There are shown to be many ε-Hermitian spaces, and an isometry criterion is stated which holds under MAℵ1 and is false under 2ℵ0< 2ℵ1.

Suppose that I is a precipitous ideal over a cardinal κ and j is a generic embedding of I. What is the nature of j? If we assume the existence of a supercompact cardinal then, by Foreman, Magidor and Shelah [FMS], it is quite unclear where some of such j's are coming from. On the other hand, if ¬∃κ0(κ) = κ++, then, by Mitchell [Mi], the restriction of j to the core model is its iterated ultrapower by measures of it. A natural question arising here is if each iterated ultrapower of can be obtained as the restriction of a generic embedding of a precipitous ideal. Notice that there are obvious limitations. Thus the ultrapower of by a measure over λ cannot be obtained as a generic embedding by a precipitous ideal over κ ≠ λ. But if we fix κ and use iterated ultrapowers of which are based on κ, then the answer is positive. Namely a stronger statement is true:

Theorem. Let τ be an ordinal and κ a measurable cardinal. There exists a generic extension V* of V so that NSℵ1 (the nonstationary ideal on ℵ1) is precipitous and, for every iterated ultrapower i of V of length ≤ τ by measures of V based on κ, there exists a stationary set forcing “the generic ultrapower restricted to V is i”.

Our aim will be to prove this theorem. We assume that the reader is familiar with the paper [JMMiP] by Jech, Magidor, Mitchell and Prikry. We shall use the method of that paper for constructing precipitous ideals. Ideas of Levinski [L] for blowing up 2ℵ1 preserving precipitousness and of our own earlier paper [Gi] for linking together indiscernibles will be used also.

We discuss saturating ultrafilters on N, relating them to other types of non-principal ultrafilter. (a) There is an (ω, c)-saturating ultrafllter on N iff 2λ ≤ c for every λ < c and there is no cover of R by fewer than c nowhere dense sets, (b) Assume Martin's axiom. Then, for any cardinal κ, a nonprincipal ultrafllter on N is (ω, κ)-saturating iff it is almost κ-good. In particular, (i) p(κ)-point ultrafilters are (ω, κ)-saturating, and (ii) the set of (ω, κ)-saturating ultrafilters is invariant under homeomorphisms of βN/N. (c) It is relatively consistent with ZFC to suppose that there is a Ramsey p(c)-point ultrafilter which is not (ω, c)-saturating.

We prove that the following are consistent with ZFC:

1. 2ω = ℵω1 + #x039A;c = ℵω1 + ΚΒ = ΚU = ω2 (for measure and category simultaneously).

2. .

This concludes the discussion about the cofinality of Κc.

The axiom of symmetry (Aℵ0) asserts that for every function F: ω2 → ω2 there is a pair of reals x and y in ω2 so that y is not in the countable set {(F(x))n: n< ω} coded by F(x) and x is not in the set coded by F(y). A(Γ) denotes axiom Aℵ0 with the restriction that graph(F) belongs to the pointclass Γ. In §2 we prove A(). In §3 we show A(), A() and ω2 ⊈ L are equivalent. In §4 several effective versions of A(REC) are examined.

The problem of characterizing the finite lattices which can be embedded into the recursively enumerable degrees has a long history, which is summarized in [AL]. This problem is an important one, as its solution is necessary if a decision procedure for the ∀∃-theory of the poset of recursively emumerable degrees is to be found. A recursive nonembeddability condition, NEC, which subsumes all known nonembeddability conditions was presented in [AL]. This paper focuses on embeddability. An embeddability condition, EC, is introduced, and we prove that every finite lattice having EC can be embedded (as a lattice) into . EC subsumes all known embeddability conditions.

EC is a Π3 condition which states that certain obstructions to proving embeddability do not exist. It seems likely that the recursive labeled trees used in EC can be replaced with trees which are effectively generated from uniformly defined finite trees, in which case EC would be equivalent to a recursive condition. We do not know whether EC and NEC are complementary. This problem seems to be combinatorial, rather than recursion-theoretic in nature. Our efforts to find a finite lattice satisfying neither EC nor NEC have, to this point, been unsuccessful. It is the second author's conjecture that the techniques for proving embeddability which are used in this paper cannot be refined very much to obtain new embeddability results.

EC is introduced in §2, and the various conditions and definitions are motivated by presenting examples of embeddable lattices and indicating how the embedding proof works in those particular cases. The embedding construction is presented in §3, and the proof in §4.

If S is a finite set, let ∣S∣ be the cardinality of S. We show that if m ∈ ω, A ⊆ ω, B ⊆ ω, and ∣i: ≤ i ≤ 2m & xi ∈ A}∣ can be computed by an algorithm which, for all x1, …, x2m, makes at most m queries to B, then A is recursive in the halting set K. If m = 1, we show that A is recursive.

We prove the following theorem. Let G be a connected simple bad group (i.e. of finite Morley rank, nonsolvable and with all the Borel subgroups nilpotent) of minimal Morley rank. Then the Borel subgroups of G are conjugate to each other, and if B is a Borel subgroup of G, then , NG(B) = B, and G has no involutions.

If κ is a measurable cardinal, let us say that a measure on κ is a κ-complete nonprincipal ultrafilter on κ. If U is a measure on κ, let jU be the canonical elementary embedding of V into its Ultrapower UltU(V). If x is a set, say that U moves x when jU(x) ≠ x; say that κ moves x when some measure on κ moves x. Recall Kunen's lemma (see [K]): “Every ordinal is moved only by finitely many measurable cardinals.” Kunen's proof (see [K]) and Fleissner's proof (see [KM, III, §10]) are essentially nonconstructive.

The following proposition can be proved by using elementary facts about iterated ultrapowers.

Proposition. Let ‹Un: n ∈ ω› be a sequence of measures on a strictly increasing sequence ‹κn: n ∈ ω› of measurable cardinals. Let U = ‹ Wα: α < ω2›, where Wωm + n = Um (m, n ∈ ω). Then, for each θ in UltU(V), if E is the (minimal) support of θ in UltU(V), then, for all m ∈ ω, Um moves θ iff E ∩ [ωm, ω(m + 1)) ≠ ∅.

Kirby and Paris have exhibited combinatorial algorithms whose computations always terminate, but for which termination is not provable in elementary arithmetic. However, termination of these computations can be proved by adding an axiom first introduced by Goodstein in 1944. Our purpose is to investigate this axiom of Goodstein, and some of its variants, and to show that these are potentially adequate to prove termination of computations of a wide class of algorithms. We prove that many variations of Goodstein's axiom are equivalent, over elementary arithmetic, and contrast these results with those recently obtained for Kruskal's theorem.

Equational logic for total functions is a remarkable fragment of first-order logic. Rich enough to lend itself to many uses, it is also quite austere. The only predicate symbol is one for a notion of equality, and there are no logical connectives. Proof theory for equational logic therefore is different from proof theory for other logics and, in some respects, more transparent. The question therefore arises to what extent a logic with a similar proof theory can be constructed when expressive power is increased.

The increase mainly studied here allows one both to consider arbitrary partial functions and to express the condition that a function be total. A further increase taken into account is equivalent to a change to universal Horn sentences for partial and for total functions.

Two ways of increasing expressive power will be considered. In both cases, the notion of equality is modified and nonlogical function symbols are interpreted as ranging over partial functions, instead of ranging only over total functions. In one case, the only further change is the addition of symbols that denote logical functions, such as the binary projection function Ae that maps each pair ‹a0, a1› of elements of a set A into the element a0. An addition of this kind results in a language, and also in a system of logic based on this language, which we call equational In the other case, instead of adding a symbol for Ae, one admits those special universal Horn sentences in which the conditions expressed by the antecedent are, in a sense, pure conditions of existence. Languages and systems of logic that result from a change of this kind will be called near-equational. According to whether the number of existence conditions that one may express in antecedents is finite or arbitrary, the resulting language and logic shall be finite or infinitary, respectively. Each of our finite near-equational languages turns out to be equivalent to one of our equational languages, and vice versa.

We prove that the cofinality of the smallest covering of R by meager sets is bigger than the additivity of measure.

In this paper we generalize the pigeonhole principle by using isols as our fundamental counting tool.

Let M be a saturated model of a superstable theory and let G = Aut(M). We study subgroups H of G which contain G(A), A the algebraic closure of a finite set, generalizing results of Lascar [L] as well as giving an alternative characterization of the simple superstable theories of [P]. We also make some observations about good, locally modular regular types p in the context of p-simple types.

We prove two results about the embeddability relation between Borel linear orders: For η a countable ordinal, let 2η (resp. 2< η) be the set of sequences of zeros and ones of length η (resp. < η), equipped with the lexicographic ordering. Given a Borel linear order X and a countable ordinal ξ, we prove the following two facts.

(a) Either X can be embedded (in a (X, ξ) way) in 2ωξ or 2ωξ + 1 continuously embeds in X.

(b) Either X can embedded (in a (X, ξ) way) in 2<ωξ or 2ωξ continuously embeds in X. These results extend previous work of Harrington, Shelah and Marker.

Let I(μ, Κ) denote the number of nonisomorphic models of power μ and IE(μ, Κ) the number of nonmutually embeddable models. We define in this paper the notion of a diverse class and use it to prove a number of results. The major result is Theorem B: For any diverse class Κ and μ greater than the cardinality of the language of Κ,

From it we deduce both an old result of Shelah, Theorem C: If T is countable and λ0 > ℵ0 then for every μ > ℵ0, IE(μ, T) ≥ min(2μ, ⊐2), and an extension of that result to uncountable languages, Theorem D: If ∣T∣ < 2ωλ0 > ∣T∣, and ∣D(T)∣ = ∣T∣ then for μ > ∣T∣,

We introduce a reducibility preordering between classes of countable structures, each class containing only structures of a given similarity type (which is allowed to vary from class to class). Though we sometimes work in a slightly larger context, we are principally concerned with the case where each class is an invariant Borel class (i.e. the class of all models, with underlying set = ω, of an Lω1ω sentence; from this point of view, the reducibility can be thought of as a (rather weak) sort of Lω1ω-interpretability notion). We prove a number of general results about this notion, but our main thrust is to situate various mathematically natural classes with respect to the preordering, most notably classes of algebraic structures such as groups and fields.

The results in this paper were motivated by the following question of Sacks. Suppose T is a recursive theory with countably many countable models. What can you say about the least ordinal α such that all models of T have Scott rank below α? If Martin's conjecture is true for T then α ≤ ω · 2.

Our goal was to look at this problem in a more abstract setting. Let E be a equivalence relation on ωω with countably many classes each of which is Borel. What can you say about the least α such that each equivalence class is ? This problem is closely related to the following question. Suppose X ⊆ ωω is and Borel. What can you say about the least α such that X is ?

In §1 we answer these questions in ZFC. In §2 we give more informative answers under the added assumptions V = L or -determinacy. The final section contains related results on the separation of sets by Borel sets.

Our notation is standard. The reader may consult Moschovakis [5] for undefined terms.

Some of these results were proved first by Sami and rediscovered by Kechris and Marker.

Recall that a theory is said to be almost strongly minimal if in every model every element is in the algebraic closure of a strongly minimal set. In 1970 Hodges and Macintyre conjectured that there is a natural number n such that every ℵ0-categorical almost strongly minimal theory is Σn axiomatizable. Recently Ahlbrandt and Baldwin [A-B] proved that if T is ℵ0-categorical and almost strongly minimal, then T is Σn axiomatizable for some n. This result also follows from Ahlbrandt and Ziegler's results on quasifinite axiomatizability [A-Z]. In this paper we will refute Hodges and Macintyre's conjecture by showing that for each n there is an ℵ0-categorical almost strongly minimal theory which is not Σn axiomatizable.

Before we begin we should note that in all these examples the complexity of the theory arises from the complexity of the definition of the strongly minimal set. It is still open whether the conjecture is true if we allow a predicate symbol for the strongly minimal set.

We will prove the following result.

Theorem. For every n there is an almost strongly minimal ℵ0-categorical theory T with models M and N such that N is Σn elementary but not Σn + 1 elementary.

To show that these theories yield counterexamples to the conjecture we apply the following result of Chang [C].

Theorem. If T is a Σn axiomatizable theory categorical in some infinite power, M and N are models of T and N is a Σn elementary extension of M, then N is an elementary extension of M.