Aspectual coercion as an intensional phenomenon. Ever since Vendler's famous classification of verbs with respect to their time schemata, linguists have distinguished at least four aspectual classes or Aktionsarten. These classes comprise states, activities, accomplishments, achievements and points, which are exemplified in (1) to (5).

1. (1) States: know, love, be beautiful.

2. (2) Activities: run, push a cart, draw.

3. (3) Accomplishments: cross the street, build a house, draw a letter.

4. (4) Achievements: begin, reach, arrive.

5. (5) Points: flash, spot, blink.

Several linguistic tests were developed to distinguish aspectual classes. For example, adverbial modification with for two hours is possible with activities (John ran for two hours) but not with accomplishments (*John built a house for two years) or achievements. Temporal modification with the adverbial in two hours shows the reverse pattern. Usually only activities and accomplishments can occur in the progressive. Expressions like *knowing the answer are ungrammatical. But there is also a crucial difference in the behaviour of activities and accomplishments with respect to their progressivised forms. From John was pushing a cart we can infer that John pushed a cart. However, the inference from John was crossing the street to John crossed the street is invalid. This phenomenon was dubbed the “imperfective paradox” by Dowty.

Introduction. Each of the classical tree forcings, such as Sacks forcing S, Laver Forcing L or Miller forcing M, is associated with a ideal on (in the case of S) or on (in the case of L or M). These can be considered forcing ideals in the sense that a real (in or) is generic for the respective forcing iff it avoids all the sets from the associated ideal defined in the ground model. Such ideals were first studied by Marczewski [9], later by Veličković [16], and Judah, Miller, Shelah [5]. They all studied the Sacks ideal. The ideals associated with Laver or Miller forcing were investigated by Brendle, Goldstern, Johnson, Repický, Shelah and Spinas (see [3], [4], [12] and [2] (chronological order)). In all these papers, among other things the additivity and covering coefficients were studied. Let J(Q) denote the ideal associated with the forcing Q. The typical problem was whether for in the trivial chain of inequalities

any of the inequalities could consistently be strict. Here add(J), cov(J) denotes the additivity, covering coefficient of the ideal J, respectively. One of the main results of [5] is that, letting and a countable support iteration of S of length, then is a model for add(J(S)) < cov(J(S)). The hard part of the argument is to show that is a model for. Building on these arguments, analogues of this result were proved in [4] for J(M) and J(L): For, letting denote a countable support iteration of length of Q and V is a model for. Actually, for Q = M this result needs a lemma from [12] which was not yet available in [4]. Again, the hard part was to show that holds in the extension. For this, it was proved that the cardinal invariant h which is defined as the distributivity number of is an upper bound of add(J(Q)) in.

As you know well, we celebrate 70th year of Gödel's “Incompleteness theorem” which he proved at the age of 25. This revolutionary theorem changed the way mathematicians think of mathematics drastically. Today, I would like to describe this theorem, its effects and expound on this subject.

Gödel's incompleteness theorem can be stated as follows.

1. 1. Let T be a consistent axiomatic theory like set theory or the theory of analysis where Peano's arithmetic (denoted simply as arithmetic from now on) is included. Then T does not prove its consistency.

2. 2. The consistency of T can be expressed as a sentence in arithmetic, therefore there exists an arithmetical sentence which cannot be proved in T.

This seemingly simple theorem changed our view of mathematics completely. Before this theorem, mathematicians believed that every problem in arithmetic could be solved by some stronger theory, e.g., set theory. But the Incompleteness theorem tells us that whatever stronger theory we use, there exists a true arithmetical sentence which cannot be proved in the theory.

After the Incompleteness theorem, the consistency statement becomes a landmark for the boundary of provable statements in the theory. Whenever we wish to show that the theory T is strictly stronger than the theory T, we first try to show that the theory T proves the consistency statement of T. In this way we can show that the theory of analysis is strictly stronger than the theory of arithmetic and that set theory is strictly stronger than the theory of analysis.

I would now like to speak of the impact that this theorem had on Hilbert's program. Hilbert was a genius to think of problems in a general setting, to find the essence of the problem and solve it. So, he was a leader of the movement of abstract systematic development and axiomatization in the 20th century mathematics. Cantor's set theory gave the ideal framework for this movement. Hilbert believed Cantor had created a new paradise for mathematicians. So, paradoxes of Cantor's set theory came as a great shock to him. Hilbert tried to save themodernmathematics and proposed the following program.

When Alfred Tarski wrote his famous definition of truth [20] (1933) for a formal language, he had several stated aims. His chief aim was to define truth of sentences. Giving correct meanings of other expressions of the language was nowhere in his list of aims at all; it was a happy accident that a general semantics fell out of his truth definition.

So the following facts, all very easily proved, came to me as a surprise. Given any notion of meaning for sentences (for example, a specification of when they are true and when not), and assuming some simple book-keeping conditions, there is a canonical way of extending this notion to a semantics for the whole language. I call it the fregean extension; it is determined up to the question which pairs of expressions have the same meaning. Tarski's semantics for first-order logic is the fregean extension of the truth conditions for sentences. Afewmore book-keeping conditions guarantee that the fregean extension can be chosen to have good functional properties of the kind often associated with Frege and with type-theoretic semantics.

Tarski himselfwas certainly interested in the question howfar his solution of his problem was canonical, and we can learn useful things from his discussion of the issue. But the main results below on fregean extensions come closer to the linguistic and logical concerns of Frege and Husserl, a generation earlier than Tarski. Husserl has been unjustly neglected by logicians, and Frege's innovations in linguistics deserve to be better known.

Introduction. This paper is about the semantic analysis of opaque verbs such as seek and owe, which allow for unspecific readings of their indefinite objects. One may be looking for a good car without there being any car that one is looking for; or, one may be looking for a good car in that a specific car exists that one is looking for. It thus appears that there are two interpretations of these verbs—a specific and an unspecific one—and one may wonder how they are related. The present paper is a contribution to this question.

History

Paris. The time of the holy inquisition. Opaque verbs differ in their semantic behaviour from ordinary verbs. This phenomenon was already known to the medieval logician Buridanus:

The followingmodern version of the opposite argument is less verbose than the original:

The two arguments are based on two different ways of reading the sentence under debate (1) — an obvious, unspecific interpretation and a somewhat remote, specific one.

From the dawn of time women and men have aspired upward. The development of determinacy proofs is no exception to this general rule. There has been a steady search for higher forms of determinacy, beginning with the results of Gale–Stewart [2] on closed length ω games and continuing to this day. Notable landmarks in this quest include proofs of Borel determinacy in Martin [5]; analytic determinacy in Martin [4]; projective determinacy in Martin–Steel [8]; and ADL(R) in Woodin [17]. Those papers consider length ω games with payoff sets of increasing complexity. One could equivalently fix the complexity of the payoff and consider games of increasing length. Such “long games” form the topic of this paper.

Long games form a natural hierarchy, the hierarchy of increasing length. This hierarchy can be divided into four categories: games of length less than ω · ω; games of fixed countable length; games of variable countable length; and games of length ω1.

Games in the first category can be reduced to standard games of length ω, at the price of increasing payoff complexity. The extra complexity only involves finitely many real quantifiers. Thus the determinacy of games of length less than ω · ω, with analytic payoff say, is the same as projective determinacy.

Games in the second category can be reduced to combinations of standard games of length ω, with increased payoff complexity, and some additional strength assumptions. The first instance of this is given in Blass [1]. The techniques presented there can be used to prove the determinacy of length ω ·ω games on natural numbers, with analytic payoff say, from ADL(R) + “R# exists.” In another, choiceless reduction to standard games, Martin and Woodin independently showed that AD + “all sets of reals admit scales” implies that all games in the second category are determined.

It is in the third category that the methods presented here begin to yield new determinacy principles. (The one previously known determinacy proof for games in the third category is a theorem of Steel [16], which applies to games of the kind described in Remark 1.1.)

Neeman [15] concentrates on third category games.

The usual model-theoretic approach to complex algebraic geometry is to view complex algebraic varieties as living definably in the structure (C,+,×). Variousmodel-theoretic properties of algebraically closed fields (such as quantifier elimination and strong minimality) are then used to obtain geometric information about the varieties. This approach extends to other geometric contexts by considering expansions of algebraically closed fields to which the methods of stability or simplicity apply. For example, differential algebraic varieties live in differentially closed fields, and difference algebraic varieties in algebraically closed fields equipped with a generic automorphism. Another approach would be to consider the variety as a structure in its own right, equipped with the algebraic (respectively differential or difference algebraic) subsets of its cartesian powers. This point of view is compatible with the theory of Zariski-type structures, developed by Hrushovski and Zilber (see [21] and [41]). While the two approaches are equivalent (i.e., bi-interpretable) in the case of complex algebraic varieties, the latter point of view extends to certain fragments of complex analytic geometry in a manner that does not seem accessible by the former.

Zilber showed in [41] that a compact complex analytic space with the structure induced by the analytic subsets of its cartesian powers is of finite Morley rank and admits quantifier elimination. Since then, there have been a number of papers investigating various aspects of the model theory of such structures, as well as possible applications to complex analytic geometry. The most notable achievement has been a kind of Chevalley Theorem due to Pillay and Scanlon [35] that classifies meromorphic groups in terms of complex tori and linear algebraic groups. It has become clear that compact complex spaces serve as a particularly rich setting in which many of the more advanced phenomena of stability theory are witnessed. For example, the full trichotomy for strongly minimal sets (trivial, not trivial but locally modular, and not locally modular) occurs in this category. It also seems reasonable to expect that a greater model-theoretic understanding of compact complex spaces can contribute to complex analytic geometry, particularly around issues concerning the bimeromorphic classification of Kähler-type spaces.

This essay owes its appearance here to the good offices of Reinhard Kahle who, at short notice, allowed me the opportunity to turn into something publishable a rat's nest of clandestine fragments hitherto available only to the author's friends and students. I had always assumed that they were in any case so heretical that nobody would publish them even if I tidied them up and I am grateful to Kahle for giving me the chance to be burnt rather than merely ignored. The one major regret I now have about the delay caused by my timidity is that in the dozen-or-so years that have passed since the first draughts of this essay were circulated David Lewis has died. Readers familiar with the literature will immediately recognise Lewis as the most important single creator of the Augean stables I am reporting on, and as the writer most likely to wish to maintain them intact as a World Heritage Site. It is true that I thought his ideas terrible, but he defended them with personal integrity and without malice: I enjoyed his company, and am sorry that he is not around to reply.

The three essays out of which it grew were entitled “The modal ather”, “Indexicality” and “The closest possible world”. In the first essay I argue that if possible worlds are to be used at all to explain necessary truth, then at least some truths (those concerning relations between worlds) are necessary in virtue of something other than truth in all those worlds. In the second I argue that the idea that actuality as indexical is in need of a lot of explanation. In the third I argue that there is no logical notion of closest possible world. The essays get progressively more technical, but I preface themwith an introductory essay whose general drift can be caught even by those with no formal background, and insert between the first and second essays a brief sketch of the topological ideas on which ideas of indexicality and proximity (of the two final essays) presumably ultimately rely.