Using an idea developed in joint work with Shelah, we show how to redefine Laver's notion of forcing making a supercompact cardinal κ indestructible under κ-directed closed forcing to give a new proof of the Kimchi-Magidor Theorem in which every compact cardinal in the universe (supercompact or strongly compact) satisfies certain indestructibility properties. Specifically, we show that if K is the class of supercompact cardinals in the ground model, then it is possible to force and construct a generic extension in which the only strongly compact cardinals are the elements of K or their measurable limit points, every κ ∈ K is a supercompact cardinal indestructible under ∈-directed closed forcing, and every κ a measurable limit point of K is a strongly compact cardinal indestructible under κ-directed closed forcing not changing ℘(κ). We then derive as a corollary a model for the existence of a strongly compact cardinal κ which is not κ+ supercompact but which is indestructible under κ-directed closed forcing not changing ℘(κ) and remains non-κ+ supercompact after such a forcing has been done.

Generalized n-gons are certain geometric structures (incidence geometries) that generalize the concept of projective planes (the nontrivial generalized 3-gons are exactly the projective planes).

In a simplified world, every generalized n-gon of finite Morley rank would be an algebraic one, i.e., one of the three families described in [9] for example. To our horror, John Baldwin [2], using methods discovered by Hrushovski [7], constructed ℵ1-categorical projective planes which are not algebraic. The projective planes that Baldwin constructed fail to be algebraic in a dramatic way.

Indeed, every algebraic projective plane over an algebraically closed field is Desarguesian [12]. In particular, an algebraically closed field (isomorphic to the base field) can be interpreted in every one of them. However, in the projective planes that Baldwin constructed, one cannot even interpret an infinite group.

In this article we show that the same phenomenon occurs for the generalized n-gons if n ≥ 3 is an odd integer. For each such n we construct many nonisomorphic generalized n-gons of finite Morley rank that do not interpret an infinite group. As one may expect, our method is inspired by Hrushovski and Baldwin, and we follow Baldwin's line of approach. Quite often our proofs are a verification of the fact that the proofs of Baldwin [2] for n = 3 carry over to an arbitrary positive odd integer n (which is sometimes far from being obvious). As in [2], we begin by defining a certain collection of finite graphs K* and a binary relation ≤ on these graphs. We show that (K*, ≤) satisfies the amalgamation property.

Before one can construct scales of minimal complexity in the Real Core Model, K(ℝ), one needs to develop the fine-structure theory of K (ℝ). In this paper, the fine structure theory of mice, first introduced by Dodd and Jensen, is generalized to that of real mice. A relative criterion for mouse iterability is presented together with two theorems concerning the definability of this criterion. The proof of the first theorem requires only fine structure; whereas, the second theorem applies to real mice satisfying AD and follows from a general definability result obtained by abstracting work of John Steel on L(ℝ). In conclusion, we discuss several consequences of the work presented in this paper relevant to two issues: the complexity of scales in K(ℝ)and the strength of the theory ZF + AD + ┐DCℝ.

We denote by KP_ the fragment of set-theory containing the axioms of extensionality, pairing, union and foundation as well as the schemas of ∆0-comprehension and ∆0-collection, that is: Kripke-Platek set-theory (KP) with the axiom of foundation in place of the ∈-induction schema. The theory KP is obtained by adding to KP_ the schema of ∈-induction

Using ∈-induction it is possible to prove the existence of the transi tive closure without appealing to the axiom of infinity (see, e.g., [1]). Vice versa, when a theory proves the existence of the transitive closure, some induction is immediately ensured (by foundation and comprehension). This is not true in general: e.g., the whole of Zermelo-Fraenkel set-theory without the axiom of infinity does not prove ∈-induction (in fact, it does not prove the existence of the transitive closure; see, e.g., [3]). Open-induction is the schema of ∈-induction restricted to open formulas. We prove the following theorem.

KP_ proves open-induction.

We reason in a fixed but arbitrary model of KP_ whom we refer to as the model. The language is extended with a name for every set in the model. We call this constants parameters. Let φ(x) be a satisfiable open-formula possibly depending on parameters and with no free variable but x. We show that φ(x) is satisfied by an ∈-minimal set, that is, a set a such that φ(a) and (∀x ∈ a) ¬φ(x). We assume that no ordinal satisfies φ(x), otherwise the existence of a ∈-minimal set follows from foundation and comprehension.

Assuming that ORD is ω + ω-Erdös we show that if a class forcing amenable to L (an L-forcing) has a generic then it has one definable in a set-generic extension of L[O#]. In fact we may choose such a generic to be periodic in the sense that it preserve the indiscernibility of a final segment of a periodic subclass of the Silver indiscernibles, and therefore to be almost codable in the sense that it is definable from a real which is generic for an L-forcing (and which belongs to a set-generic extension of L[0#]). This result is best possible in the sense that for any countable ordinal α there is an L-forcing which has generics but none periodic of period ≤ α. However, we do not know if an assumption beyond ZFC+“O# exists” is actually necessary for these results.

Let P denote a class forcing definable over an amenable ground model 〈L, A〉 and assume that O# exists.

Definition. P is relevant if P has a generic definable in L[0#]. P is almost relevant if P has a generic definable in a set-generic extension of L[0#].

Remark. The reverse Easton product of Cohen forcings 2<κ, κ regular is relevant. So are the Easton product and the full product, provided κ is restricted to the successor cardinals. See Chapter 3, Section Two of Friedman [3]. Of course any set-forcing (in L) is almost relevant.

Definition. κ is α-Erdös if whenever C is CUB in κ and f: [C]<ω → κ is regressive (i.e., f(a) < min(a)) then f has a homogeneous set of ordertype α.

We show the consistency, relative to a supercompact cardinal, of the least measurable cardinal being both strongly compact and fully Laver indestructible. We also show the consistency, relative to a supercompact cardinal, of the least strongly compact cardinal being somewhat supercompact yet not completely supercompact and having both its strong compactness and degree of supercompactness fully Laver indestructible.

An infinite field with only countably many pure types is algebraically closed.

The context for this paper is Feferman's theory of explicit mathematics, a formal framework serving many purposes. It is suitable for representing Bishop-style constructive mathematics as well as generalized recursion, including direct expression of structural concepts which admit self-application. The object of investigation here is the theory of explicit mathematics augmented by the monotone fixed point principle, which asserts that any monotone operation on classifications (Feferman's notion of set) possesses a least fixed point. To be more precise, the new axiom not merely postulates the existence of a least solution, but, by adjoining a new functional constant to the language, it is ensured that a fixed point is uniformly presentable as a function of the monotone operation.

The upshot of the paper is that the latter extension of explicit mathematics (when based on classical logic) embodies considerable proof-theoretic strength. It is shown that it has at least the strength of the subsystem of second order arithmetic based on comprehension.

We present a full completeness theorem for the multiplicative fragment of a variant of noncommutative linear logic, Yetter's cyclic linear logic (CyLL). The semantics is obtained by interpreting proofs as dinatural transformations on a category of topological vector spaces, these transformations being equivariant under certain actions of a noncocommutative Hopf algebra called the shuffle algebra Multiplicative sequents are assigned a vector space of such dinaturals, and we show that this space has as a basis the denotations of cut-free proofs in CyLL + MIX. This can be viewed as a fully faithful representation of a free *-autonomous category, canonically enriched over vector spaces.

This paper is a natural extension of the authors' previous work, “Linear Läuchli Semantics”, where a similar theorem is obtained for the commutative logic MLL + MIX. In that paper, we interpret proofs as dinaturals which are invariant under certain actions of the additive group of integers. Here we also present a simplification of that work by showing that the invariance criterion is actually a consequence of dinaturality. The passage from groups to Hopf algebras in this paper corresponds to the passage from commutative to noncommutative logic. However, in our noncommutative setting, one must still keep the invariance condition on dinaturals.

We examine two-cardinal problems for the class of O-minimal theories. We prove that an O-minimal theory which admits some (κ, λ) must admit every (κ′, λ′). We also prove that every “reasonable” variant of Chang's Conjecture is true for O-minimal structures. Finally, we generalize these results from the two-cardinal case to the δ-cardinal case for arbitrary ordinals δ.

In this paper we will present a simple condition for an ideal to be nowhere precipitous. Through this condition we show nowhere precipitousness of fundamental ideals on Pkλ, in particular the non-stationary ideal NSkλ under cardinal arithmetic assumptions.

In this section I denotes a non-principal ideal on an infinite set A. Let I+ = PA / I (ordered by inclusion as a forcing notion) and I∣X = {Y ⊂ A: Y ⋂ X ∈ I}, which is also an ideal on A for X ∈ I+. We refer the reader to [8, Section 35] for the general theory of generic ultrapowers associated with an ideal. We recall I is said to be precipitous if ⊨I+ “Ult(V, Ġ) is well-founded” [9].

The central notion of this paper is a strong negation of precipitousness [1]:

Definition. I is nowhere precipitous if I∣X is not precipitous for every X ∈ I+ i.e., ⊨I+ “Ult(V, Ġ) is ill-founded.”

It is useful to characterize nowhere precipitousness in terms of infinite games (see [11, Section 27]). Consider the following game G(I) between two players, Nonempty and Empty [5]. Nonempty and Empty alternately choose Xn ∈ I+ and Yn ∈ I+ respectively so that Xn ⊃ Yn ⊃n+1. After ω moves, Empty wins the game if⋂n<ωXn=⋂n<ωYn = Ø.

See [5, Theorem 2] for a proof of the following characterization.

Proposition. I is nowhere precipitous if and only if Empty has a winning strategy in G(I).

This paper explores algebraic aspects of two modifications of the usual account of first-order quantifiers.

Standard first-order quantificational logic is modelled algebraically by cylindric algebras. Prime examples of these are algebras whose members are sets of sequences: given a first-order model U for a language that is based on the set {υκ: κ < α} of variables, each formula φ is represented by the set

of all those α-length sequences x = 〈xκ: κ < α〉 that satisfy φ in U. Such a sequence provides a value-assignment to the variables (υκ is assigned value xκ), but it may also be viewed geometrically as a point in the α-dimensional Cartesian spaceαU of all α-length sequences whose terms come from the underlying set U of U. Then existential quantification is represented by the operation of cylindrification. To explain this, define a binary relation Tκ on sequences by putting xTκy if and only if x and y differ at most at their κth coordinate, i.e.,

Then for any set X ⊆ αU, the set

is the “cylinder” generated by translation of X parallel to the κth coordinate axis in αU. Given the standard semantics for the existential quantifier ∃υκ as

it is evident that

It is proved consistent that there be a proper σ-ideal on ω1 and an ℵ1-preserving poset ℙ such that ℙ ⊩ the σ-ideal generated by is not proper.

Starting with D. Scott's work on the mathematical foundations of programming language semantics, interest in topology has grown up in theoretical computer science, under the slogan ‘open sets are semidecidable properties’. But whereas on effectively given Scott domains all such properties are also open, this is no longer true in general. In this paper a characterization of effectively given topological spaces is presented that says which semidecidable sets are open.

This result has important consequences. Not only follows the classical Rice-Shapiro Theorem and its generalization to effectively given Scott domains, but also a recursion theoretic characterization of the canonical topology of effectively given metric spaces. Moreover, it implies some well known theorems on the effective continuity of effective operators such as P. Young and the author's general result which in its turn entails the theorems by Myhill-Shepherdson, Kreisel-Lacombe-Shoenfield and Ceĭtin-Moschovakis, and a result by Eršov and Berger which says that the hereditarily effective operations coincide with the hereditarily effective total continuous functionals on the natural numbers.

The paper describes and proves completeness theorems for a series of proof systems formalizing common sense reasoning about uncertain knowledge in the case where this consists of sets of linear constraints on a probability function.

After small forcing, any <κ-closed forcing will destroy the supercompactness and even the strong compactness of κ.

We shall prove quantifier elimination theorems for neocompact formulas, which define neocompact sets and are built from atomic formulas using finite disjunctions, infinite conjunctions, existential quantifiers, and bounded universal quantifiers. The neocompact sets were first introduced to provide an easy alternative to nonstandard methods of proving existence theorems in probability theory, where they behave like compact sets. The quantifier elimination theorems in this paper can be applied in a general setting to show that the family of neocompact sets is countably compact. To provide the necessary setting we introduce the notion of a law structure. This notion was motivated by the probability law of a random variable. However, in this paper we discuss a variety of model theoretic examples of the notion in the light of our quantifier elimination results.

Let P be any one of the following combinatorial properties: weak P-pointness, weak (semi-) Q-pointness, weak (semi-)selectivity, ω-closedness. We deal with the following two questions: (1) What is the least cardinal k such that there exists an ideal with k many generators that does not have the property P? (2) Can one extend every ideal with the property P to a prime ideal with the property P?

A definition of nonstandard universe which gets over the limitation to the finite levels of the cumulative hierarchy is proposed. Though necessarily nonwellfounded, nonstandard universes are arranged in strata in the likeness of superstructures and allow a rank function taking linearly ordered values. Nonstandard universes are also constructed which model the whole ZFC theory without regularity and satisfy the κ-saturation property.

A version of Herbrand's theorem tells us that a universal sentence of a first-order language with at least one constant is satisfiable if and only if the conjunction of all its ground instances is. In general the set of such instances is infinite, and arbitrarily large finite subsets may have to be inspected in order to detect inconsistency. Essentially, the reason that every member of such an infinite set may potentially matter, can be traced back to sentences like

(1)

Loosely put, such sentences effectively sabotage any attempt to build a model from below in a finite number of steps, since new members of the Herbrand universe are constantly brought to attention. Since they cause an indefinite expansion of the relevant part of the Herbrand universe, such sentences could quite appropriately be called expanding.

When such sentences are banned, stronger versions of Herbrand's theorem can be stated. Define a clause (disjunction of literals) to be non-expanding if every non-ground term occurring in a positive literal also occurs (possibly as an embedded subterm) in a negative literal of the same clause. Written as a disjunction of literals, the matrix of (1) clearly fails this criterion. Moreover, say that a sentence is non-expanding if it is a universal sentence with a quantifier-free matrix that is a conjunction of non-expanding clauses. Such sentences do in a sense never reach out beyond themselves, and the relevant part of the Herbrand universe is therefore drastically reduced.