The universe of sets, V, is usually seen as an entity structured in successive levels, each level being made up of objects and collections of objects belonging to the previous levels. This process of obtaining sets and axioms for set theory can be seen in Scott [74] and Shoenfield [77].

The approach we want to take differs from the previous one very strongly: the seeds from which we want to generate our universe of classes are to be the one-variable predicates (given by one-free-variable formulas) of the formal language we shall be using. In other words, any one-variable predicate of the language is to be represented as a class in our universe. In this sense, we view our theory as being about a self-referential language, a language whose predicates refer to objects which are predicates of the language itself.

We want, in short, a system such that: (i) any predicate may be represented by an object to be studied by the theory itself; (ii) the axioms for the theory may be derived from the general principle that we are dealing with a language that aims at describing its own predicates; and (iii) the theory should be strong enough to derive ZFC and suggest answers to the existence of large cardinals and to the continuum hypothesis.

An objection to such a project arises immediately: in view of the Russell-Zermelo paradox, how is it possible to have all predicates of the language as elements of the universe? This objection will be easy to deal with: we shall provide our language with a type structure to avoid paradox.

We show that the lattice L20 is not embeddable into the lattice of ideals of computably enumerable Turing degrees (ℐ), We define a structure called a pseudolattice that generalizes the notion of a lattice, and show that there is a Π2 necessary and sufficient condition for embedding a finite pseudolattice into ℐ.

According to McKinsey and Tarski [3], who introduced the notion, a closure algebra Γ = (K, ∪, ∩, –, C) is a Boolean algebra (K, ∪, ∩, –) with an additional unary operator C, closed in K, such that, for every x, y ϵ K, x ⊆ Cx, CCx = Cx, C(x∪y) = Cx∪Cy, CΛ = Λ. McKinsey [2] has proved that a matrix Γ = (K, D, ∪, ∩, –, C) is a representation (normal matrix) of S4, if and only if (1) (K, ∪, ∩, –, C) is a closure algebra, (2) D is an additive ideal of K, (3) if – Cx ϵ D then x = Λ, D being the distinguished proper subset of K and ∪, ∩, –, C having been made to correspond to ∨, ·, ∼, ◊, respectively. Following McKinsey, one proves immediately

Theorem 1. A matrix Γ is a representation of S5, if and only if it is a closure algebra and

(α) for every x ϵ K, –C–Cx = Cx, and

(β) D is an additive ideal of K such that, for every x ϵ D, Cx = V.

This raises the question of alternative characterizations of S5. Because of (β), one may without substantial loss of generality put D = {V}. The question then becomes one for conditions that are in closure algebras equivalent to (α). One such characterization has been given by McKinsey and Tarski [4]. Defining x∸y = C(x∩–y) and ¬x = V∸x, these authors have shown that the closed elements K* of a closure algebra Γ form a Brouwerian algebra Γ* = (K*, ∪, ∩, ∸), and furthermore that Γ* is a Boolean algebra under ∪, ∩, ¬, if and only if, for every x ϵ K*,

We prove that it is consistent that $\aleph _\omega $ is strong limit, $2^{\aleph _\omega } $ is large and the universality number for graphs on $\aleph _{\omega + 1} $ is small. The proof uses Prikry forcing with interleaved collapsing.

We prove that if a strongly minimal nonlocally modular reduct of an algebraically closed valued field of characteristic 0 contains +, then this reduct is bi-interpretable with the underlying field.

We further develop a previously introduced method of constructing forcing notions with the help of morasses. There are two new results: (1) If there is a simplified (ω1, 1)-morass, then there exists a ccc forcing of size ω1 that adds an ω2-Suslin tree. (2) If there is a simplified (ω1, 2)-morass, then there exists a ccc forcing of size ω1 that adds a 0-dimensional Hausdorff topology τ on ω3 which has spread s(τ) = ω1. While (2) is the main result of the paper, (1) is only an improvement of a previous result, which is based on a simple observation. Both forcings preserve GCH. To show that the method can be changed to produce models where CH fails, we give an alternative construction of Koszmider's model in which there is a chain 〈Xα ∣ α < ω2〉 such that Xα ⊆ ω1. Xβ – Xα is finite and Xα – Xβ has size ω1 for all β < α < ω2.

In [1], William H. Jobe has shown that a certain three-valued proper logic is canonically complete, i.e., each first order J is expressible by a formula that is in M-normal form. The first order operator Jk(p)1 represents the truth table of a possible formula which has the truth value k in the ith row and the truth value 1 in all other rows. “The M-normal form of a given formula is another formula which has the same truth table as the given formula but which contains no operation symbols other than those of conjunction, disjunction, and undefined unary operations, with no nonunary operation, i.e., conjunction or alternation, included in the scope of an operation which is unary, and no alternation included in the scope of a conjunction.” Here it is shown that it is possible to construct a canonically complete proper logic for each choice of M which contains, in addition to alternation and conjunction, exactly two unary operations. Moreover, it is further shown that at least two unary operations are necessary in order for such a logic to be canonically complete.

Let a be an admissible ordinal and let ∧ ≤ α be a limit ordinal. A sequence of a-r.e. degrees is said to be ascending, simultaneous and of length ∧ if (i) there is an α-recursive function t: α × ∧ → α such that, for all ϒ < ∧, A ϒ = {t(σ, ϒ)∣ σ < α} is of degree a ϒ; (ii) if ϒ < ⊤ < ∧, then a ϒ ≤α a τ and (iii) for all ϒ < ∧, there is a ⊤ > ϒ with a ϒ, >α a ϒ. Lerman [4] showed that such an exists for every ∧ ≤ α. An upper bound a of is an α-r.e. degree in which every element of is α-recursive. a is minimal if there is no α-r.e. degree b <α a which is also an upper bound of . Sacks [6] proved that every ascending sequence of simultaneously ω-r.e. degrees of length ω cannot have 0ω′, the complete ω-r.e. degree, as a minimal upper bound. In contrast, Cooper [2] showed that there exists an ascending sequence of simultaneously ω-r.e. degrees of length to having a minimal upper bound which is an ω-r.e. degree. In this paper we investigate the behavior of ascending sequences of simultaneously α-r.e. degrees for admissible ordinals α > ω. Call α Σ∞-admissibIe if it is Σ n -nadmissible for all n. Let Φ(∧) say: No ascending sequence of simultaneously α-r.e. degrees of length ∧ can have 0α′, the complete α-r.e. degree, as a minimal upper bound. Our main result in this paper is:

Let α be either a constructible cardinal with σ2ci(α) < α or Σ∞-admissible. Then σ2cf(α) is the least ordinal ν for which every ∧ ≤ α of cofinality ν (over L α) can satisfy Φ(∧).

In [1, p. 51] A. V. Arhangel'skiĭ, in connection with the problems of L-spaces and S-spaces, examined further the notions of hereditary separability and hereditary Lindelöfness. In particular he considered the following property P: “Every regular topological space has a countable net weight provided its countable product is hereditarily Lindelöf and hereditarily separable.” He noticed that the continuum hypothesis implies negation of the property P and posed a question: “Do Martin's Axiom and the negation of the continuum hypothesis imply P?” The purpose of this paper is to give a negative answer to this question.

The set-theoretical and topological notation that we use is standard and can be found in [6] and [5] respectively.

Throughout the paper we will use the notation H(X, Y) to denote the set of all finite functions from a set X to Y.

Theorem. Con(ZFC) → Con(ZFC + MA + ¬CH + there exists a 0-dimensional Hausdorff space X such that nw(X) = с and nw(Y) = ω for any Y ϵ [X]<с).

Proof. Let M be a model of ZFC satisfying CH and let F be an M-generic filter over the Cohen forcing {H(ω2 × ω2, 2), ⊃). Then f = ⋃F is a function and f: ω2 × ω2 → 2.

This paper is concerned with the combinator representation of numeral systems with logarithmic space complexity of symbols. The principle used is based on the lexicographic ordering of words over a finite alphabet.

In [S, pp. 77–88], Smullyan introduced the class of rudimentary relations, and showed that they form a basis for the recursively enumerable sets. He also asked [S, p. 81] if the addition and multiplication relations were rudimentary. In this note we answer one of these questions by showing that the addition relation is rudimentary. This result was communicated to Smullyan orally in 1960 and is announced in [S, p. 81, footnote 1]. However, the proof has not yet appeared in print. (Shortly after the publication of [S], James H. Bennett, using much more subtle methods than those of this note, showed that the multiplication relation is also rudimentary. That result appears in his doctoral dissertation [B], and is being prepared for publication.)

Let us begin by reviewing Smullyan's definition [S, p. 10] of dyadic notation for the positive integers. Each positive integer a is identified with the unique string anan−1…a1a0 of 1's and 2's such that a = Σin=0ai2i Because of this identification, we are able to speak of typographical properties of numbers.