Let L be a computable vocabulary, let XL be the space of L-structures with universe ω and let $f:{2^\omega } \to {X_L}$ be a hyperarithmetic function such that for all $x,y \in {2^\omega }$, if $x{ \equiv _h}y$ then $f\left( x \right) \cong f\left( y \right)$. One of the following two properties must hold. (1) The Scott rank of f (0) is $\omega _1^{CK} + 1$. (2) For all $x \in {2^\omega },f\left( x \right) \cong f\left( 0 \right)$.

In this paper, a simple inductive characterization of the ordinal numbers is stated and developed. The characterization forms the basis for a set of axioms for ordinal theory and also for several short explicit definitions of the ordinals. The axioms are shown to be sufficient for ordinal theory, and, subject to suitable existence assumptions, each of the definitions is shown to imply the axioms.

The present results apply to the familiar von Neumann version of the ordinals, but the methods used are easily adapted to other versions.

We show that infinite sets whose power-sets are Dedekind-finite can only carry ℵ0-categorical first order structures. We identify other subclasses of this class of Dedekind-finite sets, and discuss their possible first order structures.

We study the automorphism group of the algebraic closure of a substructure A of a pseudo-finite field F, or more generally, of a bounded PAC field F. This paper answers some of the questions of [1], and in particular that any finite group which is geometrically represented in a pseudo-finite field must be abelian.

We show that there is a cuppable c.e. degree, all of whose cupping partners are high. In particular, not all cuppable degrees are ${\operatorname {\mathrm {low}}}_3$-cuppable, or indeed ${\operatorname {\mathrm {low}}}_n$ cuppable for any n, refuting a conjecture by Li. On the other hand, we show that one cannot improve highness to superhighness. We also show that the ${\operatorname {\mathrm {low}}}_2$-cuppable degrees coincide with the array computable-cuppable degrees, giving a full understanding of the latter class.

We give a valuation theoretic characterization for a real closed field to be recursively saturated. This builds on work in [9], where the authors gave such a characterization for κ-saturation, for a cardinal $\kappa \ge \aleph _0 $. Our result extends the characterization of Harnik and Ressayre [7] for a divisible ordered abelian group to be recursively saturated.

The main result of this paper is the fact that for T a complete extension of either P or ZF + V = L, with no new symbols, there is a type which is omitted in a model of T of power ℵ1 but which is realized in all models of higher power. Therefore, the “Hanf number” for omitting types over T is greater than ℵ2.

The present section contains some basic definitions and a discussion of related results. §1 discusses generic relations on models of P and ZFC, and describes a special forcing technique due to Addison. In §2, this technique is used to obtain the main result.

The use of forcing is not essential. Those who do not wish to read a forcing argument may skip §1. At the point in §2 where forcing is used, a proof without forcing is sketched. This proof was obtained only after a careful study of the forcing argument. The forcing argument is given because it is simple, and because the technique of forcing makes intuitively clear why this and other similar results should be true.

All theories in the paper are assumed to be countable and to have infinite models. Let T be a complete theory in a language L. The Hanf number for omitting types over T, denoted by H(T), is the first infinite cardinal κ such that for all L-types, Σ, if T has models omitting Σ in all infinite powers less than κ, then T has models omitting Σ in all infinite powers.

Let L0, L1 and L2 be countable languages with L ∩ L1 = L0. Let M0 be an L0-structure and Mi, an expansion of M0 to an Li,-structure (i = 1,2). We will call an L1 ∪ L2-structure M an amalgamation of M1 and M2 if M∣Li ≅ Mi, (i = 1,2). Let's consider the following problem.

(*) Suppose that both M1 and M2 belong to the class . Can we always find an amalgamation M in ?

Of course the existence of such an amalgamation depends on the class L. Some examples of and the answers are given below.

1. = Countably saturated strongly minimal structures with the DMP In [3], Hrushovski showed that any two strongly minimal theories formulated in totally different languages have a common extension which is still strongly minimal and with the DMP (DMP is the property that states that if a point is sufficiently close to ā, then φ(, ) has the same rank and the same degree as φ(, ā).) His proof essentially shows that if L0 = ∅ then any two countably saturated strongly minimal structures with the DMP have a strongly minimal amalgamation. Also he gave an example that shows the condition L0 = ∅ is necessary.

2. = ℵ1-categorical countable structures. Let M1 be the structure (ℚ, +) and let M2 be the {E, F}-structure defined by: (i) E is an equivalence relation which divides the universe into two infinite classes A and B, (ii) F is a bijection between A and B.

This paper studies homogeneously Suslin (hom) sets of reals in tame mice. The following results are established: In 0¶ the hom sets are precisely the sets. In Mn every hom set is correctly , and (δ + 1)-universally Baire where δ is the least Woodin. In Mω, every hom set is <λ-hom, where λ is the supremum of the Woodins.

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].

We study the sets definable in an algebraic number field by first order formulas of various simple types, showing in particular that N and Z do not have very simple definitions.

In this paper, we show that the predicate logics of consistent extensions of Heyting's Arithmetic plus Church's Thesis with uniqueness condition are complete . Similarly, we show that the predicate logic of HA*. i.e. Heyting's Arithmetic plus the Completeness Principle (for HA*) is complete . These results extend the known results due to Valery Plisko. To prove the results we adapt Plisko's method to use Tennenbaum's Theorem to prove ‘categoricity of interpretations’ under certain assumptions.

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.

This article was inspired by the question: is there a definable equivalence relation on the field of complex numbers, each of whose equivalence classes has exactly two elements? The answer turned out to be no, as we now explain in greater detail.

Let Κ be an algebraically closed field and let E be a definable equivalence relation on Κ. [Note: By “definable” we will always mean “definable with parameters”.] Either E has one cofinite class, or all classes are finite and there is a number d such that all but a finite set of classes have cardinality d. In the latter case let B be the finite set of elements of Κ which are not in a class of size d. We prove the following result.

Theorem 1. a) If char(Κ) = 0 or char(Κ) = p > d, then ∣B∣ ≡ 1 (mod d).

b) If char(Κ) = 2 and d = 2, then ∣B∣ ≡ 0 (mod 2).

c) If char(Κ) = p > 2 and d = p + s, where 1 ≤ s ≤ p/2, then ∣B∣ ≡ p + 1 (mod d).

Furthermore, a)−c) are the only restrictions on ≡B≡.

If one is in the right mood, one can view this theorem as saying that the “algebraic cardinality” of the complex numbers is congruent to 1 (mod n) for every n.

§1 contains a reduction of the problem to the special case where E is induced by a rational function in one variable. §2 contains the main calculations and the proofs of a)−c). §3 contains eight families of examples showing that all else is possible. In §4 we prove an analogous result for real closed fields.

Abstract.

Banach introduced the following two-person, perfect information, infinite game on the real numbers and asked the question: For which sets A ⊆ R is the game determined?

Rules: The two players alternate moves starting with player I. Each move an is legal iff it is a real number and 0 < an, and for n > 1, an < an−1. The first player to make an illegal move loses. Otherwise all moves are legal and I wins iff exists and .

We will look at this game and some variations of it, called Banach games. In each case we attempt to find the relationship between Banach determinacy and the determinacy of other well-known and much-studied games.