When modern set theory is applied to conventional mathematical problems, it has a disconcerting tendency to produce independence results rather than theorems in the usual sense. The resulting preoccupation with “consistency ” rather than “truth ” may be felt to give the subject an air of unreality. Even elementary questions about the basic arithmetical operations of exponentiation in the context of infinite cardinalities, like the value of 2 ℵ0, cannot be settled on the basis of the usual axioms of set theory (ZFC). Although much can be said in favor of such independence results, rather than undertaking to challenge such prejudices, we have a more modest goal; we wish to point out an area of contemporary set theory in which theorems are abundant, although the conventional wisdom views the subject as dominated by independence results, namely, cardinal arithmetic. To see the subject in this light it will be necessary to carry out a substantial shift in our point of view. To make a very rough analogy with another generalization of ordinary arithmetic, the natural response to the loss of unique factorization caused

this article I will be looking at the leading question from the point of view of the logician, and for a substantial part of that, from the perspective of one supremely important logician: Kurt Godel. From the time of his stunning incompleteness results in 1931 to the end of his life, Godel called for the pursuit of new axioms to settle undecided arithmetical problems. And from 1947 on, with the publication of his unusual article, "What is Cantor's continuum problem?" [11], he called in addition for the pursuit of new axioms to settle Cantor's famous conjecture about the cardinal number of the continuum. In both cases, he pointed primarily to schemes of higher infinity in set theory as the direction in which to seek these new principles. Logicians have learned a great deal in recent years that is relevant to Godel's program, but there is considerable disagreement about what conclusions to draw from their results. I'm far from unbiased in this respect, and you'll see how I come out on these matters by the end of this essay, but I will try to give you a fair presentation of other positions along the way so you can decide for yourself which you favor.

Abstract. The Levy-Solovay Theorem [LevSol67] limits the kind of large cardinal embeddings that can exist in a small forcing extension. Here I announce a generalization of this theorem to a broad new class of forcing notions. One consequence is that many of the forcing iterations most commonly found in the large cardinal literature create no new weakly compact cardinals, measurable cardinals, strong cardinals, Woodin cardinals, strongly compact cardinals, supercompact cardinals, almost huge cardinals, huge cardinals, and so on. Large cardinal set theorists today generally look upon small forcing—that is, forcing with a poset P of cardinality less than whatever large cardinal κ is under consideration—as benign. This outlook is largely due to the Levy-Solovay theorem [LevSol67], which asserts that small forcing does not affect the measurability of any cardinal. (Specifically, the theorem says that if a forcing notion P has size less than κ, then the ground model V and the forcing extension V P agree on the measurability of κ in a strong way: the ground model measures on κ all generate as filters measures in the forcing extension, the corresponding ultrapower embeddings lift uniquely from the ground model to the forcing extension and all the measures and ultrapower embeddings in the forcing extension arise in this way.) Since the Levy-Solovay argument generalizes to the other large cardinals whose existence is witnessed by certain kinds of measures or ultrapowers, such as strongly compact cardinals, supercompact cardinals, almost huge cardinals and so on, one is led to the broad conclusion that small forcing is harmless; one can understand the measures in a small forcing extension by their relation to the measures existing already in the ground model. Here in this Communication I would like to announce a generalization of the Levy-Solovay Theorem to a broad new class of forcing notions.

Introduction The fact that small cardinals (for example @ 1 and @ 2 ) can consistently have properties similar to those of large cardinals (for example measurable or supercompact cardinals) is a recurring theme in set theory. In these notes I discuss three examples of this phenomenon; stationary reflection, saturated ideals and the tree property. These notes represent approximately the contents of a series of expository lectures given during the Set Theory meeting at CRM Barcelona in June 1996. None of the results discussed here is due to me unless I say so explicitly. I would like to express my thanks to Joan Bagaria and Adrian Mathias for organising a very enjoyable meeting. 1 2 Large cardinals and elementary embeddings We begin by reviewing the formulation of large cardinal properties in terms of elementary embeddings. See [40], [22] or [21] for more on this topic. We will write "j : V<F14.4

Abstract. In this paper I generalize the landmark Levy-Solovay Theorem [Lev-Sol67], which limits the kind of large cardinal embeddings that can exist in a small forcing extension, to a broad new class of forcing notions, a class that includes many of the forcing iterations most commonly found in the large cardinal literature. The fact is that after such forcing, every embedding satisfying a mild closure requirement lifts an embedding from the ground model. A consequence is that such forcing can create no new weakly compact cardinals, measurable cardinals, strong cardinals, Woodin cardinals, strongly compact cardinals, supercompact cardinals, almost huge cardinals, or huge cardinals, and so on. Small forcing in a large cardinal context, that is, forcing with a poset P of cardi-nality less than whatever large cardinal κ is under consideration, is today generally looked upon as benign. This outlook is largely due to the landmark Levy-Solovay theorem [LevSol67], which asserts that small forcing does not affect the measurability of any cardinal. (Specifically, the theorem says that if a forcing notion P has size less than κ, then the ground model V and the forcing extension V P agree on the

1. George Boolos has extensively investigated plural quantification in Boolos [1984] and Boolos [1985]. He observed that such locutions as the Geach and Kaplan’s sentence “There are some critics who admire only one another ” are not adequately formalized in the language of first-order predicate logic, but rather in the language of monadic second-order logic. Indeed, Boolos showed in Boolos [1984] that plural quantification is interdefinable with second-order quantification. This is an indisputable technical result, but its philosophical significance depends on the ontological status of plural quantification. What is crucial for Boolos is that plural quantification requires neither metalinguistic ascent nor ontological commitment with sets, classes or Fregean concepts. Boolos persuasively argued that plural quantification is not singular quantification in grammatical disguise. The Geach and Kaplan’s sentence involves ontological commitment with critics, but not with sets, classes, or Fregean concepts under which critics fall. The observation that second-order quantification is interdefinable with plural quantifica-tion is of special interest for the philosophy and the foundations of set theory. There are

Finite functions and the necessary use of large cardinals