In the paper we prove the following theorem:

Theorem. There is a model N of open induction in which the set of primes is bounded and N is such that its field of fractions 〈N*, +, ·, <〉 is elementarily equivalent to 〈Q, +, ·, <〉 (the standard rationals).

We fix an ω 1-saturated model 〈M, +, ·, <〉 of PA. Let 〈M*, +, ·, <〉 denote the field of fractions of M. The model N that we are looking for will be a substructure of 〈M*, +, ·, <〉.

If A ⊆ M* then let Ā denote the ring generated by A within M*, Ậ the real closure of A, and A* the field of fractions generated by A. We have

Let J ⊆ M. Then 〈M*, +, ·〉 is a linear space over J*. If x 1,…,xk ∈ M*, we shall say that x 1,…,xk are J-independent if 〈1, x 1,…, xk 〉 are J*-independent in the usual sense. As usual, we extend the notion of J-independence to the case of infinite sets.

If A ⊆ M* and X ⊆ A, then we say that X is a J-basis of A if X is a maximal subset of A which is J-independent.

Semilattice semantics for relevant logics were discovered independently by Routley and Urquhart over 10 years ago. A semilattice semantics was first published in [10], where the weak theory of implication of [8] and [3] (i.e., R →, the pure implication fragment of the system R of relevant implication) is shown to be consistent and complete with respect to it. That result was extended in [11], But the semantics is explored in greatest detail in [12]. As reported in [4], Fine outfitted the positive semilattice semantics for R+ with a suitable Hilbert-style axiomatisation. (We refer to the system as ◡R+.) In 1980 Charlwood supplied a subscripted system of natural deduction. (See [1] and [2].) A subscripted Gentzen system was devised in [5] and [6].

Obviously, the central idea of the semilattice semantics is to impose relevant-style valuations on a semilattice (with an identity) used as the underlying model structure. However, in [12] the contractionless semantics are obtained (quite reasonably) by dropping the idempotence postulate and thus changing the relatively simple semilattice structure into a commutative monoid. Here we show that the semilattice structure can be regained for positive, contractionless relevant implication. Although we have no proofs as yet, we think that this semantics will pave the way for showing completeness for the corresponding subscripted Gentzen and natural deduction systems, as well as the Hilbert-style axiomatization, ◡RW+.

Two elementary extensions of the topos axioms are given, each implying the topos has a local geometric morphism to a category of sets. The stronger one realizes sets as precisely the decidables of the topos, so there is a simple internal description of the range of validity of the law of excluded middle in the topos. It also has a natural geometric meaning. Models of the extensions in Grothendieck toposes are described.

We give a completeness theorem for a logic with probability quantifiers which is equivalent to the logics described in a recent survey paper of Keisler [K]. This result improves on the completeness theorems in [K] in that it works for languages with function symbols and produces a model whose universe is an analytic subset of the real line, and whose relations and functions are Borel relative to this universe.

Throughout this paper, B will always be a Boolean algebra and Γ an ultrafilter on B. We use + and Σ for the Boolean join operation and · and Π for the Boolean meet.

κ is always a regular cardinal. C(κ) is the full structure of κ, the structure with universe κ and whose functions and relations consist of all unitary functions and relations on κ. κB is the collection of all B-valued names for elements of κ. We use symbols f, g, h for members of κB. Formally an element f ∈ κB is a mapping κ → B with the properties that Σα∈κf(α) = 1B and that f(α) · f(β) = 0B whenever α ≠ β. We view f(α) as the Boolean-truth value indicating the extent to which the name f is equal to α, and we will hereafter write ∥f = α∥ for f(α). For every α ∈ κ there is a canonical name fα ∈ κB which has the property that ∥fα = α∥ = 1. Hereafter we identify α and fα.

If B is a κ+-complete Boolean algebra and Γ is an ultrafilter on B, then we may define the Boolean ultraproduct C(κ)B/Γ in the following manner. If ϕ(x0, x1, …, xn) is a formula of Lκ, the language for C(κ) (which has symbols for all finitary functions and relations on κ), and f0, f1, …, fn−1 are elements of κB then we define the Boolean-truth value of ϕ(f0, f1, …, fn−1) as

We will show that there is a nontrivial strongly minimal expansion of (ω, s), the natural numbers with successor. Pillay and Steinhorn [1] proved that there is no -minimal expansion of (ω, ≤). This result provides an interesting contrast.

The strongly minimal expansion of (ω, s) is very easy to describe. Consider the order-two permutation of ω, π recursively defined by

Let T be Th(ω, s, π, 0).

A subset of the Cantor space ω 2 is called monotone iff it is closed upward under the partial ordering ≤ defined by x ≤ y iff x(n) ≤ y(n) for all n ∈ ω. A set is -positive ( -positive) iff it is monotone and -positive set is a countable union of -positive sets; a -positive set is a countable intersection of -positive sets. (See Cenzer [2] for background information on these concepts.) It is clear that any -positive set is and monotone; the converse holds for n ≤ 2 [2] and was conjectured by Dyck to hold for greater n. In this note, we will disprove this conjecture by giving examples of monotone sets (for n ≥ 3) which are not even -positive.

First we note a few isomorphisms. The space ( ω 2, ≤) is isomorphic to the space ( ω 2 ≥), so instead of monotone and positive sets we may construct hereditary and negative sets (the analogous notions with “closed upward” replaced by “closed downward”). Also, ( ω 2, ≤) is isomorphic to ((ω), ⊆), where denotes the power set operator, or to ((S), ⊆) for any countably infinite set S.

In order to remove extraneous notation from the proofs, we state the results in an abstract form (whose generality is deceptive).

Much of recursion theory centers on the structures of different kinds of degrees. Classically there are the Turing degrees and r. e. Turing degrees. More recently, people have studied α-degrees for α an ordinal, and degrees over E-closed sets and admissible sets. In most contexts, deg(0) is the bottom degree and there is a jump operator' such that d' is the largest degree r. e. in d and d' > d. Both the degrees and the r. e. degrees usually have a rich structure, including a relativization to the cone above a given degree.

A natural exception to this pattern was discovered by S. Friedman [F], who showed that for certain admissible ordinals β the β-degrees ≥ 0′ are well-ordered, with successor provided by the jump.

For r. e. degrees, natural counterexamples are harder to come by. This is because the constructions are priority arguments, which require only mild restrictions on the ground model. For instance, if an admissible set has a well-behaved pair of recursive well-orderings then the priority construction of an intermediate r. e. degree (i.e., 0 < d < 0′) goes through [S]. It is of interest to see just what priority proofs need by building (necessarily pathological) admissible sets with few r. e. degrees.

Harrington [C] provides an admissible set with two r. e. degrees, via forcing. A limitation of his example is that it needs ω1 (more accurately, a local version thereof) as a parameter. In this paper, we find locally countable admissible sets, some with three r. e. degrees and some with four.

Theorem A. Let T be a small superstable theory, A a finite set, and ψ a weakly minimal formula over A which is contained in some nontrivial type which does not have Morley rank. Then ψ is contained in some nonalqebraic isolated type over A.

As an application we prove

Theorem B. Suppose that T is small and superstable, A is finite, and there is a nontrivial weakly minimal type p ∈ S(A) which does not have Morley rank. Then the prime model over A is not minimal over A.

Theorem A. Suppose that T is superstable and p is a nontrivial type of U-rank 1. Then R(p, L, ∞)= 1.

Theorem B. Suppose that T is totally transcendental and p is a nontrivial type of U-rank 1. Then p has Morley rank 1.

The principal aim of this paper is to present a construction method for nontotal extensional combinatory algebras. This is done in §2. In §0 we give definitions of some basic notions for partial combinatory algebras from which the corresponding notions for (total) combinatory algebras are obtained as specializations. In §1 we discuss some properties of nontotal extensional combinatory algebras in general. §2 describes a “partial” variant of reflexive complete partial orders yielding nontotal extensional combinatory algebras. Finally, §3 deals with properties of the models constructed in §2, such as incompletability, having no total submodel and the pathological behaviour with respect to the interpretation of unsolvable λ-terms.

The aim of this short note is to correct an error in the proof in [1], more precisely in Remark 1 on page 395, where it is stated that any improper word has rank ≥2. This claim is false; and, consequently, Remark 2, asserting that any 1,0-deduction does not contain improper Q-terms, also fails immediately. In fact, let us consider the word

It is easily seen that the rank of W is 1, but W is obviously improper. We remark that it is possible, in the same way, to give examples of improper words of rank 1 containing (without subordination) ε-subterms of arbitrarily high rank. In the following section we sketch a corrected version of the proof in order to eliminate all improper ε-terms.

We refer to the notation and definitions of [1]. A more extended description of the theories involved can be found in [2].

In much the same way that it is possible to construct a model of hyperbolic geometry in the Euclidean plane, it is possible to model quantum logic within the classical propositional calculus.

We show that either PFA+ or Martin's maximum implies Fleissner's Axiom R, a reflection principle for stationary subsets of Pℵ(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.