We prove that for every family I of closed subsets of a Polish space each set can be covered by countably many members of I or else contains a nonempty set which cannot be covered by countably many members of I. We prove an analogous result for κ-Souslin sets and show that if A# exists for any A ⊂ ωω, then the above result is true for sets. A theorem of Martin is included stating that this result is also true for weakly homogeneously Souslin sets. As an application of our results we derive from them a general form of Hurewicz's theorem due to Kechris. Louveau, and Woodin and a theorem of Feng on the open covering axiom. Also some well-known theorems on finding “big” closed sets inside of “big” and are consequences of our results.

A reader coming anew to the recent work on languages which contain their own truth predicates may be perplexed by the simple question of where to begin. A first approach to the literature suggests a field which is alive and busy with investigations heading in many different directions, but there is much less indication of how various pieces fit together. There are at least two sources of this confusion. First, the literature is large and diffuse (as befits a subject which goes back over 2000 years); Visser's survey [33] aptly describes the literature as “vast but scattered, repetitive, and disconnected.” Moreover, recent interest in the field has led to a proliferation of research and publication; it seems that almost any issue of any philosophical logic journal from the mid-1980s contains some article on the topic. The second reason, in part a consequence of the first, is that while a typical article in print usually presents a good internal motivation, with clear reference to its immediate intellectual antecedents, its place in the broader picture may not be so easily discerned. The problem can be especially acute in presentations of axiomatic approaches, because decisions on certain basic questions can lie hidden in the formal and notational details which abound in any axiomatization.

In fact, though, the recent research on methods for handling self-referential truth can be seen as a body of work which is very well structured, one in which a few fundamental decisions suffice to locate any particular approach in its appropriate place on the landscape. My goal is to describe this structure and in particular to stress a few critical forks in the road, which will be the recurring metaphor throughout this paper. I will also pay particular attention to pointing out where the interesting technical and mathematical questions lie.

In this paper, we generalize the notion of distributivity and consider some properties of distributive ideals, that is, ideals I such that the algebra P(κ)/I is distributive in our sense.

Our notation and terminology is explained in §1, while the main results of this paper begin in §2. We shall show here some relations of the distributivity and the ideal theoretic partitions. In §3, we shall study the class of distributive ideals over κ whose existence is equivalent to the ineffability of κ, and other classes. Finally, in §4, we shall consider the equivalence of the Boolean prime ideal theorem and show that the existence of certain distributive ideals characterizes several large cardinals. As a byproduct, we can give a simple proof of Ketonen's theorem that κ is strongly compact if and only if for any regular cardinal λ ≥ κ there exists a nontrivial κ-complete prime ideal over λ.

We show that Diophantine equivalence of two suitably presented countable rings implies that the existential polynomial languages of the two rings have the same “expressive power” and that their Diophantine sets are in some sense the same. We also show that a Diophantine class of countable rings is contained completely within a relative enumeration class and demonstrate that one consequence of this fact is the existence of infinitely many Diophantine classes containing holomorphy rings of ℚ.

An awkward feature of the standard fine structure theory of the Jα's (see Jensen [72]) is that special parameters are required to make good sense of the notion of “Σn Skolem hull”, parameters which may not be preserved in condensation arguments.

The purpose of this article is to indicate how a reformulation of Jensen's Σ* theory (developed for the study of core models) can be used to provide a more satisfactory treatment of uniformization, hulls, and Skolem functions for the Jα's. Then we use this approach to fine structure to formulate a principle intended to capture the combinatorial content of the axiom V = L.