We show, roughly speaking, that it requires ω iterations of the Turing jump to decode nontrivial information from Boolean algebras in an isomorphism invariant fashion. More precisely, if α is a recursive ordinal, is a countable structure with finite signature, and d is a degree, we say that has αth-jump degreed if d is the least degree which is the αth jump of some degree c such there is an isomorphic copy of with universe ω in which the functions and relations have degree at most c. We show that every degree d ≥ 0(ω) is the ωth jump degree of a Boolean algebra, but that for n < ω no Boolean algebra has nth-jump degree d < 0(n). The former result follows easily from work of L. Feiner. The proof of the latter result uses the forcing methods of J. Knight together with an analysis of various equivalences between Boolean algebras based on a study of their Stone spaces. A byproduct of the proof is a method for constructing Stone spaces with various prescribed properties.

In this paper the modal operator “x is provable in Peano Arithmetic” is incorporated into first-order theories. A provability extension of a theory is defined. Presburger Arithmetic of addition, Skolem Arithmetic of multiplication, and some first order theories of partial consistency statements are shown to remain decidable after natural provability extensions. It is also shown that natural provability extensions of a decidable theory may be undecidable.

H. Friedman proved that the following sentence SWQO() is true but not provable in the theory ATR0 (cf. [S]; ≔ the set of finite rooted trees, ∣T∣ ≔ the cardinality of T ∈ ).

Since ATR0 is stronger than PA, SWQO() is not provable in PA, either. Furthermore, SWQO() is proof-theoretically stronger than PA. A similar sentence whose proof-theoretical strength is exactly that of PA was obtained by replacing finite trees by finite sequences of bounded natural numbers under the embeddability with H. Friedman's asymmetrical gap-condition (see [SS]). The latter sentence was modified by using the weaker symmetrical gap-condition (see the corresponding particular case in [Gl]). Further improvement is indicated in [G2, Corollary 4.5]. Namely, the assertion ∣Xi∣ ≤ k · (i + 1) as in SWQO(J), above, can be weakened to

.

(in fact, to wt(Xi) ≤ k + ε · i for an arbitrary fixed ε = l−1, l < 0, where wt(X) is the weight of the “word” X).

In this paper we study the consistency strength of the theory

and we prove the consistency of this theory relative to the consistency of the existence of a supercompact cardinal and an inaccessible above it.

We analyse the Mitchell ordering in a model where κ is -hypermeasurable and > .

A realizability notion that employs only primitive recursive functions is defined, and, relative to it, the soundness of the fragment of Heyting Arithmetic (HA) in which induction is restricted to formulae is proved. A dual concept of falsifiability is proposed and an analogous soundness result is established for a further restricted fragment of HA.

Let L0 be a language on ℕ consisting of 0, 1, +, ∸, ·, ⌊½a⌋, ∣a∣, #, ∧(a,b), ∨(a,b), ¬(a), ≤ (a,b), and μx ≤ ∣s∣t(x). Here μx ≤ ∣s∣t(x) is the smallest number x ≤ ∣s∣ satisfying t(x) > 0 and 0 if there exist no such x and we stipulate that if s and t(a) are terms in Lo, then μx ≤ ∣s∣t(x) is also a term in Lo. The defining axioms of functions ∧(a,b), ∨(a,b), ¬(a), ≤ (a,b) are as follows:

Let L a language on ℕ with only predicate constant = and L0 ⊆ L. Let f (b, a1,…,am) be a function for ℕm+1 into ℕ. We say “f is weakly expressed by terms t1(b, a1,…, am),…, tr(b, a1,…, am) in L” if for every b, a1,…, am ∈ ℕ, f (b, a1,…,am) is equal to one of ti(b, a1, …, am). The critical number of b in f with respect to L is the minimum number n such that whenever f(b, a1,…, an) is weakly expressed by terms t1(b, a1, …, an),…, the number of occurrences of b in some ti(b, a1,…, an) is at least n.

As is defined in [1], a function f is defined by a limited iteration from g and h with respect to L iff the following holds: Let τ be defined as

with the condition ; and is defined by

where and are terms in L. We say “f is defined by a short limited iteration from g and h” if is defined by

where τ, s, t are the same as above satisfying the condition .

Partial models of the theory New Foundations (NF) introduced by Quine have already appeared in the literature, but in every model the membership set of NF is missing. On the other hand, Jensen showed that “NF + Urelements” is consistent with respect to ZF and, in the model built there, the membership set of the theory exists, Here we build a partial model of NF from the one of Jensen in which the membership set exists.

Using a result of Gurevich and Lewis on the word problem for finite semigroups, we give short proofs that the following theories are hereditarily undecidable: (1) finite graphs of vertex-degree at most 3; (2) finite nonvoid sets with two distinguished permutations; (3) finite-dimensional vector spaces over a finite field with two distinguished endomorphisms.

Fitting, in [1] and [2], investigated two families of many-valued modal logics. The first, which is somewhat familiar in the literature, is that of the logics characterized using a many-valued version of the Kripke model (binary modal model in his terminology) with a two-valued accessibility relation. On the other hand, those logics which are characterized using another many-valued version of the Kripke model (implicational modal model), with a many-valued accessibility relation, form the second family. Although he gave a sequent calculus for each of these logics, it is far from having the cut-elimination property (CEP) or the subformula property. So we will give a substitute for his system enjoying the subformula property, though it is not of ordinary sequent calculus but of the many-valued version of sequent calculus initiated by Takahashi [7] and Rousseau [3].

The author, unaware of the deduction systems with CEP, had given in [8] and [9], after Rousseau [4], the deduction systems for the intuitionistic many-valued logics which enjoy CEP only for a certain restricted class of proofs. Then in [10], he gave for three-valued modal logics the ones with CEP, but these systems have a rule of inference which is unnecessary if the Cut rule is present. Why are we particular about CEP? The author's answer is that a cut-free proof is easy to examine since it is composed solely of subformulas of the formulas which form its conclusion. In this direction, the author has given, for modal logics with the Brouwerian axiom [11], the ones without CEP which nevertheless enjoy the subformula property. This paper is a sequel to the study in [11].

A proof-theoretic characterization of the primitive recursive functions is the Σ1-definable functions in IΣ1 as is shown in Mints [4], Parsons [5], and [8].

Then what is a proof-theoretic characterization of Grzegorzyk's hierarchy? First we discuss a related previous work. In Clote and Takeuti [2], we introduced a theory TAC that corresponds to the computational complexity class AC. TAC has a very weak form of induction. We assign a rank to a proof in TAC in the following way. The rank of a proof P in TAC is the nesting number of inductions used in P. Then TACi is defined to be the subtheory of TAC whose proof has a rank ≤ i. We proved that TACi corresponds to the class ACi.

In this paper we introduce a theory IepΣ1 which is equivalent to IΣ1. Then we define the rank of a proof in IepΣ1 as the nesting number of inductions in the proof and prove that the proofs with rank ≤ i correspond to Grzegorcyk's hierarchy for i > 0.

We also prove that the system that has proofs with rank 0 is actually equivalent to I Δ0. These facts are interesting since it is proved in [10] that the theory isomorphic to TAC∘ by RSUV isomorphism is a conservative extension of I Δo. Therefore there is some analogy between the class AC and the primitive recursive functions.

In this paper we establish the following theorems.

Theorem A. Let T be a complete first-order theory which is uncountable, Then:

(i) I(∣T∣, T) ≥ ℵ0

(ii) If T is not unidimensional, then for any λ ≥ ∣T∣, I(λ, T) ≥ ℵ0.

Theorem B. Let T be superstable, not totally transcendental and nonmultidimensional. Let θ(x) be a formula of least R∞ rank which does not have Morley rank, and let p be any stationary completion of θ which also fails to have Morley rank. Then p is regular and locally modular.

Ramsey's Theorem states that if P is a partition of [ω]k into finitely many partition classes, then there exists an infinite set of natural numbers which is homogeneous for P. We consider the degrees of unsolvability and arithmetical definability properties of infinite homogeneous sets for recursive partitions. We give Jockusch's proof of Seetapun's recent theorem that for all recursive partitions of [ω]2 into finitely many pieces, there exists an infinite homogeneous set A such that ∅′ ≰TA. Two technical extensions of this result are given, establishing arithmetical bounds for such a set A. Applications to reverse mathematics and introreducible sets are discussed.

Embeddings into initial segments of (the lattice of ideals of r.e. degrees) are relevant for the investigation of decidability of (fragments of) the theory of (with constant-symbols denoting principal ideals). Interest in this investigation stems from open problems concerning decidability of fragments of the theory of R (the usl of r.e. degrees) and embeddings into R. Ambos-Spies and Lerman established comprehensive nonembeddability [AS/LR1] and embeddability [AS/LR2] conditions, but their complementarity is open. (For background on embeddings and decidability consult Lerman [LR1] and Lerman and Shore [LR/SH].) The first systematic investigation of embeddings into is in Calhoun [CL], where the existence of uniform bounds, in the case of embeddings into , on the number of “join traces” arising with nondistributive lattices is shown to remove “obstructions” to lattice-embeddings into R (such as typified in the nonembeddable lattice S8, see [LA/SR], [AS/LR1/). The analysis to follow extends this, by showing that it also removes the “obstruction” to lattice-embeddings into initial segments of R, arising with Lachlan's M5, which embeds into R, but not into every nontrivial initial segment R[0, a], by a result of Downey [DW]. The techniques are Lachlan's [LA], as adapted in Lerman's “pinball machine model” [LR1, LR2], combined with Fejer's “meet-trick” [FJ] as used by Downey [DW] in the proof that every distributive lattice is embeddable into every nontrivial initial segment of r.e. degrees R[0, a], a > 0. We present a sufficient condition, “allowing evasion” [Definition 4.2] for lattice embeddability into every [0, a], a > 0.

The Scott height of a structure gives ordinal measure of the inhomogeneity of the structure. The Scott specturm of a collection of structures is the set of Scott heights of structures in the collection. We will recall the precise definitions of these and related concepts in the next section. The reader thoroughly unfamiliar with these notions may want to skip ahead before reading the rest of this Introduction.

In [11] it is shown that every model of the complete theory of (, +, 1), where, as usual, denotes the integers, is ℵ0-homogeneous, and therefore has Scott height at most ω. On the other hand, a footnote in [1] gives a model of the theory of (, +) which is not ℵ0-homogeneous, while in [11] such a model is described which can be expanded to a model of the theory of (, +, 1). However, since it is also true that any model of the theory of (, +) is isomorphic to a subgroup, in fact a pure subgroup, of a direct sum of and a torsion-free divisible group, it is easy to see that any such model must be ≡∞ω to a model of cardinality at most and so must have Scott height below .

After having recalled the relevant material about Scott heights in §2, we will review the situation for torsion abelian groups in §3. In §4 we shall produce torsion-free abelian groups of high Scott height. It is the proof of Theorem 15 that was our primary motivation in writing this paper.

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.

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.

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 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.

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