A reducibility ≤p is a procedure whereby a set A can be computed from a set B. The most general and most extensively studied reducibility is Turing reducibility (≤T). However, when one analyzes effectiveness considerations in classical mathematics, one often discovers that the relevant reducibilities are stronger (i.e., more restrictive) than ≤T. To illustrate, in combinatorial group theory we find that the word problem is many-one reducible to the conjugacy problem, and that word problems occur in each r.e. truth table (tt-) degree (see, for example, Miller [17]).

In the present paper we are concerned with another strong reducibility: weak truth table (wtt-) reducibility. Here the reader should recall that A ≤wtt, β means that there is a procedure Φ and a recursive function φ such that Φ(β) = A and for all x, the u(Φ(β; X)) < φ (x). That is, the amount of information used in the computation is bounded by φ. The critical difference between truth table and weak truth table reducibilities is that for tt we will at once be “given the whole table.” Thus if Δ is a tt-procedure and δ is its use, then for all x and all strings σ of length δ(x) we can figure out Δ(σ; x). On the other hand if Δ is merely a wtt-procedure it may be that for some string σ, Δ(σ; x)↓, whilst for another string μ of the same length it may be that Δ{μ; x) ↑. We remark that wtt-reducibility arises very naturally both in effective algebra and in the structure of the r.e. T-degrees R. The reader should see, for instance, Downey and Remmel [3], where it is shown that the complexity of r.e. bases of an r.e. vector space V is characterised precisely by the wtt-degrees below V, and also Ladner and Sasso [14] or Downey [1], where the wtt-degrees are used to investigate cupping and capping in R.

We give a constructive proof of McNaughton's theorem stating that every piecewise linear function with integral coefficients is representable by some sentence in the infinite-valued calculus of Lukasiewicz. For the proof we only use Minkowski's convex body theorem and the rudiments of piecewise linear topology.

We obtain a p-adic semilinear cell decomposition theorem using methods developed by Denef in [Journal für die Reine und Angewandte Mathematik, vol. 369 (1986), pp. 154–166]. We also prove that any set definable with quantifiers in (0,1, +, —, λq, Pn){n∈ℕ,q∈ℚp} may be defined without quantifiers, where λq is scalar multiplication by q and Pn is a unary predicate which denotes the nonzero nth powers in the p-adic field ℚp. Such a set is called a p-adic semilinear set in this paper. Some further considerations are discussed in the last section.

We settle a question in the literature about degrees of models of true arithmetic and upper bounds for the arithmetic sets. We prove that there is a model of true arithmetic whose degree is not a uniform upper bound for the arithmetic sets. The proof involves two forcing constructions.

We do a quantitative analysis of modal logic. For example, for each Kripke structure M, we study the least ordinal μ such that for each state of M, the beliefs up to level μ characterize the agents' beliefs (that is, there is only one way to extend these beliefs to higher levels). As another example, we show the equivalence of three conditions, that on the face of it look quite different, for what it means to say that the agents' beliefs have a countable description, or putting it another way, have a “countable amount of information”. The first condition says that the beliefs of the agents are those at a state of a countable Kripke structure. The second condition says that the beliefs of the agents can be described in an infinitary language, where conjunctions of arbitrary countable sets of formulas are allowed. The third condition says that countably many levels of belief are sufficient to capture all of the uncertainty of the agents (along with a technical condition). The fact that all of these conditions are equivalent shows the robustness of the concept of the agents' beliefs having a “countable description”.

In [A1]–[A4] there is an abstract description, in terms of “α-systems”, of the object of a nested priority construction, and there is a metatheorem listing conditions which guarantee the success of the construction. The metatheorem has different versions, with later versions becoming simpler and more general. The information needed to meet the requirements is enumerated by a function, where α is an arbitrary recursive ordinal. All other objects associated with the α-system are r.e. In the basic metatheorem, the requirements must all be at level α, although there is a special result, for limit α, which allows one requirement at each level in an increasing sequence with limit α. There are other abstract descriptions of nested priority arguments in [K1], [K2], [K3], and [L-L].

A typical use of α-systems is that in [B], where it is shown that, for a relation R on a recursive structure , under suitable assumptions, if R does not have a “recursive Σα” definition in , then there is a recursive isomorphic copy of in which the image of R is not . There is no difficulty in generalizing this to make each of infinitely many relations not , when the assumptions are uniformly satisfied and each relation has no recursive Σα definition. However, the metatheorems for α-systems do not apply to the situation where we have several (even two) relations Ri, which do not have recursive Σβi definitions and whose images we wish to make not for distinct βi. An even more basic problem is to construct a recursive copy of a recursive structure via an isomorphism whose restriction to various sets Ui, is not If βi, = α for all i, then we can use an (α + 1)-system; otherwise, the old metatheorems do not apply.

We consider the infinite game where player ONE chooses terms of a strictly increasing sequence of first category subsets of a space and TWO chooses nowhere dense sets. If after ω innings TWO's nowhere dense sets cover ONE's first category sets, then TWO wins. We prove a theorem which implies for the real line: If TWO has a winning strategy which depends on the most recent n moves of ONE only, then TWO has a winning strategy depending on the most recent 3 moves of ONE (Corollary 3). Our results give some new information concerning Problem 1 of [S1] and clarifies some of the results in [B-J-S] and in [S1].

We study how equivalent nonisomorphic models an unsuperstable theory can have. We measure the equivalence by Ehrenfeucht-Fraisse games. This paper continues the work started in [HT].

Throughout this paper denotes a nonprincipal ultrafilter and ℐ denotes the dual ideal. ℙ(ℐ) is the poset of all partial functions p: ω → 2 such that dom(p) ∈ ℐ. In [2], Grigorieff proved that ω1 is preserved in the corresponding generic extension if and only if is a P-point. Later, when Shelah introduced the notion of a proper poset, many people observed that if is a P-point, then ℙ(ℐ) is proper. One way of proving this is to show that player II has a winning strategy in the game for ℙ(ℐ) (see [3, p. 91].)

The notion of an Axiom A poset was introduced by Baumgartner [1]. If a poset satisfies Axiom A, then player II has a winning strategy in the game for ℙ, and thus, ℙ is proper. Indeed most of the naturally occurring proper posets satisfy Axiom A (e.g., Mathias's poset and Laver's poset). Thus, it is natural to ask whether or not ℙ(ℐ) satisfies Axiom A. The main result of this paper is a negative answer to this question. We will prove this by introducing another game and showing that is a P-point if and only if the corresponding game is undetermined. We will then show that if ℙ(ℐ) satisfied Axiom A, then player II would have a winning strategy in the corresponding game .

We let [X]<ω = {s ⊆ X∣ ∣s∣ < ω}. We let Seq(X) denote the set of finite sequences of elements of X. If s = 〈x0, …, xn〉 ∈ Seq(X) and y ∈ X, then s * 〈y〉 = 〈x0,…,xn,y〉 ∈ Seq(X).

Let T be a complete O-minimal theory in a language L. We first give an elementary proof of the result (due to Marker and Steinhorn) that all types over Dedekind complete models of T are definable. Let L* be L together with a unary predicate P. Let T* be the L*-theory of all pairs (N, M), where M is a Dedekind complete model of T and N is an ⅼMⅼ+-saturated elementary extension of N (and M is the interpretation of P). Using the definability of types result, we show that T* is complete and we give a simple set of axioms for T*. We also show that for every L*-formula ϕ(x) there is an L-formula ψ(x) such that T* ⊢ (∀x)(P(x) → (ϕ(x) ↔ ψ(x)). This yields the following result:

Let M be a Dedekind complete model of T. Let ϕ(x, y) be an L-formula where l(y) – k. Let X = {X ⊂ Mk: for some a in an elementary extension N of M, X = ϕ(a, y)N ∩ Mk}. Then there is a formula ψ(y, z) of L such that X = {ψ(y, b)M: b in M}.

The Genericity Conjecture, as stated in Beller-Jensen-Welch [1], is the following:

(*) If O# ∉ L[R], R ⊆ ω, then R is generic over L.

We must be precise about what is meant by “generic”.

Definition (Stated in Class Theory). A generic extension of an inner model M is an inner model M[G] such that for some forcing notion ⊆ M:

(a) 〈M, 〉 is amenable and ⊩ is 〈M, 〉-definable for sentences.

(b) G ⊆ is compatible, closed upwards, and intersects every 〈M, 〉-definable dense D ⊆ .

A set x is generic over M if it is an element of a generic extension of M. And x is strictly generic over M if M[x] is a generic extension of M.

Though the above definition quantifies over classes, in the special case where M = L and O# exists, these notions are in fact first order, as all L-amenable classes are definable over L[O#]. From now on assume that O# exists.

Theorem A. The Genericity Conjecture is false.

The proof is based upon the fact that every real generic over L obeys a certain definability property, expressed as follows.

Fact. If R is generic over L, then for some L-amenable class A, Sat〈L, A〉 is not definable over 〈L[R],A〉, where Sat〈L,A〉 is the canonical satisfaction predicate for 〈L,A〉.

The modal predicate logic of provability identifies the “□” of modal logic with the “Bew” of proof theory, so that, where “Bew” is a formula representing, in the usual way, provability in a consistent, recursively axiomatized theory Γ extending Peano arithmetic (PA), an interpretation of a language for the modal predicate calculus is a map * which associates with each modal formula an arithmetical formula with the same free variables which commutes with the Boolean connectives and the quantifiers and which sets (□ϕ)* equal to Bew(⌈ϕ*⌉). Where Δ is an extension of PA (all the theories we discuss will be extensions of PA), MPL(Δ) will be the set of modal formulas ϕ such that, for every interpretation *, ϕ* is a theorem of Δ. Most of what is currently known about the modal predicate logic of provability consists in demonstrations that MPL(Δ) must be computationally highly complex. Thus Vardanyan [11] shows that, provided that Δ is 1-consistent and recursively axiomatizable, MPL(Δ) will be complete , and Boolos and McGee [5] show that MPL({true arithmetical sentences}) is complete in {true arithmetical sentences}. All of these results take as their starting point Artemov's demonstration in [1] that {true arithmetical sentences} is 1-reducible to MPL({true arithmetical sentences}).

The aim here is to consolidate these results by providing a general theorem which yields all the other results as special cases. These results provide a striking contrast with the situation in modal sentential logic (MSL); according to fundamental results of Solovay [8], provided Γ does not entail any falsehoods, MSL({true arithmetical sentences}) and MSL(PA) (which is the same as MSL(Γ)) are both decidable.

A stationary subset S of a regular uncountable cardinal κreflects fully at regular cardinals if for every stationary set T ⊆ κ of higher order consisting of regular cardinals there exists an α Є T such that S ∩ α is a stationary subset of α. Full Reflection states that every stationary set reflects fully at regular cardinals. We will prove that under a slightly weaker assumption than κ having the Mitchell order κ++ it is consistent that Full Reflection holds at every λ ≤ κ and κ is measurable.

We construct a weak second-order theory of arithmetic which includes Weak König's Lemma (WKL) for trees defined by bounded formulae. The provably total functions (with -graphs) of this theory are the polynomial time computable functions. It is shown that the first-order strength of this version of WKL is exactly that of the scheme of collection for bounded formulae.

The following fairly elementary result seems to raise possibilities for the study of countable models of a theory in a countable language. For the terminology of model theory we refer to [CK].

Let L be a countable first-order language. Let L′ be the relational language having, for each formula φ of L and each sequence υ1,…,υn of variables including the free variables of φ, an n-ary relation symbol Pφ. For any L-structure and any formula Ψ(υ) of L, we define the Ψ-fraction of to be the L′-structure Ψ whose universe consists of those elements of satisfying Ψ(υ) and whose relations {Rφ}φϵL are defined by letting a1,…,an satisfy Rφ in Ψ if, and only if, a1,…, an satisfy φ in .

An L-elementary class means the class of all L-structures satisfying each of some set of sentences of L. The countable part of an L-elementary class K means the class of all countable L-structures from K.

Theorem. Let K be an L-elementary class and let Ψ(υ) be a formula of L. Then the class of countable Ψ-fractions of structures in K is the countable part of some L′-elementary class.

Comment. By the downward Löwenheim-Skolem theorem, the countable Ψ-fractions of structures in K are the same as the Ψ-fractions of countable structures in K.

Proof. We give a set Σ′ of L′-sentences whose countable models are exactly the countable Ψ-fractions of structures in K.

Laver [L] and others [G-S] have shown how to make the supercompactness or strongness of κ indestructible by a wide class of forcing notions. We show, alternatively, how to make these properties fragile. Specifically, we prove that it is relatively consistent that any forcing which preserves κ<κ and κ+, but not P(κ), destroys the measurability of κ, even if κ is initially supercompact, strong, or if I1(κ) holds. Obtained as an application of some general lifting theorems, this result is an “inner model” type of theorem proved instead by forcing.

Let X be a set, and let be the superstructure of X, where X0 = X and is the power set of X) for n ∈ ω. The set X is called a flat set if and only if for each x ∈ X, and x ∩ ŷ = ø for x, y ∈ X such that x ≠ y. where is the superstructure of y. In this article, it is shown that there exists a bijection of any nonempty set onto a flat set. Also, if is an ultrapower of (generated by any infinite set I and any nonprincipal ultrafilter on I), it is shown that is a nonstandard model of X: i.e., the Transfer Principle holds for and , if X is a flat set. Indeed, it is obvious that is not a nonstandard model of X when X is an infinite ordinal number. The construction of flat sets only requires the ZF axioms of set theory. Therefore, the assumption that X is a set of individuals (i.e., x ≠ ϕ and a ∈ x does not hold for x ∈ X and for any element a) is not needed for to be a nonstandard model of X.

We give new proofs of both incompleteness theorems. We do not use the diagonalization lemma, but work with some quickly growing functions instead.

One of the purposes of this paper is to prove a partial Schur-Zassenhaus Theorem for groups of finite Morley rank.

Theorem 2. Let G be a solvable group of finite Morley rank. Let π be a set of primes, and let H ⊲ G a normal π-Hall subgroup. Then H has a complement in G.

This result has been proved in [1] with the additional assumption that G is connected, and thought to be generalized in [2] by the authors of the present article. Unfortunately in the last section of the latter paper there is an irrepairable mistake. Here we give a new proof of the Schur-Zassenhaus Theorem using the results of [2] up to the last section and a new result that we are going to state below.

The second author has shown in [11] that a nilpotent ω-stable group is the central product of a divisible subgroup and a subgroup of bounded exponent, generalizing a well-known result of Angus Macintyre about abelian groups [8]. One could ask a similar question for solvable groups: are they a product of two subgroups, one divisible, one of bounded exponent? One is allowed to be hopeful because of the well-known decomposition of the connected solvable algebraic groups over algebraically closed fields as the product of the unipotent radical and a torus.

We shall show that certain natural and interesting intervals in the lattice of varieties of representable relation algebras embed the lattice of all subsets of the natural numbers, and therefore must have a very complicated lattice-theoretic structure.