Suppose that U and U′ are normal ultrafilters associated with some supercompact cardinal. How may we compare U and U′? In what ways are they similar, and in what ways are they different? Partial answers are given in [1], [2], [3], [5], [6], and [7]. In this paper, we continue this study.

In [6], Menas introduced a combinatorial principle χ(U) of normal ultrafilters U associated with supercompact cardinals, and showed that normal ultrafilters satisfying this property also satisfy a partition property. In [5], Kunen and Pelletier showed that this partition property for U does not imply χ(U). Using results from [3], we present a different method of finding such normal ultrafilters which satisfy the partition property but do not satisfy χ(U). Our method yields a large collection of such normal ultrafilters.

The idea of this paper is to approach linear orderings as generalized ordinals and to study how they are made from their initial segments. First we look at how the equality of two linear orderings can be expressed in terms of equality of their initial segments. Then we shall use similar methods to define functions by recursion with respect to the initial segment relation. Our method is based on the use of a game where smaller and smaller initial segments of linear orderings are considered. The length of the game is assumed to exceed that of the descending sequences of elements of the linear orderings considered. By use of such game-theoretical methods we can for example extend the recursive definitions of the operations of sum, product and exponentiation of ordinals in a unique and natural way for arbitrary linear orderings. Extensions coming from direct limits do not satisfy our game-theoretic requirements in general. We also show how our recursive definitions allow very simple constructions for fixed points of functions, giving rise to certain interesting linear orderings.

This paper deals with relation, cylindric and polyadic equality algebras. First of all it addresses a problem of B. Jónsson. He asked whether relation set algebras can be expanded by finitely many new operations in a “reasonable” way so that the class of these expansions would possess a finite equational base. The present paper gives a negative answer to this problem: Our main theorem states that whenever Rs+ is a class that consists of expansions of relation set algebras such that each operation of Rs+ is logical in Jónsson's sense, i.e., is the algebraic counterpart of some (derived) connective of first-order logic, then the equational theory of Rs+ has no finite axiom systems. Similar results are stated for the other classes mentioned above. As a corollary to this theorem we can solve a problem of Tarski and Givant [87], Namely, we claim that the valid formulas of certain languages cannot be axiomatized by a finite set of logical axiom schemes. At the same time we give a negative solution for a version of a problem of Henkin and Monk [74] (cf. also Monk [70] and Németi [89]).

Throughout we use the terminology, notation and results of Henkin, Monk, Tarski [71] and [85]. We also use results of Maddux [89a].

Notation. RA denotes the class of relation algebras, Rs denotes the class of relation set algebras and RRA is the class of representable relation algebras, i.e. the class of subdirect products of relation set algebras. The symbols RA, Rs and RRA abbreviate also the expressions relation algebra, relation set algebra and representable relation algebra, respectively.

For any class C of similar algebras EqC is the set of identities that hold in C, while Eq1C is the set of those identities in EqC that contain at most one variable symbol. (We note that Henkin et al. [85] uses the symbol EqC in another sense.)

We show how to extract effective bounds Φ for ⋀u1⋀v ≤ytu⋁wηG0-sentences which depend on u only (i.e. ⋀u⋀v ≤y, tu⋁w ≤η ΦuG0) from arithmetical proofs which use analytical assumptions of the form

(ϒ, δ, ρ, and τ are arbitrary finite types, η ≤ 2, G0 and F0 are quantifier-free, and s and t are closed terms). If τ ≤ 2, (*) can be weakened to

This is used to establish new conservation results about weak Konig's lemma. Applications to proofs in classical analysis, especially uniqueness proofs in approximation theory, will be given in subsequent papers.

We investigate various ways of introducing axioms for randomness in set theory. The results show that these axioms, when added to ZF, imply the failure of AC. But the axiom of extensionality plays an essential role in the derivation, and a deeper analysis may ultimately show that randomness is incompatible with extensionality.

In this paper we study differential fields of characteristic 0 (with perhaps additional structure) whose theory is superstable. Our main result is that such a differential field has no proper strongly normal extensions in the sense of Kolchin [K1]. This is an approximation to the conjecture that a superstable differential field is differentially closed (although we believe the full conjecture to be false). Our result improves earlier work of Michaux [Mi] who proved that a (plain) differential field with quantifier elimination has no proper Picard-Vessiot extension. Our result is a generalisation of Michaux's, due to the fact that any plain differential field K with quantifier elimination is ω-stable. (Any quantifier free type over K defines a unique type over K in the sense of dc(k), the differential closure of K, and as we mention below the theory of differentially closed fields is ω-stable.) The proof of our main result depends on (i) Kolchin's theory [K3] which states that any strongly normal extension L of an algebraically closed differential field K is generated over K by an element η of some algebraic group G defined over CK, the constants of K, where η satisfies some specific differential equations over K related to invariant differential forms on G (η is “G-primitive” over K), and (ii) the fact that a superstable field has a unique generic type which is semiregular.

In Parikh [71] it is shown that, if T is an r.e. consistent extension of Peano arithmetic PA, then, for each primitive recursive function g, there is a formula φ of PA such that

(In the following, Proof T(z, φ) and Prov T(φ) denote the metalinguistic assertions that z codes a proof of φ in T and that φ is provable in T respectively, where ProofT(z, ┌φ┐) and ProvT(┌φ┐) are the formalizations of Proof T(z,φ) and ProvT(φ) respectively in the language of PA, ┌φ┐ denotes the Gödel number of φ and ┌φ┐ denotes the corresponding numeral. Also, for typographical reasons, subscripts will not be made boldface.) If g is a rapidly increasing function, we express (1) by saying that ProvT(┌φ┐) has a much shorter proof modulo g than φ. Parikh's result is based on the fact that a suitable formula A(x), roughly asserting that (1) holds with x in place of φ, has only provable fixed points. In de Jongh and Montagna [89], this situation is generalized and investigated in a modal context. There, a characterization is given of arithmetical formulas arising from modal formulas of a suitable modal language which have only provable fixed points, and Parikh's result is obtained as a particular case.

This note deals with a proof-theoretic characterisation of certain complexity classes of functions in fragments of intuitionistic bounded arithmetic. In this Introduction we survey the background and state our main result.

We follow Buss [B1] and consider a language for arithmetic whose nonlogical symbols are 0, S (the successor operation Sx = x + 1), +, ·, ∣ ∣ (∣x∣ being the number of digits in the binary notation for x), rounded down to the nearest integer), # (x#y = 2∣x∣∣y∣) and ≤. We define 1 = S0, 2 = S1, s0x = 2x and s1x = 2x + 1. In Buss's approach the functions s0 and s1 play a special role. Notice that six is the number obtained from x by suffixing the digit i to its binary representation, and thus the natural numbers are generated from 0 by repeated applications of the operations s0 and s1. This means that they satisfy the induction scheme

Using the fact that is x with its last binary digit deleted, this can be stated more compactly in the following form, called by Buss the polynomial induction or PIND schema:

Buss defined a theory S2 consisting of a finite set BASIC of open axioms and the PIND-schema restricted to bounded formulas ϕ. The topic of bounded arithmetic is concerned with S2 and its fragments.

In [MP] Marker and Pillay showed that if X ⊂ Cn is constructible but (C, +, X) is not locally modular, then multiplication is definable in the structure (C, +,X). That result extended earlier results of Martin [M] and Rabinovich and Zil'ber [RZ]. Here we will examine additive reducts of R and Qp.

Definition. A subset X of Rn is called semialgebraic if it is definable in the structure (R, +,·). A subset X of Rn is called semilinear if it is definable in the structure (R, +, <,λr)r∈b, where λr is the function x ↦ rx [scalar multiplication by r].

Every semilinear set is a Boolean combination of sets of the form {: p () = 0} and {: q() > 0}, where p() and q() are linear polynomials.

Van den Dries asked the following question: if X is semialgebraic but not semilinear, can we define multiplication in (R, +, <,X)? This was answered negatively by Pillay, Scowcroft and Steinhorn.

Theorem 1.1 [PSS]. Suppose X ⊂ Rnis semialgebraic and X ⊂ Infor some bounded interval I. Then multiplication is not definable in (R, +, <,X,λr)r∈R.

In particular if X = · ∣ [0, l ]2, the graph of multiplication restricted to the unit interval, then X is not semilinear so we have a negative answer to van den Dries' question. Peterzil showed that this is the only restriction.

We show that the partial order of -sets under inclusion is elementarily definable with parameters in the semilattice of r.e. wtt-degrees. Using a result of E. Herrmann, we can deduce that this semilattice has an undecidable theory, thereby solving an open problem of P. Odifreddi.

Let be an admissible structure. A cPCd() class is the class of all models of a sentence of the form , where is an -r.e. set of relation symbols and Φ is an -r.e. set of formulas of ℒ∞,ω that are in . The main theorem is a generalization of the following: Let be a pure countable resolvable admissible structure such that is not Σ-elementarily embedded in HYP(). Then a class K of countable structures whose universes are sets of urelements is a cPCd() class if and only if for some Σ formula σ (with parameters from ), is in K if and only if is a countable structure with universe a set of urelements and σ, where , the smallest admissible set above relative to , is a generalization of HYP to structures with similarity type Σ over that is defined in this article. Here we just note that when Lα is admissible, HYPLα() is Lβ() for the least β ≥ α such that Lβ() is admissible, and so, in particular, that is just HYP() in the usual sense when has a finite similarity type.

The definition of is most naturally formulated using Adamson's notion of a +-admissible structure (1978). We prove a generalization from admissible to +-admissible structures of the well-known truncation lemma. That generalization is a key theorem applied in the proof of the generalized Spector-Gandy theorem.

In this paper we consider properties, related to model-completeness, of the theory of integrally closed commutative regular rings. We obtain the main theorem claiming that in a Boolean algebra B, the truth of a prenex Σn-formula whose parameters ai, partition B, can be determined by finitely many conditions built from the first entry of Tarski invariant T(ai)'s, n-characteristic D(n, ai)'s and the quantities S(ai, l) and S′(ai, l) for l < n. Then we derive two important theorems. One claims that for any Boolean algebras A and B, an embedding of A into B preserving D(n, a) for all a ϵ A is a Σn-extension. The other claims that the theory of n-separable Boolean algebras admits elimination of quantifiers in a simple definitional extension of the language of Boolean algebras. Finally we translate these results into the language of commutative regular rings.

Our theme is that not every interesting question in set theory is independent of ZFC. We give an example of a first order theory T with countable D(T) which cannot have a universal model at ℵ1; without CH; we prove in ZFC a covering theorem from the hypothesis of the existence of a universal model for some theory; and we prove—again in ZFC—that for a large class of cardinals there is no universal linear order (e.g. in every regular ). In fact, what we show is that if there is a universal linear order at a regular λ and its existence is not a result of a trivial cardinal arithmetical reason, then λ “resembles” ℵ1—a cardinal for which the consistency of having a universal order is known. As for singular cardinals, we show that for many singular cardinals, if they are not strong limits then they have no universal linear order. As a result of the nonexistence of a universal linear order, we show the nonexistence of universal models for all theories possessing the strict order property (for example, ordered fields and groups, Boolean algebras, p-adic rings and fields, partial orders, models of PA and so on).

The so-called Curry-Howard interpretation (Curry [1934], Curry and Feys [1958], Howard [1969], Tait [1965]) is known to provide a rather neat term-functional account of intuitionistic implication. Could one refine the interpretation to obtain an almost as good account of other neighbouring implications, including the so-called ‘resource’ implications (e.g. linear, relevant, etc.)?

We answer this question positively by demonstrating that just by working with side conditions on the rule of assertability conditions for the connective representing implication (‘→’) one can characterise those ‘resource’ logics. The idea stems from the realisation that whereas the elimination rule for conditionals (of which implication is a particular case) remains virtually unchanged no matter what kind of conditional one has (i.e. linear, relevant, intuitionistic, classical, etc., all have modus ponens), the corresponding introduction rule carries an element of vagueness which can be explored in the characterisation of several sorts of conditionals. The rule of →-introduction is classified as an ‘improper’ inference rule, to use a terminology from Prawitz [1965]. Now, the so-called improper rules leave room for manoeuvre as to how a particular logic can be obtained just by imposing conditions on the discharge of assumptions that would correspond to the particular logical discipline one is adopting (linear, relevant, ticket entailment, intuitionistic, classical, etc.). The side conditions can be ‘naturally’ imposed, given that a degree of ‘vagueness’ is introduced by the presentation of those improper inference rules, such as the rule of →-introduction:

which says: starting from assumption ‘A’, and arriving at ‘B’ via an unspecified number of steps, one can discharge the assumption and conclude that ‘A’ implies ‘B’.

Our purpose in this note is to study countable ℵ0-categorical structures whose theories are tree-decomposable in the sense of Baldwin and Shelah. The permutation group corresponding to such a structure can be decomposed in a canonical manner into simpler permutation groups in the same class. As an application of the analysis we show that these structures are finitely homogeneous.

The global function on finite systems is a new concept defined by Gurevich in [1] and discussed in [2] and [3]. In the last ten years this concept has become more and more useful in computer science and logic. Gurevich also pointed out the importance of global functionals on finite systems. In this paper we will give a brief introduction to the concepts of global functions and global functionals on finite systems.

In studying the natural number system N = 〈N, +,0〉 we often refer to its functions and functionals. There are a lot of books and papers in this area. Kleene in [4] gave a detailed introduction to the recursive functions of N. The functionals of N are normally very difficult to compute because here we need to tell the machine what the input function is, which is not very easy to do. In developing the theory of finite systems the functions and functionals are also very useful. For computing the functionals in finite systems we can take the entire graph of a function as the input, which is not possible in N. We will discuss recursive functions and functionals for finite systems. The definitions of recursive functions are very similar to the case in N, but we will have a very different situation. In N the number of elements is infinite. The number of all possible functions from N to N is the continuum. In a finite system the number of all possible functions is finite. It seems that there is no necessity to define the global functions.

We consider arbitrary stratified languages. We study structures which satisfy the same stratified sentences and we obtain an extension of Keisler's Isomorphism Theorem to this situation. Then we consider operations which are definable by a stratified formula and modify the ‘type’ of their argument by one; we prove that for such an operation F the sentence c = F(c) and the scheme φ(c) ↔ (F(c)), where ↔(x) varies among all the stratified formulas with no variable other than x free, imply the same stratified {c}-sentences.

In this paper I give a characterization of the closed fragment of the provability logic of I Δ0 + EXP with a propositional constant for EXP. In three appendices many details on arithmetization are provided.

In §3 we construct a universal, ℵ0-categorical recursively presented partial order with greatest lower bound operator. This gives us the unique structure which embeds every countable lower semilattice. In §§5 and 6 we investigate the recursive and recursively enumerable substructures of this structure, in particular finding a suitable definition for the simple-maximal hierarchy and giving an example of an infinite recursively enumerable substructure which does not contain any infinite recursive substructure.

The idea of looking at the lattice of recursively enumerable substructures of some recursive algebraic structure was introduced by Metakides and Nerode in [5], and since then many different kinds of algebraic structures have been studied in this way, including vector spaces, Boolean algebras, groups, algebraically closed fields, and equivalence relations. Since different algebraic structures have different recursion theoretic properties, one natural question is whether an algebraic structure with relatively little structure (such as a partial order or an equivalence relation) exhibits behavior more like classical recursion theory than one with more structure (such as vector spaces or algebraically closed fields).

In [6] and [7], Metakides and Remmel studied recursion theory on orderings, and, as they point out in [6], orderings differ from most other algebraic structures in that the algebraic closure operation on orderings is trivial; but this does not present a problem for them, given the questions they explore. Moreover, they take an approach of proving general theorems which can then be applied to specific orderings. Our tack is different, although also well-established (see, for example, [3]), in which a “largest” structure is defined (in §3) which corresponds to the natural numbers in classical recursion theory. In order to distinguish substructures from subsets, a function symbol is added, namely greatest lower bound. The greatest lower bound function is fundamental to the study of orderings and occurs naturally in many of them, and thus is an appropriate addition to the theory of orderings. In §4 we redefine the concepts of simple and maximal in a manner appropriate to this structure, and prove several existence theorems.

We generalize to theories of modules (complete or not) a result of U. Felgner stating that a complete theory of abelian groups is a Horn theory if and only if it is closed under products. To prove this we show that a reduced product of modules ΠFMi (i ϵ I) is elementarily equivalent to a direct product of ultraproducts of the modules Mi(i ϵ I).