In this paper we will study four forcing notions, two of them giving a minimal degree of constructibility. These constructions give answers to questions in [Ih].

We study function fields over p-adically closed fields in the first-order language of fields. Using ideas of Duret [D], we show that the field of constants is definable, and that the genus is an elementary property.

In the paper [S1] I introduced a game, denoted by MG(J) (where J is a free ideal on some infinite set S) and called “the meager nowhere dense game for J”. The special case when J is the collection of finite subsets of the set S is called the countable-finite game on S. It proceeds as follows.

First player ONE picks a countable set C 1, then player TWO picks a finite set F 1. Then in the second inning ONE picks a countable set C 2 with C 1 ⊂ C 2 (unless explicitly indicated otherwise, “⊂” means “is a proper subset of”) and TWO responds with a finite set F 2, and so on. The players construct a sequence (C1,F1,C2,F2,…,Ck,Fk,…) where for each positive integer k

(i) Ck denotes ONE's countable set picked during the kth inning,

(ii) Fk denotes TWO's finite set picked during the kth inning, and

(iii) Ck ⊂ Ck + 1.

Such a sequence is a play of the countable-finite game on S, and TWO wins this play if is contained in . The notion of a winning perfect information strategy is defined as usual (see, for example, [S1]). Zermelo-Fraenkel set theory together with the axiom of choice (denoted by ZFC; for a statement of the axioms see pp. xv–xvi of [K]) is a strong enough theory to build a winning perfect information strategy for player TWO in this game.

Does TWO have a winning strategy requiring less than perfect information? Fix a positive integer k. A strategy of TWO which requires knowledge of only at the most the k most recent moves of ONE is said to be a k-tactic. For the countable-finite game on an infinite set S the following facts about the existence of winning k-tactics for TWO are proved in [S1]:

1) TWO does not have a winning 1-tactic (Theorem 1 of [S1]).

2) If the cardinality of S is less than ℵ2 then TWO has a winning 2-tactic (Corollary 4 of [S1]).

3) If TWO has a winning k-tactic in the countable-finite game on an infinite set S, then TWO has a winning 3-tactic (Proposition 15 of [SI]).

We prove that in many situations it is consistent with ZFC that part of the invariants involved in Cichoń's diagram are equal to κ while the others are equal to λ, where κ < λ are both arbitrary regular uncountable cardinals. We extend some of these results to the case when λ is singular. We also show that cf is consistent with ZFC.

Given a subset A of a fixed saturated model , we let denote the algebra of definable subsets of the domain of M of with parameters from A. Then a complete type p over A can be regarded as a measure on , assigning the value 1 to members of p and 0 to nonmembers. In [5] and [6], Keisler developed a theory of forking concerning probability measures. Therefore it generalizes the ordinary theory. On a different track, we can view the complement of the type p, or the collection of null sets of any measure on , as ideals on . Moreover, ideals and the pseudometric of a measure form examples of the so-called FN topology. Roughly speaking, an FN topology is one that topologizes the ring structure of a Boolean algebra. Associated with this topology is a special basis, called an FN basis. For , it is natural to identify such a basis with a collection of partial types approximating a certain type.

In this paper, we extend the theory of forking and deal with FN bases (hence ideals in particular). By proving a few results here, we hope to indicate that this extension could become as fruitful as Keisler's theory and hence provide an alternative. Our approach here is more algebraic and less analytic. Unlike [5], methods from nonstandard analysis are not used.

There are two fundamental reasons why the classical theory should be generalized. One is to extend our investigation to unstable theories. (There are no stability assumptions on the theory in the present work.) The other is to study different and more flexible ways of collecting formulas, such as FN basis.

After fixing our notation in §1, we first deal with ideals in §2. We study certain nice extensions. Specifically, we prove the existence of flat nonforking ideals. We then provide a normalization theorem for ideals.

This paper is a sequel to [4]. Our main concern here is to study the FN bases, which correspond to classical notions like heirs and definable extensions. We also attempt to characterize these notions in our context. This leads to the use of tools like ultrapower constructions.

We continue to work under the assumptions and using the notation of [4, §1]. For example, we require a sufficiently saturated model of a fixed complete theory T in a language , and ℒ and A and B are used to denote (not necessarily small) subsets of M. The concept of FN basis is defined in [4, Definition 3.2]. Nonforking is defined in [4, Definitions 2.2 and 3.7].

This paper is organized as follows: §1 deals with heirs and definability. The classical result stating that definability is equivalent to the uniqueness of an heir is generalized (Theorem 1.7). §2 uses an ultrapower to prove the existence of an heir (Corollary 2.6). We also determine exactly when embeddability into an ultrapower is possible. §3 relates ultrapowers to flatness and nonforking. In sharp contrast to the classical case, ultrapowers of FN bases need not be nonforking; but some criteria are established (Propositions 3.2 and 3.3, and Theorem 3.4). Finally, definability is characterized as the flatness of all ultrapowers (Theorem 3.6).

By a variety, we mean a class of structures in some language containing only function symbols which is equationally defined or equivalently is closed under homomorphisms, submodels and products.

If K is a class of -structures then I(K, λ) denotes the number of nonisomorphic models in K of cardinality λ. When we say that K has few models, we mean that I(K,λ) < 2λ for some λ > ∣∣. If I(K,λ) = 2λ for all λ > ∣∣, then we say K has many models. In [9] and [10], Shelah has shown that for an elementary class K, having few models is a strong structural condition.

A chain-closed field is defined as a chainable field (i.e. a real field such that, for all n ∈ N, ΣK 2n+2 ≠ ΣK 2n ) which does not admit any “faithful” algebraic extension, and can also be seen as a field having a Henselian valuation ν such that the residue field K/ν is real closed and the value group νK is odd divisible with ∣νK/2νK∣ = 2. If K admits only one such valuation, we show that f ∈ K(X) is in ΣK(X)2n for any real algebraic extension L of K,“f(L) ⊆ ΣL 2n ” holds. The conclusion is also true for K = R((t))(a chainable but not chain-closed field), and in the case n = 1 it holds for several variables and any real field K.

We study the following conjecture.

Conjecture. Let T be an ω-stable theory with continuum many countable models. Then either i) T has continuum many complete extensions in L 1(T), or ii) some complete extension of T in L 1 has continuum many L 1-types without parameters.

By Shelah's proof of Vaught's conjecture for ω-stable theories, we know that there are seven types of ω-stable theory with continuum many countable models. We show that the conjecture is true for all but one of these seven cases. In the last case we show the existence of continuum many L 2-types.

For a structure let φ() be the number of nonisomorphic, countably infinite substructures of . The problem considered here, suggested by M. Pouzet, is that of characterizing those countable for which φ() ≤ ℵ0. In this paper we will deal exclusively with structures in a finite, binary relational language L. The characterization of those L-structures for which φ() ≤ ℵ0 (which turns out to be equivalent to ) is given in Theorem 3. It is the culmination of a three-step process. The first step, resulting in Theorem 1, shows that for a countable stable L-structure , φ() ≤ ℵ0 iff is cellular. (See Definition 0.1.) In the second step we consider linearly ordered sets = (A, ≤ ℵ0), and characterize in Theorem 2 the order types of those for which φ() ≤ ℵ0. Finally, in Theorem 3, we amalgamate Theorems 1 and 2 to get the classification of all countable L-structures for which φ() ≤ ℵ0.

Let S be a recursive theory. Let a theory T* consisting of Σ1 sentences be called maximal (with respect to S) if T* is maximal consistent with S, i.e. there is no Σ1 sentence consistent with T* + S which is not in T*.

A maximal theory with respect to IΔ0 was considered by Wilkie and Paris in [WP] in connection with the end-extension problem.

Let us recall that IΔ0 is the fragment of Peano arithmetic consisting of the finite collection of algebraic axioms PA− together with the induction scheme restricted to bounded formulas.

The main open problem concerning the end-extendability of models of IΔ0 is the following:

(*) Does every model of IΔ0 + BΣ1 have a proper end-extension to a model of IΔ0?

Here BΣ1 is the following collection scheme:

where φ runs over bounded formulas and may contain parameters.

It is well known(see [KP]) that if I is a proper initial segment of a model M of IΔ0, then I satisfies IΔ0 + BΣ1.

For a wide discussion of the problem (*) see [WP]. Wilkie and Paris construct in [WP] a model M of IΔ0 + BΣ1 which has no proper end-extension to a model of IΔ0 under the assumption IΔ0 ⊢¬Δ0 H (see [WP] for an explanation of this assumption). Their model M is a model of a maximal theory T* where S = IΔ0.

Moreover, T*, which is the set Σ1(M) of all Σ1 sentences true in M, is not codable in M.

It is shown that many different problems have the same degree of unsolvability. Among these problems are:

The Inductive Inference Problem. Infer in the limit an index for a recursive function f presented as f(0), f(1),f(2),….

The Recursive Index Problem. Decide in the limit if i is the index of a total recursive function.

The Zero Nonvariant Problem. Decide in the limit if a recursive function f presented as f(0), f(1), f(2),… has value unequal to zero for infinitely many arguments.

Finally, it is shown that these unsolvable problems are strictly easier than the halting problem.

We axiomatize all quantifier properties which can be expressed by a universal condition on the class of algebras of sets.

We prove that every countably determined set C is U-meager if and only if every internal subset A of C is U-meager, provided that the cofinality and coinitiality of the cut U are both uncountable. As a consequence we prove that for such cuts a countably determined set C which intersects every U-monad in at most countably many points is U-meager. That complements a similar result in [KL]. We also give some partial solutions to some open problems from [KL]. We prove that the set , where H is an infinite integer, cannot be expressed as a countable union of countably determined sets each of which is U-meager for some cut U with min{cf(U), ci(U)} ≥ ω1. Also, every Borel, or countably determined set C which is U-meager for every cut U is a countable union of Borel, or countably determined sets respectively, which are U-nowhere dense for every cut U. Further, the class of Borel U-meager sets for min{cf(U),ci(U)} ≥ ω1 coincides with the least family of sets containing internal U-meager sets and closed with respect to the operation of countable union and intersection. The same is true if the phrase “U-meager sets” is replaced by “U-meager for every cut U.”

Cherlin introduced the concept of bad groups (of finite Morley rank) in [Ch1]. The existence of such groups is an open question. If they exist, they will contradict the Cherlin-Zil'ber conjecture that states that an infinite simple group of finite Morley rank is a Chevalley group over an algebraically closed field. In this paper, we prove that bad groups of finite Morley rank 3 act on a natural geometry Γ (namely on a special pseudoplane; see Corollary 20) sharply flag-transitively.We show that Γ is not very far from being a projective plane and when it is so rk(Γ) = 2 and Γ is not Desarguesian (Theorem 2). Baldwin [Ba] recently discovered non-Desarguesian projective planes of Morley rank 2. This discovery, together with this paper, makes the existence of bad groups (also of bad fields) more plausible. A bad field is a pair (K, A) of finite Morley rank, where K is an algebraically closed field, A <≠ K* and A is infinite. There existence is also unknown.

In this paper, we define the concept of a sharp-field as a pair (K, A), where K is a field, A < K*and

1. K = A − A,

2. If a + b − 1 ∈ A, a ∈ A, b ∈ A, then either a = 1 or b = 1.

If K is finite this is equivalent to 1 and

2.′ ∣K∣ = ∣A∣2 ∣A∣ + 1.

Finite sharp-fields are special cases of difference sets [De]

Sometimes it is said that mathematics talks only about truths that are invariant under isomorphisms. Thus, a syntactical characterization of sentences preserved under isomorphisms seems to be important to understand what mathematics is.

The importance of such sentences is underlined, for instance, by Bourbaki's requirement in [1, Chapter IV] that all axioms for mathematical structures satisfy the condition of being what they call transportable, that is, preserved under isomorphisms. In [2, §1], a formalization of Bourbaki's notion of mathematical structure and of the notion of transportable sentence was given and discussed. We adopt here, as in [2], Bourbaki's terminology, and call a sentence φ transportable if φ is true in a structure E if and only if φ is true in any structure F isomorphic to E.

An alternative formulation for Bourbaki's notion of structure is offered in [2, §3], which is also intended as a formalization of Suppes' notion of set-theoretical predicate, as formulated, for instance, in [3]. It is this alternative formulation that we adopt here.

In [2, p. 106], it was conjectured that a set-theoretical sentence is transportable if and only if it is equivalent to a sentence of type theory. One direction of the conjecture is obvious, namely, that any sentence of type theory is transportable. It is also clear that there are sentences of set theory which are not transportable. The main purpose of the present paper is to prove the rest of the conjecture. That is, we prove here that any transportable set-theoretical sentence is equivalent to a sentence of type theory.

Let = {0,1, +,·,<} be the usual first-order language of arithmetic. We show that Peano arithmetic is the least first-order -theory containing IΔ0 + exp such that every complete extension T of it has a countable model K satisfying

(i) K has no proper elementary substructures, and

(ii) whenever L ≻ K is a countable elementary extension there is and such that .

Other model-theoretic conditions similar to (i) and (ii) are also discussed and shown to characterize Peano arithmetic.

F. Richman raised the question of whether the following principle of second order arithmetic is valid in intuitionistic higher order arithmetic HAH:

and if not, whether assuming Church's Thesis CT and Markov's Principle MP would help. Blass and Scedrov gave models of HAH in which this principle, which we call RP, is not valid, but their models do not satisfy either CT or MP.

In this paper a realizability topos Lif is constructed in which CT and MP hold, but RP is false. (It is shown, however, that RP is derivable in HAH + CT + MP + ECT0, so RP holds in the effective topos.) Lif is a generalization of a realizability notion invented by V. Lifschitz. Furthermore, Lif is a subtopos of the effective topos.

We study the structures (K ⊂ K r), where K is an ordered field and K r its real closure, in the language of ordered fields with an additional unary predicate for the subfield K. Two such structures (K ⊂ K r) and (L ⊂ L r) are not necessarily elementary equivalent when K and L are. But with some saturation assumption on K and L, then the two structures become equivalent, and we give a description of the complete theory.