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].

In 1970, Kenneth Kunen showed that there is no non-trivial elementary embedding of the universe V into itself [2] using the axiom of choice. Kunen remarked in his paper that the result can be formalized in Morse-Kelley theory of sets and classes. In this paper, we will work within ZF, Zermelo-Fraenkel axioms, and deal with embeddings definable with a formula and a parameter.

In ZF, a “class” is usually synonymous with “property”, that is a class definable with a parameter, C = {x: φ(x,p)}, where φ is a formula in the language [∈}. Using this convention, let j be a class. Then “j is an elementary embedding of V into V” is not a single statement but a schema of statements “j preserves ψ” for each formula ψ. We prove that this schema is expressible in the language {∈} by a single formula:

Lemma. An embedding j: V → V is elementary iff j preservesψ.

Here ψ(α, ψ, a) is the property “a is an ordinal, φ is a formula and Vα.”

The lemma is of course a schema of lemmas, one for each formula denning j and for each ψ to be preserved.

Using this we prove our theorem in ZF (again, a schema of theorems.):

Theorem 1.1. There is no nontrivial definable elementary embedding j: V → V.

Many symbols and their definitions follow those used by Drake's book [1]. The formula Sat expresses the satisfaction relation . The formula Fmla(u) expresses that u is the Gödel-set for a formula.

It is proved that all the isomorphisms in the cartesian category freely generated by a set of objects (i.e., a graph without arrows) can be written in terms of arrows from the symmetric monoidal category freely generated by the same set of objects. This proof yields an algorithm for deciding whether an arrow in this free cartesian category is an isomorphism.

We analyze combinatorial properties of open covers of sets of real numbers by using filters on the natural numbers. In fact, the goal of this paper is to characterize known properties related to ω-covers of the space in terms of combinatorial properties of filters associated with these ω-covers. As an example, we show that all finite powers of a set of real numbers have the covering property of Menger if, and only if, each filter on ω associated with its countable ω-cover is a P+ filter.

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.

Using a combinatorial theorem of Herwig on extending partial isomorphisms of relational structures, we give a simple proof that certain classes of algebras, including Crs, polyadic Crs, and WA, have the ‘finite base property’ and have decidable universal theories, and that any finite algebra in each class is representable on a finite set.

Assume T is a small superstable theory. We introduce the notion of a flat Morley sequence, which is a counterpart of the notion of an infinite Morley sequence in a type p, in case when p is a complete type over a finite set of parameters. We show that for any flat Morley sequence Q there is a model M of T which is τ-atomic over {Q}. When additionally T has few countable models and is 1-based, we prove that within M there is an infinite Morley sequence I, with I ⊂ dcl(Q), such that M is prime over I.

We show that a formula φ(x, y) is stable if and only if φ is the pairing map on the unit ball of E × E*, where E is a reflexive Banach space. The result remains true if the formula φ is replaced by a set of formulas .

For a canonical model of set theory whose projective theory of the real numbers is stable under set forcing extensions, a set of reals of minimal complexity is constructed which fails to be universally Baire. The construction uses a general method for generating non-universally Baire sets via the Levy collapse of a cardinal, as well as core model techniques. Along the way it is shown (extending previous results of Steel) how sufficiently iterable fine structure models recognize themselves as global core models.

We note here, in answer to a question of Poizat, that the Morley and Lascar ranks need not coincide in differentially closed fields. We will approach this through the (perhaps) more fundamental issue of the variation of Morley rank in families. We will be interested here only in sets of finite Morley rank. Section 1 consists of some general lemmas relating the above issues. Section 2 points out a family of sets of finite Morley rank, whose Morley rank exhibits discontinuous upward jumps. To make the base of the family itself have finite Morley rank, we use a theorem of Buium.

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 continue the work of [14, 3, 1, 19, 16, 4, 12, 11, 20] investigating the strength of set existence axioms needed for separable Banach space theory. We show that the separation theorem for open convex sets is equivalent to WKL0 over RCA0. We show that the separation theorem for separably closed convex sets is equivalent to ACA0 over RCA0. Our strategy for proving these geometrical Hahn–Banach theorems is to reduce to the finite-dimensional case by means of a compactness argument.

In [13], M. Tokarz specified some infinite family of consequence operations among all ones associated with the relevant logic RM or with the extensions of RM and proved that each of them admits a deduction theorem scheme. In this paper, we show that the family is complete in a sense that if C is a consequence operation with CRM ≤ C and C admits a deduction theorem scheme, then C is equal to a consequence operation specified in [13]. In algebraic terms, this means that the only quasivarieties of Sugihara algebras with the relative congruence extension property are the quasivarieties corresponding, via the algebraization process, to the consequence operations specified in [13].

Let CH be the class of compacta (i.e., compact Hausdorff spaces), with BS the subclass of Boolean spaces. For each ordinal α and pair (K, L) of subclasses of CH, we define Lev≥α(K, L), the class of maps of level at least α from spaces in K to spaces in L, in such a way that, for finite α, Lev≥α(BS, BS) consists of the Stone duals of Boolean lattice embeddings that preserve all prenex first-order formulas of quantifier rank α. Maps of level ≥ 0 are just the continuous surjections, and the maps of level ≥ 1 are the co-existential maps introduced in [8]. Co-elementary maps are of level ≥ω a for all ordinals α: of course in the Boolean context, the co-elementary maps coincide with the maps of level ≥ ω. The results of this paper include:

(i) every map of level ≥ ωis co-elementary;

(ii) the limit maps of an co-indexed inverse system of maps of level ≥ α are also of level ≥ α; and

(iii) if K is a co-elementary class, k > ω and Lev≥k(K,K) = Lev≥k+1(K,K), then Lev≥1(K,K) = Lev≥(K,K).

A space X ∈ K is co-existentially closed inK if Lev≥0(K, X) = Lev≥1(K,X). Adapting the technique of “adding roots,” by which one builds algebraically closed extensions of fields (and, more generally, existentially closed extensions of models of universal-existential theories), we showed in [8] that every infinite member of a co-inductive co-elementary class (such as CH itself, BS, or the class CON of continua) is a continuous image of a space of the same weight that is co-existentially closed in that class. We show here that every compactum that is co-existentially closed in CON (a co-existentially closed continuum) is both indecomposable and of covering dimension one.

We show that every sufficiently iterable countable mouse has a unique iteration strategy whose associated iteration maps are lexicographically minimal. This enables us to extend the results of [3] on the good behavior of the standard parameter from tame mice to arbitrary mice.

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.

We present a new proof for the standardization theorem in λ-calculus, which is largely built upon a structural induction on λ-terms. We then extract some bounds for the number of β-reduction steps in the standard β-reduction sequence obtained from transforming a given β-reduction sequence, sharpening the standardization theorem. As an application, we establish a super exponential bound for the lengths of β-reduction sequences from any given simply typed λ-terms.

In a previous paper—[17]—we characterized strong measure zero sets of reals in terms of a Ramseyan partition relation on certain subspaces of the Alexandroff duplicate of the unit interval. This framework gave only indirect access to the relevant sets of real numbers. We now work more directly with the sets in question, and since it costs little in additional technicalities, we consider the more general context of metric spaces and prove:

1. If a metric space has a covering property of Hurewicz and has strong measure zero, then its product with any strong measure zero metric space is a strong measure zero metric space (Theorem 1 and Lemma 3).

2. A subspace X of a σ-compact metric space Y has strong measure zero if, and only if, a certain Ramseyan partition relation holds for Y (Theorem 9).

3. A subspace X of a σ-compact metric space Y has strong measure zero in all finite powers if, and only if, a certain Ramseyan partition relation holds for Y (Theorem 12).

Then 2 and 3 yield characterizations of strong measure zeroness for σ-totally bounded metric spaces in terms of Ramseyan theorems.

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.

A bounded ultrasheaf is a nonstandard universe constructed from a superstructure in a Boolean valued model of set theory. We consider the bounded elementary embeddings between bounded ultrasheaves. Then the standardization principle is true if and only if the ultrafilters are comparable by the Rudin-Frolik order. The base concept is that the bounded elementary embeddings correspond to the complete Boolean homomorphisms. We represent this by the Rudin-Keisler order of ultrafilters of Boolean algebras.