Annals of Pure and Applied Logic最新文献

For every number $n\ge 2$, let ${\Gamma }_n$ be the hypergraph on $R^n$ of arity four consisting of all non-degenerate Euclidean rectangles. It is consistent with ZF+DC set theory that the chromatic number of ${\Gamma }_n$ is countable while that of ${\Gamma }_{n+1}$ is not.

We study the reverse mathematics of infinitary extensions of the Hales-Jewett theorem, due to Carlson and Simpson. These theorems have multiple applications in Ramsey's theory, such as the existence of finite big Ramsey degrees for the triangle-free graph, or the Dual Ramsey theorem. We show in particular that the Open Dual Ramsey theorem holds in ${\mathrm{ACA}}_0^{+}$.

We present a detailed formalization of Lipschitz and Wadge games in the context of second order arithmetic and we investigate the logical strength of Lipschitz and Wadge determinacy, and the tightly related Semi-Linear Ordering principle, for the first levels of the Hausdorff difference hierarchy in the Cantor space. As a result, we obtain characterizations of ${\mathrm{WKL}}_0$ and ${\mathrm{ACA}}_0$ in terms of these determinacy principles.

Choice and independence of premise principles play an important role in characterizing Kreisel's modified realizability and Gödel's Dialectica interpretation. In this paper we show that a great many intuitionistic set theories are closed under the corresponding rules for finite types over $N$. It is also shown that the existence property (or existential definability property) holds for statements of the form $\exists y^{\sigma }\phi \left(y\right)$, where the variable y ranges over objects of finite type σ. This applies in particular to $\mathrm{CZF}$ (Constructive Zermelo-Fraenkel set theory) and $\mathrm{IZF}$ (Intuitionistic Zermelo-Fraenkel set theory), two systems known not to have the general existence property. On the technical side, the paper uses a method that amalgamates generic realizability for set theory with truth, whereby the underlying partial combinatory algebra is required to contain all objects of finite type.

In this paper, it is shown that the fact that the BPI holds in the Cohen's symmetric model can be used as an equal substitute for the Halpern-Läuchli theorem. Also, some alternatives to the Halpern-Läuchli theorem in the form of absoluteness theorems for a certain class of statements are consequences¹ of such consistency results. The article also contains a new proof of the Halpern-Läuchli theorem.

We show that Morley's theorem on the number of countable models of a countable first-order theory becomes an undecidable statement when extended to second-order logic. More generally, we calculate the number of equivalence classes of equivalence relations obtained by countable intersections of projective sets in several models of set theory. Our methods include random and Cohen forcing, Woodin cardinals and Inner Model Theory.

Based on the well-known cardinal characteristics $s$, $r$ and $i$, we introduce nine related cardinal characteristics by using the notion of asymptotic density to characterise different intersection properties of infinite sets. We prove several bounds and consistency results, e.g. the consistency of $s<s_{\frac{1}{2}}$, as well as several results about possible values of $i_{\frac{1}{2}}$.

We investigate computable metrizability of Polish spaces up to homeomorphism. In this paper we focus on Stone spaces. We use Stone duality to construct the first known example of a computable topological Polish space not homeomorphic to any computably metrized space. In fact, in our proof we construct a right-c.e. metrized Stone space which is not homeomorphic to any computably metrized space. Then we introduce a new notion of effective categoricity for effectively compact spaces and prove that effectively categorical Stone spaces are exactly the duals of computably categorical Boolean algebras. Finally, we prove that, for a Stone space X, the Banach space $C(X;R)$ has a computable presentation if, and only if, X is homeomorphic to a computably metrized space. This gives an unexpected positive partial answer to a question recently posed by McNicholl.

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.