Aczel's theory of hypersets provides an interesting alternative to the standard view of sets as inductively constructed, well-founded objects, thus providing a convienent formalism in which to consider non-well-founded versions of classically well-founded constructions, such as the “circular logic” of [3], This theory and ZFC are mutually interpretable; in particular, any model of ZFC has a canonical “extension” to a non-well-founded universe. The construction of this model does not immediately generalize to weaker set theories such as the theory of admissible sets. In this paper, we formulate a version of Aczel's antifoundation axiom suitable for the theory of admissible sets. We investigate the properties of models of the axiom system KPU−, that is, KPU with foundation replaced by an appropriate strengthening of the extensionality axiom. Finally, we forge connections between “non-wellfounded sets over the admissible set A” and the fragment LA of the modal language L∞.

One of the early partition relation theorems which include ordinals was the observation of Erdös and Rado [7] that if κ = cf(κ) > ω then the Dushnik–Miller theorem can be sharpened to κ→(κ, ω + 1)2. The question on the possible further extension of this result was answered by Hajnal who in [8] proved that the continuum hypothesis implies ω1 ↛ (ω1, ω + 2)2. He actually proved the stronger result ω1 ↛ (ω: 2))2. The consistency of the relation κ↛(κ, (ω: 2))2 was later extensively studied. Baumgartner [1] proved it for every κ which is the successor of a regular cardinal. Laver [9] showed that if κ is Mahlo there is a forcing notion which adds a witness for κ↛ (κ, (ω: 2))2 and preserves Mahloness, ω-Mahloness of κ, etc. We notice in connection with these results that λ→(λ, (ω: 2))2 holds if λ is singular, in fact λ→(λ, (μ: n))2 for n < ω, μ < λ (Theorem 4).

In [11] Todorčević proved that if cf(λ) > ω then a ccc forcing can add a counter-example to λ→(λ, ω + 2)2. We give an alternative proof of this (Theorem 5) and extend it to larger cardinals: if GCH holds, cf (λ) > κ = cf (κ) then < κ-closed, κ+-c.c. forcing adds a counter-example to λ→(λ, κ + 2)2 (Theorem 6).

Erdös and Hajnal remarked in their problem paper [5] that Galvin had proved ω2→(ω1, ω + 2)2 and he had also asked if ω2→(ω1, ω + 3)2 is true. We show in Theorem 1 that the negative relation is consistent.

Let κℝ be the least ordinal κ such that Lκ (ℝ) is admissible. Let A = {x ϵ ℝ ∣ (∃α < κℝ) such that x is ordinal definable in Lα (ℝ)}. It is well known that (assuming determinacy) A is the largest countable inductive set of reals. Let T be the theory: ZFC − Replacement + “There exists ω Woodin cardinals which are cofinal in the ordinals.” T has consistency strength weaker than that of the theory ZFC + “There exists ω Woodin cardinals”, but stronger than that of the theory ZFC + “There exists n Woodin Cardinals”, for each n ϵ ω. Let M be the canonical, minimal inner model for the theory T. In this paper we show that A = ℝ ∩ M. Since M is a mouse, we say that A is a mouse set. As an application, we use our characterization of A to give an inner-model-theoretic proof of a theorem of Martin which states that for all n, every real is in A.

In classical recursion theory, the jump operator is an important concept fundamental in the study of degrees. In particular, it provides a way of defining the hyperarithmetic hierarchy by iterating over Kleene's O. In subrecursion theories, hierarchies (variants of the fast growing hierarchy) are also important in underlying the central concepts, e.g. in classifying provably recursive functions and associated independence results for given theories (see, e.g. [BW87], [HW96], [R84] and [Z77]). A major difference with the hyperarithmetic hierarchy is in the way each level of a subrecursive hierarchy is crucially dependent upon the system of ordinal notations used (see [F62]). This has been perhaps the major stumbling block in finding a classification of all general recursive functions using such hierarchies.

Here we present a natural subrecursive analogue of the jump operator and prove that the hierarchy based on the ”subrecursive jump” is elementarily equivalent to the fast growing hierarchy.

The paper is organised as follows. First the preliminary definitions are given together with a statement of the main theorem and a brief outline of its proof. The proof of the theorem is then given, with the more technical parts separated out as facts which are proven afterwards.

We let {e}g(x) denote computation of the e-th partial recursion with oracle g, on input x. Furthermore [e] denotes the e-th elementary recursive function, defined so that

Similarly, for a given oracle g the e-th relativized elementary function is denoted by [e]g.

Two countable filters on ω are incompatible if they have no common infinite pseudointersection. Letting (Pf) denote the minimal size of a maximal uncountable family of pairwise incompatible countable filters on ω, we prove the consistency of t < a(Pf).

We present a cut-elimination proof for simple type theory with an axiom of choice formulated in the language with an epsilon-symbol. The proof is modeled after Takahashi's proof of cut-elimination for simple type theory with extensionality. The same proof works when types are restricted, for example for second-order classical logic with an axiom of choice.

By Frege's Theorem is meant the result, implicit in Frege's Grundlagen, that, for any set E, if there exists a map υ from the power set of E to E satisfying the condition

then E has a subset which is the domain of a model of Peano's axioms for the natural numbers. (This result is proved explicitly, using classical reasoning, in Section 3 of [1].) My purpose in this note is to strengthen this result in two directions: first, the premise will be weakened so as to require only that the map υ be defined on the family of (Kuratowski) finite subsets of the set E, and secondly, the argument will be constructive, i.e., will involve no use of the law of excluded middle. To be precise, we will prove, in constructive (or intuitionistic) set theory, the following

Theorem. Let υ be a map with domain a family of subsets of a set E to E satisfying the following conditions:

(i) ø ϵdom(υ)

(ii)∀U ϵdom(υ)∀x ϵ E − UU ∪ x ϵdom(υ)

(iii)∀UV ϵdom(5) υ(U) = υ(V) ⇔ U ≈ V.

Then we can define a subset N of E which is the domain of a model of Peano's axioms.

We study some generalized notions of cohesiveness which arise naturally in connection with effective versions of Ramsey's Theorem. An infinite set A of natural numbers is n-cohesive (respectively, n-r-cohesive) if A is almost homogeneous for every computably enumerable (respectively, computable) 2-coloring of the n-element sets of natural numbers. (Thus the 1-cohesive and 1-r-cohesive sets coincide with the cohesive and r-cohesive sets, respectively.) We consider the degrees of unsolvability and arithmetical definability levels of n-cohesive and n-r-cohesive sets. For example, we show that for all n ≥ 2, there exists a n-cohesive set. We improve this result for n = 2 by showing that there is a 2-cohesive set. We show that the n-cohesive and n-r-cohesive degrees together form a linear, non-collapsing hierarchy of degrees for n ≥ 2. In addition, for n ≥ 2 we characterize the jumps of n-cohesive degrees as exactly the degrees ≥ 0(n+1) and also characterize the jumps of the n-r-cohesive degrees.

This paper continues investigations of the monotone fixed point principle in the context of Feferman's explicit mathematics begun in [14]. Explicit mathematics is a versatile formal framework for representing Bishop-style constructive mathematics and generalized recursion theory. 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 constant to the language, it is ensured that a fixed point is uniformly presentable as a function of the monotone operation. Let T0 + UMID denote this extension of explicit mathematics. [14] gave lower bounds for the strength of two subtheories of To + UMID in relating them to fragments of second order arithmetic based on comprehension. [14] showed that To ↾ + UMID and To ↾ + INDk + UMID have at least the strength of ( – CA), and ↾ and – CA), respectively.

Here we are concerned with the exact reversals. Let UMIDN be the monotone fixed-point principle for subclassifications of the natural numbers. Among other results, it is shown that To ↾ + UMIDN and To ↾ + INDN + UMIDN have the same strength as ( – CA) ↾ and ( – CA), respectively.

The results are achieved by constructing set-theoretic models for the aforementioned systems of explicit mathematics in certain extensions of Kripke-Platek set theory and subsequently relating these set theories to subsystems of second arithmetic.

We prove that if I is a partially ordered set in a countable transitive model of ZFC then can be extended by a generic sequence of reals ai, i ∈ I, such that is preserved and every ai is Sacks generic over [〈aj: j < i〉]. The structure of the degrees of -constructibility of reals in the extension is investigated.

As applications of the methods involved, we define a cardinal invariant to distinguish product and iterated Sacks extensions, and give a short proof of a theorem (by Budinas) that in ω2-iterated Sacks extension of L the Burgess selection principle for analytic equivalence relations holds.

Tarski [19] proved the important theorem that the class of representable relation algebras is equationally axiomatizable. One of the key steps in his proof is showing that the class of (isomorphs of) simple set relation algebras—that is, algebras of binary relations with a unit of the form U × U for some non-empty set U —is universal, i.e., is axiomatizable by a set of universal sentences. In the same paper Tarski observed that the class of (isomorphs of) relation algebras constructed from groups (so-called group relation algebras) is also universal.

We shall abstract the essential ingredients of Tarski's method (in Corollary 2.4), and then combine them with some observations about atom structures, to establish (in Theorem 2.6) a rather general method for showing that certain classes of simple relation algebras—and, more generally, certain classes of simple algebras in a discriminator variety V—are universal, and consequently that the collections of (isomorphs of) subdirect products of algebras in such classes form subvarieties of V. As applications of the method we show that two well-known classes of simple relation algebras, those constructed from projective geometries (sometimes called Lyndon algebras) and those constructed from modular lattices with a zero (sometimes called Maddux algebras), are universal. In the process we prove that these two classes consist precisely of all (isomorphs of) complex algebras over the respective geometries and modular lattices, provided that we choose the primitive notions of the latter structures in an appropriate fashion. We also derive Tarski's theorems and a related theorem of the author as easy corollaries of Theorem 2.6.

We extend work of H. Friedman, L. Harrington and P. Welch to the third level of the projective hierarchy. Our main theorems say that (under appropriate background assumptions) the possibility to select definable elements of non-empty sets of reals at the third level of the projective hierarchy is equivalent to the disjunction of determinacy of games at the second level of the projective hierarchy and the existence of a core model (corresponding to this fragment of determinacy) which must then contain all real numbers. The proofs use Sacks forcing with perfect trees and core model techniques.

Modules over a ring R, when tensored with an (R, S)-bimodule, are converted to S-modules. Here I consider, from the standpoint of the model theory of modules, the effect of this operation. The primary motivation arises from questions concerning representation type of algebras and interpretability of modules, where such tensor functors play a key role, but this paper is devoted to more general considerations. For instance, the elementary duality of [2] and [1] is generalised here. It is also shown that, although tensor product does not preserve elementary equivalence, one can define the tensor product of two complete theories of R-modules. The results in Section 1 grew out of a number of discussions with T. Kucera.

This is our generic situation. We have an (R, S)-bimodule B and we are considering the functor –⊗RBS from Mod – R (the category of right R-modules) to Mod – S which is given on objects by MR ↦ (M ⊗RB)s and has the obvious action on morphisms. There is a somewhat more general situation: namely we may consider the effect of tensoring (T, R)-bimodules over R with B to obtain (T, S)-bimodules. For the intended applications this case is not needed. Moreover, although some results extend to this more general case, there are some which definitely do not (see the example after 2.1). Therefore we confine ourselves to the case first described. We will also consider B itself as a variable and so ask “What is the effect of tensoring R-modules with (R, S)-bimodules?”.

Let CR denote the first-order theory of commutative rings with unity, formulated in the language L = 〈 +, •, 0, 1〉. Virtually everything that is known about existentially complete (e.c.) models of CR is contained in Cherlin's paper [2], where it is shown, in particular, that the e.c. models are not first-order axiomatizable. The purpose of this note is to show that, in analogy with the case of fields, there exists a unique prime e.c. model of CR in each characteristic n > 2. As a consequence we settle Problem 8 in the list of open questions at the end of Hodges' book Building models by games ([5], p. 278).

By a “prime” e.c. model of characteristic n ≥ 2 we mean one that embeds in every e.c. model of characteristic n. (The embedding is not always elementary, since [2] not all e.c. models of characteristic n are elementarily equivalent.) The prime model is characterized by the fact that it is the union of a chain of finite subrings each of which is an amalgamation base for CR. In §1 we describe the finite amalgamation bases for CR and show that every finite model embeds in a finite amalgamation base; in §2 we use this information to obtain prime e.c. models and answer Hodges' question.

Our results on prime e.c. models were obtained some years ago, during the fall term of 1982, while the author was a visitor at Wesleyan University. The author wishes to take this opportunity to thank the mathematics department at Wesleyan for its hospitality during that visit, and subsequent ones.

In this paper we prove a strong nonstructure theorem for κ(T)-saturated models of a stable theory T with dop. This paper continues the work started in [1].

It will be proved that the Shoenfield cupping conjecture holds in R/M, the quotient of the recursively enumerable degrees modulo the cappable r.e. degrees. Namely, for any [a], [b] ∈ R/M such that [0] ≺ [b] ≺ [a] there exists [c] ∈ R/M such that [c] ≺ [a] and [a] = [b] ∨ [c].

We construct a decidable first-order theory T such that the theory of its finite models is undecidable. Moreover, T will be equationally axiomatizable and of finite type.

Saturation is (μ, κ)-transferable in T if and only if there is an expansion T1 of T with |T1| = |T| such that if M is a μ-saturated model of T1 and |M| ≥ κ then the reduct M|L(T) is κ-saturated. We characterize theories which are superstable without f.c.p., or without f.c.p. as, respectively those where saturation is (ℵ0, λ)-transferable or (κ(T), λ)-transferable for all λ. Further if for some μ ≥ |T|,2μ > μ+, stability is equivalent to for all μ ≥ |T|, saturation is (μ, 2μ)-transferable.

We show the faithful embedding of common knowledge logic CKL into game logic GL, that is, CKL is embedded into GL and GL is a conservative extension of the fragment obtained by this embedding. Then many results in GL are available in CKL, and vice versa. For example, an epistemic consideration of Nash equilibrium for a game with pure strategies in GL is carried over to CKL. Another important application is to obtain a Gentzen-style sequent calculus formulation of CKL and its cut-elimination. The faithful embedding theorem is proved for the KD4–type propositional CKL and GL, but it holds for some variants of them.

To any metric space it is possible to associate the cardinal invariant corresponding to the least number of rectifiable curves in the space whose union is not meagre. It is shown that this invariant can vary with the metric space considered, even when restricted to the class of convex subspaces of separable Banach spaces. As a corollary it is obtained that it is consistent with set theory that any set of reals of size ℵ1 is meagre yet there are ℵ1 rectifiable curves in ℝ3 whose union is not meagre. The consistency of this statement when the phrase “rectifiable curves” is replaced by “straight lines” remains open.