The problem of characterizing the finite lattices which can be embedded into the recursively enumerable degrees has a long history, which is summarized in [AL]. This problem is an important one, as its solution is necessary if a decision procedure for the ∀∃-theory of the poset of recursively emumerable degrees is to be found. A recursive nonembeddability condition, NEC, which subsumes all known nonembeddability conditions was presented in [AL]. This paper focuses on embeddability. An embeddability condition, EC, is introduced, and we prove that every finite lattice having EC can be embedded (as a lattice) into . EC subsumes all known embeddability conditions.

EC is a Π3 condition which states that certain obstructions to proving embeddability do not exist. It seems likely that the recursive labeled trees used in EC can be replaced with trees which are effectively generated from uniformly defined finite trees, in which case EC would be equivalent to a recursive condition. We do not know whether EC and NEC are complementary. This problem seems to be combinatorial, rather than recursion-theoretic in nature. Our efforts to find a finite lattice satisfying neither EC nor NEC have, to this point, been unsuccessful. It is the second author's conjecture that the techniques for proving embeddability which are used in this paper cannot be refined very much to obtain new embeddability results.

EC is introduced in §2, and the various conditions and definitions are motivated by presenting examples of embeddable lattices and indicating how the embedding proof works in those particular cases. The embedding construction is presented in §3, and the proof in §4.

For the reader's convenience, we begin Part II by restating the rules for constructing Sq derivations given in §2:

We let R be the set of 14 rules for Sq, R* the set of pure *-rules in R, R≃ the set of pure ≃-rules in R, and R− the set R ∩ − {Exs≃}. The closure of any F ⊆ Sq under R will be denoted by csR F, and the direct image of F under Cut*, for example, by Cut*(F).

No theory of a partially ordered set of finite width has the independence property, generalizing Poizat's corresponding result for linearly ordered sets. In fact, a question of Poizat concerning linearly ordered sets is answered by showing, moreover, that no theory of a partially ordered set of finite width has the multi-order property. It then follows that a distributive lattice is not finite-dimensional iff its theory has the independence property iff its theory has the multi-order property.

In this paper, we show that non-well-founded sets can be defined constructively by formalizing Hallnäs' limit definition of these within Martin-Löfs theory of types. A system is a type W together with an assignment of and to each ∝ ∈ W. We show that for any system W we can define an equivalence relation =w such that ∝ =w ß ∈ U and =w is the maximal bisimulation. Aczel's proof that CZF can be interpreted in the type V of iterative sets shows that if the system W satisfies an additional condition (*), then we can interpret CZF minus the set induction scheme in W. W is then extended to a complete system W* by taking limits of approximation chains. We show that in W* the antifoundation axiom AFA holds as well as the axioms of CFZ−.

If S is a finite set, let ∣S∣ be the cardinality of S. We show that if m ∈ ω, A ⊆ ω, B ⊆ ω, and ∣i: ≤ i ≤ 2m & xi ∈ A}∣ can be computed by an algorithm which, for all x1, …, x2m, makes at most m queries to B, then A is recursive in the halting set K. If m = 1, we show that A is recursive.

This is the first of a sequence of papers in which we will develop a foundation for the theory of computation based on a precise, mathematical notion of abstract algorithm. To understand the aim of this program, one should keep in mind clearly the distinction between an algorithm and the object (typically a function) computed by that algorithm. The theory of computable functions (on the integers and on abstract structures) is obviously relevant to this work, but we will focus on making rigorous and identifying the mathematical properties of the finer (intensional) notion of algorithm.

It is characteristic of this approach that we take recursion to be a fundamental (primitive) process for constructing algorithms, not a derived notion which must be reduced to others—e.g. iteration or application and abstraction, as in the classical λ-calculus. We will model algorithms by recursors, the set-theoretic objects one would naturally choose to represent (syntactically described) recursive definitions. Explicit and iterative algorithms are modelled by (appropriately degenerate) recursors.

The main technical tool we will use is the formal language of recursion, FLR, a language of terms with two kinds of semantics: on each suitable structure, the denotation of a term t of FLR is a function, while the intension of t is a recursor (i.e. an algorithm) which computes the denotation of t. FLR is meant to be intensionally complete, in the sense that every (intuitively understood) “algorithm” should “be” (faithfully modelled, in all its essential properties by) the intension of some term of FLR on a suitably chosen structure.

Among the symbolic languages used most frequently in the indirect interpretation of natural language are Montague's Intensional Logic IL [5, 384ff.] and its extensional counterpart, the language Ty2 of two-sorted type theory. The question of which of these two formal languages is to be preferred has been obscured by lack of knowledge about the exact relation between them. The present paper is an attempt to clarify the situation by showing that, modulo a small, decidable class of formulas irrelevant to these applications, IL and Ty2 are equivalent in the strong sense that there exists a reversible translation between the terms of either language.

In [3, 6Iff.] Gallin has shown that there exists a simple and natural translation * of IL into Ty2. Following Gallin's translation procedure, it is even possible to conceive of IL as a highly restricted sublanguage of Ty2, viz. as that part which only contains expressions of certain intensional types plus one variable of the basic type of indices or worlds. In an obvious sense, this sublanguage has less expressive power than the whole of Ty2, where it is possible to express conditions on entities that do not even exist in IL's ontology. However, by a certain amount of coding, one can translate Ty2 into IL [3, 105]. Conditions on nonintensional entities then become conditions on corresponding intensional objects; and these paraphrases preserve (standard) validity and entailment. On the other hand, this retranslation of Ty2 into IL is not an inversion of *, as can be seen from a simple example.

We show that the DA-uniflcation problem is undecidable. That is, given two binary function symbols ⊕ and ⊗, variables and constants, it is undecidable if two terms built from these symbols can be unified provided the following DA-axioms hold:

Two terms are DA-unifiable (i.e. an equation is solvable in DA) if there exist terms to be substituted for their variables such that the resulting terms are equal in the equational theory DA.

This is the smallest currently known axiomatic subset of Hilbert's tenth problem for which an undecidability result has been obtained.

We deal with the consistency strength of ZFC + variants of MA + suitable sets of reals are measurable (and/or Baire, and/or Ramsey). We improve the theorem of Harrington and Shelah [2] repairing the asymmetry between measure and category, obtaining also the same result for Ramsey. We then prove parallel theorems with weaker versions of Martin's axiom (MA(σ-centered), (MA(σ-linked)), , MA(K)), getting Mahlo, inaccessible and weakly compact cardinals respectively. We prove that if there exists r ∈ R such that and MA holds, then there exists a -selective filter on ω, and from the consistency of ZFC we build a model for ZFC + MA(I) + every -set of reals is Lebesgue measurable, has the property of Baire and is Ramsey.

The failure of a large and diverse body of work to settle some of the now-classical questions of computational complexity (notably P =? NP) suggests that they might not, in fact, be resolvable by established proof techniques.

Hartmanis and Hopcroft [HH] raised the issue of independent statements about computational complexity in 1976, constructing, for any consistent r.e. theory T capable of expressing statements about Turing machines, a Turing machine MT such that statements which intuitively express the computational complexity of MT are independent of T. Their technique involves a simple diagonalizing search over the theorems of T. In this paper we prove a constructive version of their independence result in the context of a generalization of a hierarchy of free variable logics defined by Rose [R1]. These logics are based on an axiomatized treatment of an extension by Löb and Wainer [LW] of Grzegorczyk's [G] hierarchy into the transfinite. Associated with each extended Grzegorczyk class (relative to an ordinal notation system S satisfying certain conditions) is the logic -arithmetic.

Free variable logics are interesting from the perspective of theoretical computer science. We may construe the equational definition of functions as a form of programming system. Each -arithmetic, then, has the nice property of containing both a programming system and a logic for stating and proving facts about programs.

Using coding devices based on a theorem due to Zsigmondy, Birkhoff and Vandiver, we first define in terms of successor S and coprimeness predicate ⊥ a full arithmetic over the set of powers of some fixed prime, then we define in the same terms a restriction of the exponentiation. Hence we prove the main result insuring that all arithmetical relations and functions over prime powers and their opposite are {S, ⊥}-definable over Z. Applications to definability over Z and N are stated as corollaries of the main theorem.

We prove the following theorem. Let G be a connected simple bad group (i.e. of finite Morley rank, nonsolvable and with all the Borel subgroups nilpotent) of minimal Morley rank. Then the Borel subgroups of G are conjugate to each other, and if B is a Borel subgroup of G, then , NG(B) = B, and G has no involutions.

Let We be the eth recursively enumerable (r.e.) set in a standard enumeration. The fixed point form of Kleene's recursion theorem asserts that for every recursive function f there exists e which is a fixed point of f in the sense that We = W f(e). In this paper our main concern is to study the degrees of functions with no fixed points. We consider both fixed points in the strict sense above and fixed points modulo various equivalence relations on recursively enumerable sets.

Our starting point for the investigation of the degrees of functions without (strict) fixed points is the following result due to M. M. Arslanov [A1, Theorem 1] and known as the Arslanov completeness criterion. Proofs of this result may also be found in [So1, Theorem 1.3] and [So2, Chapter 12], and we will give a game version of the proof in §5 of this paper.

Theorem 1.1 (Arslanov). Let A be an r.e. set. Then A is complete (i.e. A has degree 0′) iff there is a function f recursive in A with no fixed point.

Throughout this paper κ denotes an infinite cardinal, S = Sym(κ) and G is a subgroup of S. We shall be seeking the subgroups G with [S: G] < 2κ. In [2], the following result was proved.

Theorem 1. If [S: G] ≤ κthen there exists a subset Δ of k such that ∣Δ∣ < k and S(Δ) ≤ G.

Here S(Δ) = Sym(K/⊿) is the pointwise stabilizer of Δ in S.

However, the converse of Theorem 1 is not true. For if cf(κ) ≤ ∣Δ∣ < κ, then [S: S(Δ)] ≥ κcf(κ) > κ. This suggests that a substantially sharpened version of Theorem 1 may be true.

Question 1 [2]. Is it provable in ZFC, or even in ZFC with GCH, that if [S: G] ≤ κ then there is a subset Δ of κ such that ∣Δ∣ < cf(κ) and S(Δ) ≤ G?

At least two of the authors of [2] made a serious attempt to answer the above question positively. In §3, we shall see that they were essentially trying to prove that measurable cardinals do not exist.

The following result, due independently to Semmes [5] and Neumann [2], suggests a second way in which Theorem 1 might be improved.

Theorem 2. If k = ℵ0andthen there is a finite subset Δ of k such thatS(Δ) ≤ G.

Question 2 [2]. Is it provable in ZFC that if [S: G] < 2κ then there is a subset Δ of κ such that ∣Δ∣ < κ and S(Δ) < G?

This question will be answered negatively in §4.

If κ is a measurable cardinal, let us say that a measure on κ is a κ-complete nonprincipal ultrafilter on κ. If U is a measure on κ, let jU be the canonical elementary embedding of V into its Ultrapower UltU(V). If x is a set, say that U moves x when jU(x) ≠ x; say that κ moves x when some measure on κ moves x. Recall Kunen's lemma (see [K]): “Every ordinal is moved only by finitely many measurable cardinals.” Kunen's proof (see [K]) and Fleissner's proof (see [KM, III, §10]) are essentially nonconstructive.

The following proposition can be proved by using elementary facts about iterated ultrapowers.

Proposition. Let ‹Un: n ∈ ω› be a sequence of measures on a strictly increasing sequence ‹κn: n ∈ ω› of measurable cardinals. Let U = ‹ Wα: α < ω2›, where Wωm + n = Um (m, n ∈ ω). Then, for each θ in UltU(V), if E is the (minimal) support of θ in UltU(V), then, for all m ∈ ω, Um moves θ iff E ∩ [ωm, ω(m + 1)) ≠ ∅.

The notion of ∀-recursiviness, introduced by Lacombe [1], is intended to describe the effectively definable functions and predicates in abstract structures with equality and denumerable domains. The fact that on every such structure ∀-recursiviness and search computability are equivalent is proved by Moschovakis in [2].

The definition of search computability [3] does not require the presence of the equality among the basic predicates of the structure. There exist abstract structures where the equality is not search-computable and even not semicomputable. On the other hand, in some structures the equality is not an “effective” predicate. Consider, for example, a structure whose domain consists of all partial recursive functions.

A notion of relative computability in abstract structures with denumerable domains, which we shall call here ∀-admissibility, was introduced by D. Skordev in 1977. The notion of ∀-admissibility is a generalization of Lacombe's ∀-recursiviness and does not require the presence of the equality among the basic predicates. In 1977 Skordev conjectured that, in every partial structure with denumerable domain, ∀-admissibility and search computability are equivalent.

Since 1977 some attempts have been made to establish Skordev's conjecture. It is proved in [4] for structures with total basic functions and without basic predicates, and in [5] for structures with finite domains. The proofs in [4] and [5] make use of the priority method and are very complicated.

Kirby and Paris have exhibited combinatorial algorithms whose computations always terminate, but for which termination is not provable in elementary arithmetic. However, termination of these computations can be proved by adding an axiom first introduced by Goodstein in 1944. Our purpose is to investigate this axiom of Goodstein, and some of its variants, and to show that these are potentially adequate to prove termination of computations of a wide class of algorithms. We prove that many variations of Goodstein's axiom are equivalent, over elementary arithmetic, and contrast these results with those recently obtained for Kruskal's theorem.

The paper [Schütte + Simpson] deals with the following one-dimensional case of Friedman's extension (see in [Simpson 1]) of Kruskal's theorem ([Kruskal]). Given a natural number n, let Sn+1 be the set of all finite sequences of natural numbers <n + 1. If s1 = (a0,…,ak) ∈Sn+1 and s2 = (b0,…,bm) ∈Sn + 1, then a strictly monotone function f: {0,…, k} → {0,…, m} is called an embedding of s1 into s2 if the following two assertions are satisfied:

1) ai, = bf(i), for all i < k;

2) if f(i) < j < f(i + 1) then bj > bf(i+1), for all i < k, j < m.

Then for every infinite sequence s1, s2,…,sk,… of elements of Sn + 1 there exist indices i < j and an embedding of si into Sj. That is, Sn+1 forms a well-quasi-ordering (wqo) with respect to embeddability. For each n, this statement W(Sn+1) is provable in the standard second order conservative extension of Peano arithmetic. On the other hand, the proof-theoretic strength of the statements W(Sn+1) grows so fast that this formal theory cannot prove the limit statement ∀nW(Sn+1). The appropriate first order -versions of these combinatory statements preserve their proof-theoretic strength, so that actually one can speak in terms of provability in Peano arithmetic. These are the main conclusions from [Schütte + Simpson].

We wish to extend this into the transfinite. That is, we take an arbitrary countable ordinal τ > 0 instead of n + 1 and try to obtain an analogous “strong” combinatory statement about finite sequences of ordinals < τ.

A function is recursive (in given operations) if its values are computed explicitly and uniformly in terms of other “previously computed” values of itself and (perhaps) other “simultaneously computed” recursive functions. Here, “explicitly” includes definition by cases.

We investigate those recursive functions on the structure N = 〈ω, 0, succ, pred〉 that are computed in terms of themselves only, without other simultaneously computed recursive functions.

It is consistent that for many cardinals λ there is a family of at least λ+ unbounded subsets of λ which have uniformization properties. In particular if it is consistent that a supercompact cardinal exists, then it is consistent that ℵω has such a family. We have applications to point set topology, Whitehead groups and reconstructing separable abelian p-groups from their socles.