It was shown by Taiclin [6], and independently announced by Tarski [7], that the elementary theory of commutative cancellation semigroups is hereditarily undecidable. In his proof Tarski exhibited a subsemigroup of 〈N, ·〉, the natural numbers with multiplication, whose theory is both hereditarily and essentially undecidable. (The details of his construction were published by V. H. Dyson [1].) In connection with these results, Tarski suggested to the author that it would be of interest to solve the decision problem for the theory K which consists of all elementary sentences which are true in every subalgebra (i.e. every subsemigroup) of 〈N, +〉. The object of this note is to prove that the theory K is hereditarily undecidable.

A compact complex space X is viewed as a 1-st order structure by taking predicates for analytic subsets of X, X x X, … as basic relations. Let f: X → Y be a proper surjective holomorphic map between complex spaces and set Xy ≔ f−1(y). We show that the set

is analytically constructible, i.e.. is a definable set when X and Y are compact complex spaces and f: X → Y is a holomorphic map. The analogous result in the context of algebraic geometry gives rise to the definability of Morley degree.

Dual combinators emerge from the aim of assigning formulas containing ← as types to combinators. This paper investigates formally some of the properties of combinatory systems that include both combinators and dual combinators. Although the addition of dual combinators to a combinatory system does not affect the unique decomposition of terms, it turns out that some terms might be redexes in two ways (with a combinator as its head, and with a dual combinator as its head). We prove a general theorem stating that no dual combinatory system possesses the Church-Rosser property. Although the lack of confluence might be problematic in some cases, it is not a problem per se. In particular, we show that no damage is inflicted upon the structurally free logics, the system in which dual combinators first appeared.

E. Specker has proved that the axiom of choice (AC) is false in NF [6]. Since AC is stratified, one can, according to another famous result of Specker [7], prove directly ¬AC in type theory (TT) plus some finite set of ambiguity axioms, i.e. sentences of the form φ ↔ φ+, where φ+ results from φ by adding one to its type indices.

We shall in §2 of this paper give a disproof of AC directly in TT plus some axioms of ambiguity. The argument will be split into two parts. The first one (contained in Proposition 2) concerns cardinal arithmetic and has nothing to do with typical ambiguity. Though carried out in TT, it could have been done in other set theories such as Zermelo's Z or ZF. The second part is an application of this to the cardinals of the universes at different types. This is made possible through the introduction of an appropriate definition of 2α in §1 enabling one to express shifting sentences as “typed properties” of the universe, in Boffa's sense. The disproof of AC is then completed in TT plus two extra ambiguity axioms. In §3, we show that this is in a sense the best possible result: that means that every single ambiguity axiom is consistent with TT plus AC, thus giving a positive solution to a conjecture of Specker [7, p. 119].

We show that there exist downwards properly (in fact noncuppable) e-degrees that are not high. We also show that every high e-degree bounds a noncuppable e-degree.

We consider a well-known partial order of Prikry for producing a collapsing function of minimal degree. Assuming MA + ¬CH, every new real constructs the collapsing map.

It is shown that the low basis theorem is meaningful and provable in IΣ1 and that the priority-free solution to Post's problem formalizes in this theory.

Let S and T be countable complete theories. We assume that T is superstable without the dimensional order property, and S is interpretable in T in such a way that every model of S is coded in a model of T. We show that S does not have the dimensional order property, and we discuss the question of whether Depth(S) ≤ Depth(T). For Mekler's uniform interpretation of arbitrary theories S of finite similarity type into suitable theories TS of groups we show that Depth(S) ≤ Depth(TS) ≤ 1 + Depth(S).

In [1], Degtëv constructed a nonzero r.e. tt-degree containing a single r.e. m-degree. It is not difficult to construct an r.e. tt-degree containing infinitely many r.e. m-degrees (Fischer [6]); indeed, in [3], the author constructed an r.e. tt-degree with no greatest r.e. m-degree. Odifreddi [12, Problem 10] asked if every r.e. tt-degree contains either one or infinitely many r.e. m-degrees. The goal of this paper is to solve Odifreddi's question by showing:

Theorem. There exists a nonzero r.e. tt-degree containing exactly 3 r.e. m-degrees.

This theorem can be extended to show that there exist r.e. tt-degrees with arbitrarily large finite numbers of r.e. m-degrees.

We remark that save for the aforementioned results, very little is known about the structures that can be realized as the collection of r.e. m-degrees within an r.e. tt-degree. It seems conceivable that the methods of the present paper may be useful in, for example, embedding distributive (semi) lattices into such structures.

In part II of this paper [4], we continue our analysis of r.e. m- and tt-degrees. We define an r.e. tt-degree to be singular if it contains a single r.e. m-degree, and an r.e. T-degree a to be singular if a contains a singular r.e. tt-degree.

In [4] we study the distribution (in the r.e. T-degrees) of singular tt-degrees. We show that 0′T is singular (solving a question of Odifreddi [11]), and that the singular T-degrees are dense, but also we construct a nonsingular T-degree. The techniques used for the first results extend those of §2 of the present paper.

We say that a theory T is intermediate under effective reducibility if the isomorphism problems among its computable models is neither hyperarithmetic nor on top under effective reducibility. We prove that if an infinitary sentence T is uniformly effectively dense, a property we define in the paper, then no extension of it is intermediate, at least when relativized to every oracle in a cone. As an application we show that no infinitary sentence whose models are all linear orderings is intermediate under effective reducibility relative to every oracle in a cone.

We prove that the set of properties describable by a uniform sequence of first-order sentences using at most k + 1 distinct variables is exactly equal to the set of properties checkable by a Turing machine in DSPACE[nk] (where n is the size of the universe). This set is also equal to the set of properties describable using an iterative definition for a finite set of relations of arity k. This is a refinement of the theorem PSPACE = VAR[O[1]] [8]. We suggest some directions for exploiting this result to derive trade-offs between the number of variables and the quantifier depth in descriptive complexity.

In examples like the total recursive functions or the computable real numbers the canonical indexings are only partial maps. It is even impossible in these cases to find an equivalent total numbering. We consider effectively given topological T0-spaces and study the problem in which cases the canonical numberings of such spaces can be totalized, i.e., have an equivalent total indexing. Moreover, we show under very natural assumptions that such spaces can effectively and effectively homeomorphically be embedded into a totally indexed algebraic partial order that is closed under the operation of taking least upper bounds of enumerable directed subsets.

We show that either PFA+ or Martin's maximum implies Fleissner's Axiom R, a reflection principle for stationary subsets of P ℵ(λ). In fact, the “plus version” (for one term denoting a stationary set) of Martin's axiom for countably closed partial orders implies Axiom R.

This paper introduces, as an alternative to the (absolutely) lawless sequences of Kreisel and Troelstra, a notion of choice sequence lawless with respect to a given class of lawlike sequences. For countable , the class of -lawless sequences is comeager in the sense of Baire. If a particular well-ordered class of sequences, generated by iterating definability over the continuum, is countable then the -lawless sequences satisfy the axiom of open data and the continuity principle for functions from lawless to lawlike sequences, but fail to satisfy Troelstra's extension principle. Classical reasoning is used.

Let α be an admissible ordinal. In this paper we study the structure of the upper semilattice of α-recursively enumerable degrees. Various results about the structure which are of fundamental importance had been obtained during the past two years (Sacks-Simpson [7], Lerman [4], Shore [9]). In particular, the method of finite priority argument of Friedberg and Muchnik was successfully generalized in [7] to an α-finite priority argument to give a solution of Post's problem for all admissible ordinals. We refer the reader to [7] for background material, and we also follow closely the notations used there.

Whereas [7] and [4] study priority arguments in which the number of injuries inflicted on a proper initial segment of requirements can be effectively bounded (Lemma 2.3 of [7]), we tackle here priority arguments in which no such bounds exist. To this end, we focus our attention on the fine structure of Lα , much in the fashion of Jensen [2], and show that we can still use a priority argument on an indexing set of requirements just short enough to give us the necessary bounds we seek.

Assume that all algebras are atomless. (1) Spind(A × B) = Spind(A) ∪ Spind(B). (2) Spind(Ai). Now suppose that κ and λ are infinite cardinals, with κ uncountable and regular and with κ < λ. (3) There is an atomless Boolean algebra A such that u(A) = κ and i(A) = λ. (4) If λ is also regular, then there is an atomless Boolean algebra A such that t(A) = s(A) = κ and α (A) = λ. All results are in ZFC, and answer some problems posed in Monk [01] and Monk [∞].

A new method is presented for constructing models of set theory, using a technique of forming pseudo-ultrapowers. In the presence of the axiom of choice, the traditional ultrapower construction has proven to be extremely powerful in set theory and model theory; if the axiom of choice is not assumed, the fundamental theorem of ultrapowers may fail, causing the ultrapower to lose almost all of its utility. The pseudo-ultrapower is designed so that the fundamental theorem holds even if choice fails; this is arranged by means of an application of the omitting types theorem. The general theory of pseudo-ultrapowers is developed. Following that, we study supercompactness in the absence of choice, and we analyze pseudo-ultrapowers of models of the axiom of determinateness and various infinite exponent partition relations. Relationships between pseudo-ultrapowers and forcing are also discussed.

In this paper we consider the problem of lifting properties of the Fréchet ideal Iκ = {X ⊆ κ: ∣X∣ < κ} on a regular uncountable cardinal κ, to an analogue about Iκλ , the ideal of not unbounded subsets of Pκλ. With this in mind, in §1 we introduce and study the class of seminormal λ-generated ideals on Pκλ. We shall see that ideals belonging to this class display properties which are clearly analogous to those of the Fréchet ideal on κ (for instance, with regard to saturation, normality and weak selectivity) and yet are closely related to Iκλ . Our results here show that if λ<λ = λ, then many restrictions of Iκλ are weakly selective, nowhere precipitous and, quite suprisingly, seminormal (but nowhere normal). These latter two results suggest the question of whether any restriction of Iκλ can ever be normal. In §2 we prove that if κ is strongly inaccessible, λ<κ = 2 λ and NS κλ , the ideal of nonstationary subsets of Pκλ, has a mild selective property, then NS κλ ∣A = Iκλ ∣A for some stationary A ⊆ Pκλ.

In [1] Baumgartner showed that if κ is weakly compact and P is the collection of indescribable subsets of κ, then P → (P, κ)2. As a Pκλ analogue of indescribability, Carr (see [3]–[5]) introduced the λ-Shelah property, but was unable to derive the natural Pκλ analogue of Baumgartner's result, (where NSh κλ is the normal ideal on Pκλ induced by the λ-Shelah property). In §3 we show that the problem lies in the fact that, as far as we know, NSh κλ is not sufficiently distributive, and derive conditions which are sufficient and, in a sense, necessary to yield partitions related to .