For countable structures and , let abbreviate the statement that every sentence true in also holds in . One can define a back and forth game between the structures and that determines whether . We verify that if θ is an Lω,ω sentence that is not equivalent to any Lω,ω sentence, then there are countably infinite models and such that ⊨ θ, ⊨ ¬θ, and . For countable languages ℒ there is a natural way to view ℒ structúres with universe ω as a topological space, Xℒ. Let [] = { ∊ Xℒ∣ ≅ } denote the isomorphism class of . Let and be countably infinite nonisomorphic ℒ structures, and let C ⊆ ωω be any subset. Our main result states that if , then there is a continuous function f: ωω → Xℒ with the property that x ∊ C ⇒ f(x) ∊ [] and x ∉ C ⇒ f(x) ∊ f(x) ∈ []. In fact, for α ≤ 3, the continuous function f can be defined from the relation.

In [1, §14E], a sequent calculus formulation (L.-formulation) of type assignment (theory of functionality) is given for a system based either on a system of combinators with strong reduction or on a system of λη-calculus provided that the rule for subject conversion (which says that if X has type α and X cnv Y then Y has type α) is postulated for the system. This sequent calculus formulation does not work for a system based on the λβ-calculus. In [2] I introduced a sequent calculus formulation for a system without the rule of subject conversion based on any of the three systems mentioned above. Further, in [2, §5] I pointed out that if proper inclusions of the form of the statement that λx·x is a function from α to β are postulated, then functions are identified with their restrictions in the λη-calculus but not in the λβ-calculus, and that therefore there is some interest in having a sequent calculus formulation of type assignment with the rule of subject conversion for systems based on the λβ-calculus. In this paper, such a system is presented, the elimination theorem (Gentzen's Hauptsatz) is proved for it, and it is proved equivalent to the natural deduction formulation of [1, §14D].

I shall assume familiarity with the λβ-calculus, and shall use (with minor modifications) the notational conventions of [1]. Hence, the theory of type assignment (theory of functionality) will be based on an atomic constant F such that if α and β are types then Fαβ represents roughly the type of functions from α to β (more exactly it represents the type of functions whose domain includes α and under which the image of α is included in β).

Let S be the set of those α ∈ ω2 that have cofinality ω1. It is consistent relative to a measurable that the nonempty player wins the pressing down game of length ω1, but not the Banach-Mazur game of length ω + 1 (both games starting with S).

We develop a version of Namba forcing which is useful for constructing models with no good scale on ℵω . A model is produced in which holds for all finite n ≥ 1, but there is no good scale on ℵω ; this strengthens a theorem of Cummings, Foreman, and Magidor [3] on the non-compactness of square.

We show that it is not possible to construct a Fraenkel-Mostowski model in which the axiom of choice for well-ordered families of sets and the axiom of choice for sets of well-orderable sets are both true, but the axiom of choice is false.

Let R be a commutative integral unital domain and L a free noncommutative Lie algebra over R. In this article we show that the ring R and its action on L are 0-interpretable in L, viewed as a ring with the standard ring language $+ , \cdot ,0$. Furthermore, if R has characteristic zero then we prove that the elementary theory $Th\left( L \right)$ of L in the standard ring language is undecidable. To do so we show that the arithmetic ${\Bbb N} = \langle {\Bbb N}, + , \cdot ,0\rangle $ is 0-interpretable in L. This implies that the theory of $Th\left( L \right)$ has the independence property. These results answer some old questions on model theory of free Lie algebras.

At the end of his paper [2], Silver shows how methods developed by Jensen and Solovay in order to prove results about the constructible universe L may be adapted to prove corresponding results for the universe L[μ], where μ is a normal measure on some uncountable cardinal ρ. In this paper we pursue this in greater depth. Jensen proved, in fact, much stronger results than those considered in [2], and we shall show that all of these carry over from L to L[μ], More precisely, we show that if V = L[μ] is assumed, then for any regular uncountable cardinal κ and any uncountable λ < κ, + (κ, λ) holds, and that + (κ, κ) holds just in the case κ is not ineffable. This result was proved to hold in L by Jensen, who first formulated the principles + (κ, λ).

Our proof differs in detail from Jensen's, and at one point (in choosing the set B of + ) differs fundamentally from his argument. However, the fact remains that our argument is modelled closely upon Jensen's, and it should be made clear that in many parts it is a straightforward adaption of his proof to the L[μ] situation. It is regrettable that, at the time of our writing this, Jensen's proof still only exists in the rough, handwritten form of [1]; so we shall give our argument in some detail, even those parts which are merely “translations” of Jensen's original arguments.

In this paper we prove some theorems concerning measures on complete Boolean algebras. Among other things, in §I of this paper, we construct a counterexample to the following conjecture of W. Luxemburg: Every measure on a nonatomic hyperstonian Boolean algebra is normal. (See [3, p. 57].) This result is expressed by Theorem 1, §I. In order to construct this example we have to suppose that a real-valued measurable cardinal exists. This hypothesis is independent of the usual axioms of set theory. Luxemburg proved that our assumption is necessary. Our second result is stated in Theorem 2 near the end of the paper.

This paper was inspired by Lerman [14] in which he proved various properties of upper bounds for the arithmetical degrees. The degrees which are upper bounds for the arithmetical degrees were first studied by Hodes [5] and Enderton and Putnam [5]. Enderton and Putnam [5] showed that if a is an upper bound for the arithmetical degrees, then a (2) ≥ 0(ω). This area was further studied by Hodes [5], Knight, Lachlan and Soare [12], Jockusch and Simpson [12], and Sacks [17].

Enderton and Putnam [5] and Sacks [17] have combined to show that 0(ω) is the least degree in {a (2): a is an upper bound for the arithmetical degrees}. So there seem to be some analogies between the degrees of upper bounds for the arithmetical degrees and the degrees below 0(2). But Sacks [17] showed an important difference between these two structures; namely, the Turing jumps of upper bounds for the arithmetical degrees have no least element. In Lerman [14], a systematic investigation of properties of the jumps of upper bounds for the arithmetical degrees was suggested, which could lead to a definition of the jump operator over the elementary theory of the partial ordering of the degrees. Although Cooper [5] found a degree-theoretic definition of the jump operator, Lerman's plan is still interesting.

We say a degree a is generic if there is a representative A of a such that A is Cohen generic for the arithmetical sentences. By Jockusch [5], the statement that A is Cohen generic for the arithmetical sentences is equivalent to saying that for any arithmetical set S of binary strings, there is a σ extended by A such that σ is in S or no extension of σ is in S. First, we investigate the relation between generic degrees and upper bounds for the arithmetical degrees. In the case D(≤ 0′), the set of degrees below 0′, it is well known that any nonrecursive r.e. degree bounds a 1-generic degree (Cohen generic for Σ1 sentences). Jockusch and Ponser [12] showed that any degree a with a″ >T (a ∪ 0′)′ bounds a 1-generic degree.

We assume the existence of a supercompact cardinal and produce a model with weak square but no very good scale at a particular cardinal. This follows work of Cummings, Foreman, and Magidor, but uses a different approach. We produce another model, starting from countably many supercompact cardinals, where □K,<K holds but □K, λ fails for λ < K.

It is proved consistent that there be a proper σ-ideal on ω 1 and an ℵ1-preserving poset ℙ such that ℙ ⊩ the σ-ideal generated by is not proper.

We study the position of the computable setting in the “common theory of locality” developed in [4, 5] for local problems on $\Delta $-regular trees, $\Delta \in \omega $. We show that such a problem admits a computable solution on every highly computable $\Delta $-regular forest if and only if it admits a Baire measurable solution on every Borel $\Delta $-regular forest. We also show that if such a problem admits a computable solution on every computable maximum degree $\Delta $ forest then it admits a continuous solution on every maximum degree $\Delta $ Borel graph with appropriate topological hypotheses, though the converse does not hold.

The Gödel translation provides an embedding of the intuitionistic logic $\mathsf {IPC}$ into the modal logic $\mathsf {Grz}$, which then embeds into the modal logic $\mathsf {GL}$ via the splitting translation. Combined with Solovay’s theorem that $\mathsf {GL}$ is the modal logic of the provability predicate of Peano Arithmetic $\mathsf {PA}$, both $\mathsf {IPC}$ and $\mathsf {Grz}$ admit provability interpretations. When attempting to ‘lift’ these results to the monadic extensions $\mathsf {MIPC}$, $\mathsf {MGrz}$, and $\mathsf {MGL}$ of these logics, the same techniques no longer work. Following a conjecture made by Esakia, we add an appropriate version of Casari’s formula to these monadic extensions (denoted by a ‘+’), obtaining that the Gödel translation embeds $\mathsf {M^{+}IPC}$ into $\mathsf {M^{+}Grz}$ and the splitting translation embeds $\mathsf {M^{+}Grz}$ into $\mathsf {MGL}$. As proven by Japaridze, Solovay’s result extends to the monadic system $\mathsf {MGL}$, which leads us to a provability interpretation of both $\mathsf {M^{+}IPC}$ and $\mathsf {M^{+}Grz}$.

In this paper, we study the employment of $\Sigma _1$-sentences with certificates, i.e., $\Sigma _1$-sentences where a number of principles is added to ensure that the witness is sufficiently number-like. We develop certificates in some detail and illustrate their use by reproving some classical results and proving some new ones. An example of such a classical result is Vaught’s theorem of the strong effective inseparability of $\mathsf {R}_0$.

We also develop the new idea of a theory being $\mathsf {R}_{0\mathsf {p}}$-sourced. Using this notion, we can transfer a number of salient results from $\mathsf {R}_0$ to a variety of other theories.

In this paper we shall prove that any 2-transitive finitely homogeneous structure with a supersimple theory satisfying a generalized amalgamation property is a random structure. In particular, this adapts a result of Koponen for binary homogeneous structures to arbitrary ones without binary relations. Furthermore, we point out a relation between generalized amalgamation, triviality and quantifier elimination in simple theories.