorical and C-minimal C-sets is done via finite trees with labeled vertices and edges, where labels are natural numbers, or infinity and complete theories of indiscernible, ${\aleph }_0$-categorical or finite, and C-minimal C-sets.

We develop a duality for (modal) lattices that need not be distributive, and use it to study positive (modal) logic beyond distributivity, which we call weak positive (modal) logic. This duality builds on the Hofmann, Mislove and Stralka duality for meet-semilattices. We introduce the notion of ${\Pi }_1$-persistence and show that every weak positive modal logic is ${\Pi }_1$-persistent. This approach leads to a new relational semantics for weak positive modal logic, for which we prove an analogue of Sahlqvist's correspondence result.¹

Towards combining “compactness” and “hugeness” properties at ${\omega }_2$, we investigate the relevance of side-conditions forcing. We reduce the upper bound on the consistency strength of the weak Chang's Conjecture at ${\omega }_2$ using Neeman's forcing. On the other hand, we find a barrier to the applicability of these methods to our problem and give a counterexample to a claim of Neeman about the effects of iterating such forcing.

We prove that the theory of the models constructible using finitely many cofinality quantifiers – $C_{{\lambda }_1,\dots ,{\lambda }_n}^{⁎}$ and $C_{<{\lambda }_1,\dots ,<{\lambda }_n}^{⁎}$ for ${\lambda }_1,\dots ,{\lambda }_n$ regular cardinals – is set-forcing absolute under the assumption of class many Woodin cardinals, and is independent of the regular cardinals used. Towards this goal we prove some properties of the generic embedding induced from the stationary tower restricted to <μ-closed sets.

Chagrov and Zakharyaschev posed the problem of existence of extensions of Solovay's system S, which is a non-normalizable quasi-normal modal logic, that do not admit deductively independent sets of axioms. This paper gives a solution by exhibiting countably many extensions of S without deductively independent sets of axioms.

We show that $\mathrm{PFA}$ (Proper Forcing Axiom) implies that adding any number of Cohen subsets of ω will not add an ${\omega }_2$-Aronszajn tree or a weak ${\omega }_1$-Kurepa tree, and moreover no σ-centered forcing can add a weak ${\omega }_1$-Kurepa tree (a tree of height and size ${\omega }_1$ with at least ${\omega }_2$ cofinal branches). This partially answers an open problem whether ccc forcings can add ${\omega }_2$-Aronszajn trees or ${\omega }_1$-Kurepa trees (with $\neg {\square }_{{\omega }_1}$ in the latter case).

We actually prove more: We show that a consequence of $\mathrm{PFA}$, namely the guessing model principle, $\mathrm{GMP}$, which is equivalent to the ineffable slender tree property, $\mathrm{ISP}$, is preserved by adding any number of Cohen subsets of ω. And moreover, $\mathrm{GMP}$ implies that no σ-centered forcing can add a weak ${\omega }_1$-Kurepa tree (see Section 2.1 for definitions).

For more generality, we study variations of the principle $\mathrm{GMP}$ at higher cardinals and the indestructibility consequences they entail, and as applications we answer a question of Mohammadpour about guessing models at weakly but not strongly inaccessible cardinals and show that there is a model in which there are no weak ${\aleph }_{\omega +1}$-Kurepa trees and no ${\aleph }_{\omega +2}$-Aronszajn trees.

We study a strengthening of the notion of a universally meager set and its dual counterpart that strengthens the notion of a universally null set.

We say that a subset A of a perfect Polish space X is countably perfectly meager (respectively, countably perfectly null) in X, if for every perfect Polish topology τ on X, giving the original Borel structure of X, A is covered by an $F_{\sigma }$-set F in X with the original Polish topology such that F is meager with respect to τ (respectively, for every finite, non-atomic, Borel measure μ on X, A is covered by an $F_{\sigma }$-set F in X with $\mu \left(F\right)=0$).

We prove that if $2^{{\aleph }_0}\le {\aleph }_2$, then there exists a universally meager set in $2^N$ which is not countably perfectly meager in $2^N$ (respectively, a universally null set in $2^N$ which is not countably perfectly null in $2^N$).