Since about 1955, recursive, (hyper-)arithmetic and well-orderings have been investigated by many authors. Now we shall generally consider analytic well-orderings and compare their representation-capacities for ordinals.

Let K be a class of predicates. Then, R is called a Kwell-ordering if R is a binary relation of natural numbers belonging to the class K and satisfies the following conditions:

(i) R(x, y) ∧ R(y, x) → x = y;

(ii) x, y ∈ D(R) → R(x, y) ∨ R(y, x), where D(R) is the domain of R, that is to say, the set {x ∣ (∃y)[R(x, y) ∨ R(y, x)]};

(iii) R(x, y) ∧ R(y, z) → R(x, z);

(iv) .

In this paper it will be shown that the three primitive symbols used by Lewis are independent in each one of his systems S1–S6.

Lemma I. “·” is not definable in terms of “∼” and “◊” in any one of the systems S1–S6.

Proof. This is obvious, since no binary operation can be defined by unary operations alone.

Lemma II. “∼” is not definable in terms of “˙” and “◊” in any one of the systems S1–S5.

In this paper the symbolism, definitions and results of my two papers, namely A test for the existence of tautologies according to many-valued truth-tables (hereafter referred to as ET) and Note on truth-tables (hereafter referred to as TT) will be presupposed.

The problem whether two arbitrary truth-tables with finite number of elements are equal or not was reduced in TT to the question of equality of two identical tables in which different elements have been designated. However, no general method for testing the equality of truth-tables was given there, although some cases were discussed for which such a method is available. In the present paper we shall describe a general method using the considerations of ET and TT.

It has been proved by Pepis that any formula of the first-order predicate calculus is equivalent (in respect of being satisfiable) to another with a prefix of the form

containing a single existential quantifier. In this paper, we shall improve this theorem in the like manner as the Ackermann and the Gödel reduction theorems have been improved in the preceding papers of the same main title. More explicitly, we shall prove the

Theorem 1. To any given first-order formula it is possible to construct an equivalent one with a prefix of the form (1) and a matrix containing no other predicate variable than a single binary one.

An analogous theorem, but producing a prefix of the form

has been proved in the meantime by Surányi; some modifications in the proof, suggested by Kalmár, led to the above form.

In a recent paper [3] H. Gaifman investigated some model theoretic consequences of Matijasevič's theorem [5], and posed some further problems which naturally arise. We provide here partial answers to two of these problems, the results having been previously announced in the postscript of [3].

Firstly, it is shown in [3] that if M1 and M2 are models of the Peano axioms P and M1 ⊆ M2, then M1 is closed under the recursive functions of M2. The converse of this statement is false. Moreover, Gaifman asks: Is every initial segment of a model M of P which is closed under the recursive functions of M (or the ∑n-definable functions) also a model of P? We show that this is false and our method gives, en route, another proof of a theorem of Rabin [7] stating the P is not implied by any consistent set of ∑n sentences for any n.

Secondly, we partially answer a question posed on p. 129 of [3] by proving (some-what more than) every countable nonstandard model of P has an end extension in which a diophantine equation, not solvable in the original model, has a solution. We can, in fact, take the new model to be isomorphic to the original one. This generalises (apart from the countability restriction) a theorem of Rabin [6].

There has been increasing interest in intuitionistic methods over the years. Still, there has been relatively little work on intuitionistic set theory, and most of that has been on intuitionistic ZF. This investigation is about intuitionistic admissibility and theories of similar strength.

There are several more particular goals for this paper. One is just to get some more Kripke models of various set theories out there. Those papers that have dealt with IZF usually were more proof-theoretic in nature, and did not provide models. Furthermore, the inspirations for many of the constructions here are classical forcing arguments. Although the correspondence between the forcing and the Kripke constructions is not made tight, the relationship between these two methods is of interest (see [6] for instance) and some examples, even if only suggestive, should help us better understand the relationship between forcing and Kripke constructions. Along different lines, the subject of least and greatest fixed points of inductive definitions, while of interest to computer scientists, has yet to be studied constructively, and probably holds some surprises. Admissibility is of course the proper set-theoretic context for this study. Finally, while most of the classical material referred to here has long been standard, some of it has not been well codified and may even be unknown, so along the way we'll even fill in a gap in the classical literature.

The next section develops the basics of IKP, including some remarks on fixed points of inductive definitions.

The Sikorski Extension Theorem [6] states that, for any Boolean algebra A and any complete Boolean algebra B, any homomorphism of a subalgebra of A into B can be extended to the whole of A. That is,

Inj: Any complete Boolean algebra is injective (in the category of Boolean algebras).

The proof of Inj uses the axiom of choice (AC); thus the implication AC → Inj can be proved in Zermelo-Fraenkel set theory (ZF). On the other hand, the Boolean prime ideal theorem

BPI: Every Boolean algebra contains a prime ideal (or, equivalently, an ultrafilter)

may be equivalently stated as:

The two element Boolean algebra 2 is injective,

and so the implication Inj → BPI can be proved in ZF.

In [3], Luxemburg surmises that this last implication cannot be reversed in ZF. It is the main purpose of this paper to show that this surmise is correct. We shall do this by showing that Inj implies that BPI holds in every Boolean extension of the universe of sets, and then invoking a recent result of Monro [5] to the effect that BPI does not yield this conclusion.

The purpose of this paper is to construct a non-finitary system of logic S, based on the theory of types, in which classical quantification theory holds without restriction and the axiom of reducibility holds with such slight restriction as is perhaps unlikely to interfere with the construction of classical mathematics. The system will be shown to be consistent.

For purposes of orientation, a rough description of the ontology of the system will first be given. The ‘individuals’ are expressions built up in the following way: ‘0’ is an individual, and if ϕ and ψ are individuals, so is ˹*ϕψ˺. The usual apparatus of truth functional connectives and quantifiers is employed, and one primitive four place predicate ‘S’. ˹Sϕψχω˺ says that ϕ is the result of writing ψ for all occurrences of χ in ω, where ϕ, ψ, χ and ω may be formulae, individuals, or members of a wider category called ‘expressions’ which includes individuals, formulae, and other things as well. (At first sight this looks like a confusion of use and mention, but the formation rules and rules of inference are so phrased that no ambiguity arises.)

The individuals are divided into types concentrically, so that type 0 has them all as members and type n includes type m for m > n. There is no highest type, but we shall find it convenient from the point of view of exposition to construct an infinity of ‘auxiliary’ systems Q(k), restricted to types ≦ k, before presenting the system S.

The study of effectiveness in classical mathematics is rapidly expanding, through recent research in algebra, topology, model theory, and functional analysis. Well-known contributors are Barwise (Wisconsin), Crossley (Monash), Dekker (Rutgers), Ershoff (Novosibirsk), Feferman (Stanford), Harrington (Berkeley), Mal′cev (Novosibirsk), Morley (Cornell), Nerode (Cornell), Rabin (Hebrew University), Shore (Cornell). Further interesting work is due to Kalantari (University of California, Santa Barbara), Metakides (Rochester), Millar (Wisconsin), Remmel (University of California, San Diego), Nurtazin (Novosibirsk). Areas investigated include enumerated algebras, models of complete theories, vector spaces, fields, orderings, Hilbert spaces, and boolean algebras.

We investigate the effective content of the structure theory of p-groups. Recall that a p-group is a torsion abelian group in which the (finite) order of each element is some power of a fixed prime p. (In the sequel, “group” = “additively written abelian group”.)

The structure theory of p-groups is based on the two elementary notions of order and height. Recall that the order of x is the least integer n such that nx = 0. The height of x is the number of times p divides x, that is, the least n such that x = pny for some y in the group but x ≠ pn+1y for any y. If for each n ∈ ω there is a “pnth-root” yn, so that x = pnyn, then we say that x has infinite height. In 1923, Prüfer related the two notions as criteria for direct sum decomposition, proving

Theorem. Every group of bounded order is a direct sum of cyclic groups, and

Theorem. Every countable primary group with no (nonzero) elements of infinite height is a direct sum of cyclic groups.

This paper looks at linear orders in the following way. A preordering is given, which is linear and recursively enumerable. By performing the natural identification, one obtains a linear order for which equality is not necessarily recursive. A format similar to Metakides and Nerode's [3] is used to study these linear orders. In effective studies of linear orders thus far, the law of antisymmetry (x ≦ y ∧ y ≦ x ⇒ y) has been assumed, so that if the order relation x ≦ y is r.e. then x < y is also r.e. Here the assumption is dropped, so that x < y may not be r.e. and the equality relation may not be recursive; the possibility that equality is not recursive leads to new twists which sometimes lead to negative results.

Reported here are some interesting preliminary results with simple proofs, which are obtained if one looks at these objects with a view to doing recursion theory in the style of Metakides and Nerode. (This style, set in [3], is seen in many subsequent papers by Metakides and Nerode, Kalantari, Remmel, Retzlaff, Shore, and others, e.g. [1], [4], [6], [7], [8], [11]. In a sequel, further investigations will be reported which look at r.e. presented linear orders in this fashion and in the context of Rosenstein's comprehensive work [10].

Obviously, only countable linear orders are under consideration here. For recursion-theoretic notation and terminology see Rogers [9].

(Xα: α < ω2) ⊂ ℘(ω1) is a strong chain in ℘(ω1)/Fin if and only if Xβ – Xα is finite and Xα – Xβ is uncountable for each β < α < ω1. We show that it is consistent that a strong chain in ℘(ω1) exists. On the other hand we show that it is consistent that there is a strongly almost-disjoint family in ℘(ω1) but no strong chain exists: is used to construct a c.c.c forcing that adds a strong chain and Chang's Conjecture to prove that there is no strong chain.

We prove that the Wadge order on the Borel subsets of the Scott domain is not a well-quasi-order, and that this feature even occurs among the sets of Borel rank at most 2. For this purpose, a specific class of countable 2-colored posets $\mathbb{P}_{emb} $ equipped with the order induced by homomorphisms is embedded into the Wadge order on the $\Delta _2^0 $-degrees of the Scott domain. We then show that $\mathbb{P}_{emb} $ admits both infinite strictly decreasing chains and infinite antichains with respect to this notion of comparison, which therefore transfers to the Wadge order on the $\Delta _2^0 $-degrees of the Scott domain.

If a uniform ultrafilter U over an uncountable cardinal κ is not outright countably complete, probably the next best thing is that it have a finest partition: a master function f:κ → ω with ƒ −({n}) ∉ U for each n ϵ ω such that for any g: κ → κ, either (a) it is one-to-one on a set in U, or (b) it factors through ƒ (mod U), i.e. for some function h, {α < κ ∣ h(f(α)) = g(α)} ϵ U. In this paper, it is shown that recent contructions of irregular ultrafilters over ω 1 can be amplified to incorporate a finest partition.

Henceforth, let us assume that all ultrafilters are uniform.

There has been an extensive study of substantial hypotheses, which are nonetheless weaker than countable completeness, on ultrafilters over uncountable cardinals. To survey some results and to establish a context, let us first recall the Rudin-Keisler (RK) ordering on ultrafilters: If U i is an ultrafilter over I i i for i = 1, 2, then U 1 ≤RK U 2 iff there is a projecting function Ψ:I 2 → I 1 such that U 1 = Ψ *(U 2) = {X ⊆, I 1∣ Ψ −1(X) ϵ U 2}· U 1, =RK U 2 iff U 1, ≤RK and U 2 and U 2≤RK U 1; and U 1<RK U 2 iff U 1≤RK U 2 yet U 1 ≠RK U 2. In terms of this ordering, if an ultrafilter U has a finest partition ƒ, then ƒ *(U) over ω is maximum amongst all RK predecessors of U: for any g:κ → κ, if g *(U) <RK U, then g is not one-to-one on a set in U, so since g factors through ƒ with some h,g *(U) = h *(ƒ *(U)). Say now that an ultrafilter U over κ > ω is indecomposable iff whenever ω < λ < κ, there is no V ≤RK U such that V is a (uniform) ultrafilter over λ.

Let X be a standard Borel space (i.e., a Polish space with the associated Borel structure), and let E be a countable Borel equivalence relation on X, i.e., a Borel equivalence relation E for which every equivalence class [x]E is countable. By a result of Feldman-Moore [FM], E is induced by the orbits of a Borel action of a countable group G on X.

The structure of general countable Borel equivalence relations is very little understood. However, a lot is known for the particularly important subclass consisting of hyperfinite relations. A countable Borel equivalence relation is called hyperfinite if it is induced by a Borel ℤ-action, i.e., by the orbits of a single Borel automorphism. Such relations are studied and classified in [DJK] (see also the references contained therein). It is shown in Ornstein-Weiss [OW] and Connes-Feldman-Weiss [CFW] that for every Borel equivalence relation E induced by a Borel action of a countable amenable group G on X and for every (Borel) probability measure μ on X, there is a Borel invariant set Y ⊆ X with μ(Y) = 1 such that E ↾ Y (= the restriction of E to Y) is hyperfinite. (Recall that a countable group G is amenable if it carries a finitely additive translation invariant probability measure defined on all its subsets.) Motivated by this result, Weiss [W2] raised the question of whether every E induced by a Borel action of a countable amenable group is hyperfinite. Later on Weiss (personal communication) showed that this is true for G = ℤn. However, the problem is still open even for abelian G. Our main purpose here is to provide a weaker affirmative answer for general amenable G (and more—see below). We need a definition first. Given two standard Borel spaces X, Y, a universally measurable isomorphism between X and Y is a bijection ƒ: X → Y such that both ƒ, ƒ-1 are universally measurable. (As usual, a map g : Z → W, with Z and W standard Borel spaces, is called universally measurable if it is μ-measurable for every probability measure μ on Z.) Notice now that to assert that a countable Borel equivalence relation on X is hyperfinite is trivially equivalent to saying that there is a standard Borel space Y and a hyperfinite Borel equivalence relation F on Y, which is Borel isomorphic to E, i.e., there is a Borel bijection ƒ: X → Y with xEy ⇔ ƒ(x)F ƒ(y). We have the following theorem.

We work throughout in a finite relational language L. This paper is built on [2] and [3]. We repeat some of the basic notions and results from these papers for the convenience of the reader but familiarity with the setup in the first few sections of [3] is needed to read this paper. Spencer and Shelah [6] constructed for each irrational α between 0 and 1 the theory Tα as the almost sure theory of random graphs with edge probability n−α. In [2] we proved that this was the same theory as the theory Tα built by constructing a generic model in [3]. In this paper we explore some of the more subtle model theoretic properties of this theory. We show that Tα has the dimensional order property and does not have the finite cover property.

We work in the framework of [3] so probability theory is not needed in this paper. This choice allows us to consider a wider class of theories than just the Tα. The basic facts cited from [3] were due to Hrushovski [4]; a full bibliography is in [3]. For general background in stability theory see [1] or [5].

We work at three levels of generality. The first is given by an axiomatic framework in Context 1.10. Section 2 is carried out in this generality. The main family of examples for this context is described in Example 1.3. Sections 3 and 4 depend on a function δ assigning a real number to each finite L-structure as in these examples. Some of the constructions in Section 3 (labeled at the time) use heavily the restriction of the class of examples to graphs. The first author acknowledges useful discussions on this paper with Sergei Starchenko.

A computer handles λ-terms more easily if these are translated into combinatory terms. This translation process is called bracket abstraction. The simplest abstraction algorithm—the (fab) algorithm of Curry (see Curry and Feys [6])—is lengthy to implement and produces combinatory terms that increase rapidly in length as the number of variables to be abstracted increases.

There are several ways in which these problems can be alleviated:

(1) A change in order of the clauses in the algorithm so that (f) is performed as a last resort.

(2) The use of an extra clause (c), appropriate to βη reduction.

(3) The introduction of a finite number of extra combinators.

The original 1924 form of bracket abstraction of Schönfinkel [17], which in fact predates λ-calculus, uses all three of these techniques; all are also mentioned in Curry and Feys [6].

A technique employed by many computing scientists (Turner [20], Peyton Jones [16], Oberhauser [15]) is to use the (fab) algorithm followed by certain “optimizations” or simplifications involving extra combinators and sometimes special cases of (c).

Another is either to allow a fixed infinite set of (super-) combinators (Abdali [1], Kennaway and Sleep [10], Krishnamurthy [12], Tonino [19]) or to allow new combinators to be defined one by one during the abstraction process (Hughes [7] and [8]).

A final method encodes the variables to be abstracted as an n-tuple—this requires only a finite number of combinators (Curien [5], Statman [18]).

Eine auf neue Art und mit neuen symbolischen Mitteln durchgeführte Analyse des Prozesses der Prädikation ergibt, daβ das für die logisch-mengentheoretischen Paradoxa nach Art des Russellschen essentielle Moment weder das der Selbstprädikation als solches ist noch irgendetwas, was man anderweitig bisher für sie verantwortlich gemacht hat, sondern vielmehr der Umstand, daβ der Algorithmus der Prädikation ebensowohl wie zu einem natürlichen Abschluβ auch auf einen unendlichen Regreβ führen kann. Damit werden die genannten Paradoxa als den bekannten erledigten der mathematischen unendlichen Prozesse wesentlich gleichartig und der gleichen Behandlung zugänglich erkannt. Das neue Kalkülmittel der limitierten Variablen bewirkt – ohne irgendwelche metalogische Regulative ad hoc – die automatische Sicherung des logischen Kalküls gegen die früheren Verführungen.

This paper deals primarily with countable, simple, connected graphs and the following two conditions which are trivially satisfied if the graphs are finite:

(a) there is an edge-recognition algorithm, i.e., an effective procedure which enables us, given two distinct vertices, to decide whether they are adjacent,

(b) there is a shortest path algorithm, i.e., an effective procedure which enables us, given two distinct vertices, to find a minimal path joining them.

A graph G = ‹ν, η› with ν as set of vertices and η as set of edges is an α-graph if it satisfies (a); it is an ω-graph if it satisfies (b). G is called r.e. (isolic) if the sets ν and η are r.e. (isolated). While every ω-graph is an α-graph, the converse is false, even if G is r.e. or isolic. Several basic properties of finite graphs do not generalize to ω-graphs. For example, an ω-tree with more than one vertex need not have two pendant vertices, but may have only one or none, since it may be a 1-way or 2-way infinite path. Nevertheless, some elementary propositions for finite graphs G = ‹ν, η› with n = card(ν), e = card(η), do generalize to isolic ω-graphs, e.g., n − 1 ≤ e ≤ n(n − 1)/2. Let N and E be the recursive equivalence types of ν and η respectively. Then we have for an isolic α-tree G = ‹ν, η›: N = E + 1 iff G is an ω-tree. The theorem that every finite graph has a spanning tree has a natural, effective analogue for ω-graphs.

The structural difference between isolic α-graphs and isolic ω-graphs will be illustrated by: (i) every infinite graph is isomorphic to some isolic α-graph; (ii) there is an infinite graph which is not isomorphic to any isolic ω-graph.

An isolic α-graph is also called a twilight graph. There are c such graphs, c denoting the cardinality of the continuum.

We point out that a group first order definable in a differentially closed field K of characteristic 0 can be definably equipped with the structure of a differentially algebraic group over K. This is a translation into the framework of differentially closed fields of what is known for groups definable in algebraically closed fields (Weil's theorem).

I restrict myself here to showing (Theorem 20) how one can find a large “differentially algebraic group chunk” inside a group defined in a differentially closed field. The rest of the translation (Theorem 21) follows routinely, as in [B].

What is, perhaps, of interest is that the proof proceeds at a completely general (soft) model theoretic level, once Facts 1–4 below are known.

Fact 1. The theory of differentially closed fields of characteristic 0 is complete and has quantifier elimination in the language of differential fields (+, ·,0,1, −1,d).

Fact 2. Affine n-space over a differentially closed field is a Noetherian space when equipped with the differential Zariski topology.

Fact 3. If K is a differentially closed field, k ⊆ K a differential field, and a and are in k, then a is in the definable closure of k ◡ iff a ∈ ‹› (where k ‹› denotes the differential field generated by k and).

Fact 4. The theory of differentially closed fields of characteristic zero is totally transcendental (in particular, stable).