This note is concerned with the old topic, initiated by Kleene, of the connections between recursive function theory and provability in intuitionistic arithmetic. More specifically, we are interested in the relationship between the hierarchy of degrees of unsolvability and the interdeducibility of cases of excluded middle. The work described below was motivated by a counterexample, to be given presently, which shows that that relationship is more complicated than one might suppose.

Let HA be first-order intuitionistic arithmetic. Let the symbol ⊢ mean derivability in HA. For each natural number n, let n¯ be the corresponding numeral. Let Ω be the standard model of arithmetic. Say that a sentence ϕ is true iff Ω⊨ ϕ. Now suppose ϕ(x) and Ψ(x) are formulas with only the variable x free. Suppose

Then it is natural to conjecture that {n∣Ω⊨Ψ(n¯)} is recursive in {n∣Ω⊨ϕ(n¯)}.

However, this conjecture is false. Consider the formula is a formalization of Kleene's T-predicate.

This paper concerns certain relationships between the ordering of degrees of unsolvability and the jump operation. It is shown that every minimal degree a < 0′ satisfies a″ = 0″. To restate this result in more suggestive language and compare it with related results, we shall use notation based on the now standard terminology of “high” and “low” degrees. Let Hn be the class of degrees a < 0′ such that a(n) = 0(n+1), and let Ln be the class of degrees a ≤ 0′ such that an = 0n. (Observe that Hi ⊆ Lj and Li ⊆ Lj, whenever i ≤ j, and Hi ∩ Lj = ∅ for all i andj.) The result mentioned above may now be restated in the form that every minimal degree a ≤ 0′ is in L2. This extends an earlier result of S. B. Cooper ([1], see also [4]) that no minimal degree a < 0′ is in H1. In the other direction, Sasso, Epstein, and Cooper ([10], [15]) have shown that there is a minimal degree a < 0′ which is not in L1. Also, C.E.M. Yates [14, Corollary 11.14], showed the existence of a minimal degree a < 0′ in L1. Thus each minimal degree a < 0′ lies in exactly one of the two classes L1L2 – L1 and each of the classes contains minimal degrees.

Our results are not restricted to the degrees below 0′. We show in fact that every minimal degree a satisfies a″ = (a ∪ 0′)′. To restate this result and discuss extensions of it, we extend the “high-low” classification of degrees from the degrees below 0′ to degrees in general. There are a number of fairly plausible ways of doing this, but we choose the one way we know of doing so which leads to interesting results. Let GHn be the class of degrees a such that a(n) = (a ∪ 0′)(n), and let GLn be the class of degrees a such that a(n) = (a ∪ 0′)(n−1).

Let θ(ν) be a formula in the first-order language of arithmetic and let

In this note we study the relationship between the schemas I′ and I+.

Our interest in I+ lies in the fact that it is ostensibly a more reasonable schema than I′. For, if we believe the hypothesis of I+(θ) then to verify θ(n) only requires at most 2log2(n) steps, whereas assuming the hypothesis of I′(θ) we require n steps to verify θ(n). In the physical world naturally occurring numbers n rarely exceed 10100. For such n applying 2log2(n) steps is quite feasible whereas applying n steps may well not be.

Of course this is very much an anthropomorphic argument so we would expect that it would be most likely to be valid when we restrict our attention to relatively simple formulas θ. We shall show that when restricted to open formulas I+ does not imply I′ but that this fails for the classes Σn, Πn, n ≥ 0.

We shall work in PA−, where PA− consists of Peano's Axioms less induction together with

∀u, w(u + w = w + u ∧ u · w = w · u),

∀u, w, t ((u + w) + t = u + (w + t) ∧ (u · w) · t = u · (w · t)),

∀u, w, t(u · (w + t) = u · w + u · t),

∀u, w(u ≤ w ↔ ∃t(u + t = w)),

∀u, w(u ≤ w ∨ w ≤ u),

∀u, w, t(u + w = u + t → w = t).

The reasons for working with PA− rather than Peano's Axioms less induction is that our additional axioms, whilst intuitively reasonable, will not necessarily follow from some of the weaker forms of I+ which we shall be considering. Of course PA− still contains those Peano Axioms which define + and

Notice that, trivially, PA− ⊦ I′(θ) → I+(θ) for any formula θ.

We present two theorems whose applications are to eliminate diagonalization arguments from a variety of constructions of degrees of unsolvability.

All definitions and notations come from [1, Chapter 1]. We give a brief resumé of them here.

We identify a set with its characteristic function. (A(x)= 1 if x ∈ A and A (x) = 0 if x ∉ A.) A string σ is the restriction of a characteristic function to a finite initial segment of natural numbers, lh(σ) = length of σ = n + 1 if σ = A[n] for some set A. (A [n] is the restriction of A to {m: m ≤ n}.) If i = 0 or 1, σ * i is defined as the string of length lh(σ) + 1 such that σ * i ⊇ σ and σ * i(lh(σ)) = i. We write σ ∣ τ if σ ⊉ τ and τ ⊉ σ.

{Φn} is a listing of the partial recursive functionals. We write “A ≤TB” (“A is Turing reducible to B”) if ∃n∀xΦn(B)(x) = A(x).

A partial function, T, from strings to strings is a tree if T is order preserving and for all strings, σ, if one of T(σ * 0), T(σ * 1) is defined then T(σ), T(σ * 0), T(σ * 1) are all defined and T(σ * 0)∣(σ * 1).

Let us consider the following infinite game between two players, Empty and Nonempty. We are given a large set S. Empty opens the game by choosing a large subset S0 of S; then Nonempty chooses a large set S1 ⊆ S0; then Empty chooses large S2 ⊆ S, etc. The game is over after ω moves. If ⋂n=0xSn is empty then Empty wins, and if ⋂n=0∞Sn is nonempty then Nonempty wins.

If “large” means “infinite”, then Empty can beat Nonempty rather easily: he chooses So countable, S0 = {a0, a1,…, an,…}, and then he chooses S2 such that a0 ∉ S2, S4 such that a1, ∉ S4 and so on.

Next we assume that S is a set of uncountable cardinality, and that “large” means “of cardinality ∣S∣”. Then still Empty can win, but his winning strategy is somewhat more sophisticated: Let us identify S with a cardinal number κ. Thus each subset of S of size κ is a set of ordinals below κ. For each X ⊆ κ of size κ, let fx be the unique order-preserving mapping of X onto κ, and let F(X) = {x ϵ X: f(x) is a successor ordinal}. Empty's strategy is to play S0 = F(K), and when Nonempty plays S2k − 1, let S2k = F(S2k − 1).

By [12] we know that transfinite induction up to ΘεΩN+10 is not provable in IDN, the theory of N-times iterated inductive definitions. In this paper we will show that conversely transfinite induction up to any ordinal less than ΘεΩN+10 is provable in IDNi, the intuitionistic version of IDN, and extend this result to theories for transfinitely iterated inductive definitions.

In [14] Schütte proves the wellordering of his notational systems using predicates is wellordered) with Mκ ≔ {x ∈ and 0 ≤ κ ≤ N. Obviously the predicates are definable in IDNi with the defining axioms:

where Prog [Mκ, X] means that X is progressive with respect to Mκ, i.e.

The crucial point in Schütte's wellordering proof is Lemma 19 [14, p. 130] which can be modified to

where TI[Mκ + 1, a] is the scheme of transfinite induction over Mκ + 1 up to a.

In this paper we shall outline a purely model theoretic method for obtaining independence results for Peano's first order axioms (P). The method is of interest in that it provides for the first time elementary combinatorial statements about the natural numbers which are not provable in P. We give several examples of such statements.

Central to this exposition will be the notion of an indicator. Indicators were introduced by L. Kirby and the author in [3] although they had occurred implicitly in earlier papers, for example Friedman [1]. The main result on indicators which we shall need (Lemma 1) was proved by Laurie Kirby and the author in the summer of 1976 but it was not until early in the following year that the author realised that this lemma could be used to give independence results.

The first combinatorial independence results obtained were essentially statements about certain finite games and consequently were not immediately meaningful (see Example 2). This shortcoming was remedied by Leo Harrington who, upon hearing an incorrect version of our results, noticed a beautifully simply independent combinatorial statement. We outline this result in Example 3. An alternative, more detailed, proof may be found in [5].

Clearly Laurie Kirby and Leo Harrington have made a very significant contribution to this paper and we wish to express our sincere thanks to them.

It is proved here, assuming Projective Determinacy, that every ascending sequence of -degrees has a minimal strict upper bound but no least strict upper bound. This generalizes a result of Friedman for n = 1.

Our general notation and terminology will be that of [Ke1] and [Mo1]. Letters i,j, k,… denote members of ω and α, β, ϒ,… members of ωω i.e. reals. Projective Determinacy (PD) is the hypothesis that every projective set of reals in determined, while in general for a collection of sets of reals Γ, Determinacy (Γ) abbreviates the statement that every set in Γ is determined.

§1. -degrees. For each m ≥ 1 and α, β Є ωω let α ≤mβ ⇔ α Є (β), α <mβ ⇔ α ≤mβ ∧ β ≰mα, and α ≡mβ ⇔ α ≤mβ ∧ β ≤mα.Clearly ≡m is an equivalence relation on ωω. The ≡m -equivalence class of α Є ωω is called its -degree, in symbols

If d = [α]m, e = [β]m then we define

The main point of this paper is a further development of some aspects of the recent theory of recursively enumerable (r.e.) algebraic structures. Initial work in this area is due to Frölich and Shepherdson [4] and Rabin [10]. Here we are only concerned with vector space structure. The previous work on r.e. vector spaces is due to Dekker [2], [3], Metakides and Nerode [8], Remmel [11], Retzlaff [13], and the author [5].

Our object of study is V∞ a countably infinite dimensional fully effective vector space over a countable recursive field . By fully effective we mean that V∞. under a fixed Godel numbering has the following properties:

(i) Operations of vector addition and scalar multiplication on V∞ are presented by partial recursive functions on the Gödel numbers of elements of V∞.

(ii) V∞ has a dependence algorithm, i.e., there is a uniform effective procedure which applied to any n vectors of V∞ determines whether or not they are linearly independent.

We also study , the lattice of r.e. subspaces of V∞ (under the operations of intersection, ⋂ and (weak) sum, +). We note that if is not distributive and is merely modular (see [1]). This fact indicates the essential difference between the lattice of r.e. sets and .

This paper continues the study of existentially complete nilpotent groups initiated in [6]. Following [6], we let Kn denote the theory of groups nilpotent of class ≤ n and let Kn+ denote the theory of torsion-free groups nilpotent of class ≤ n. The principal results of [6] were that for n ≥ 2, neither Kn nor Kn+ has a model companion, and the classes E, F, and G of existentially complete, finitely generic and infinitely generic models of Kn are all distinct. The question of the relationships between these classes in the context of Kn was left open, however, and the proof of their distinctness for Kn+ obviously did not carry over to Kn+, because it made strong use of torsion elements.

In this paper we establish the relationships between E, F, and G for K2+. We show that all three classes are distinct. We also show that there is only one countable finitely generic model, and only one countable infinitely generic model, and that all the countable existentially complete models can be arranged in a sequence N1 ⊆ N2 ⊆ N3 ⊆ … ⊆ Nω, where Z(Nn) is the direct sum of n copies of Q. Another result is that the finite and infinite forcing companions of K2+ differ by an ∀∃∀ sentence. Finally, we show that there exist finitely generic models of K2+ in all infinite cardinalities.

One of the main results of Barwise [2] (see also [7, Chapter VIII]) showed that the s – reflection principle for a set A is equivalent to Σ1-compactness of . Here A is any transitive p.r. closed set, and is the infinitary language on A which allows conjunction and disjunction over arbitrary sets Φ Є A, and finite quantification.

In this paper we consider languages , where B is a Δ0 subset of A, which is like but we allow quantifiers ∀x and ∃x where x is any set of variables indexed by an element of B. A treatment similar to that of [2] for establishes a sufficient, and in some cases necessary, condition for to be Σ1-compact. The use of infinitary Skolem functions is intrinsic to the method, so to avoid a separate development of the rudiments of the Skolem language we actually define to have b-ary relation and function symbols for every b Є B.

It is the purpose of this paper to investigate the model theory of logic with a generalized quantifier; in particular the logic L(Q1) where Q1xφ(x) has the intended meaning “there exist uncountably many x such that φ(x)”. We do this from the point of view that the best way to study what happens in the so-called “ω1-standard” models of L(Q1) is to examine the countable ideal models of L(Q) that satisfy all of the axioms for L(Q1) (see definitions of ω1-standard and ideal models in §1). We believe that this study can be as fruitful for L(Q1) as the study of countable models of ZF has been for set theory.

A major problem is formulating an adequate definition of submodel for countable ideal models that is compatible with that for ω1-standard models. Thus we begin the paper by discussing several possible definitions of the notion of submodel. We then adopt a particular definition of submodel and investigate model-completeness in L(Q). We define model-completeness both for ω1-standard models and for countable ideal models and compare the two notions. We also examine elimination of quantifiers, as well as investigating formulas preserved under submodels, again both for ω1-standar d and countable ideal models.

We show that several theorems on ordinal bounds in different parts of logic are simple consequences of a basic result in the theory of global inductive definitions.

Let A be a subset of ω, the set of natural numbers. The degree of A is its degree of recursive unsolvability. We say that A is rich if every degree above that of A is represented by a subset of A. We say that A is poor if no degree strictly above that of A is represented by a subset of A. The existence of infinite poor (and hence nonrich) sets was proved by Soare [9].

Theorem 1. Suppose that A is infinite and not rich. Then every hyperarith-metical subset H of ω is recursive in A.

In the special case when H is arithmetical, Theorem 1 was proved by Jockusch [4] who employed a degree-theoretic analysis of Ramsey's theorem [3]. In our proof of Theorem 1 we employ a similar, degree-theoretic analysis of a certain generalization of Ramsey's theorem. The generalization of Ramsey's theorem is due to Nash-Williams [6]. If A ⊆ ω we write [A]ω for the set of all infinite subsets of A. If P ⊆ [ω]ω we let H(P) be the set of all infinite sets A such that either [A]ω ⊆ P = ∅. Nash-Williams' theorem is essentially the statement that if P ⊆ [ω]ω is clopen (in the usual, Baire topology on [ω]ω) then H(P) is nonempty. Subsequent, further generalizations of Ramsey's theorem were proved by Galvin and Prikry [1], Silver [8], Mathias [5], and analyzed degree-theoretically by Solovay [10]; those results are not needed for this paper.

Chang, in [1], proves an interpolation theorem (Theorem I, remark b)) for a first-order language. The proof of Chang's theorem uses essentially nonsimple devices, like special and ω1-saturated models.

In remark e) in [1], Chang asks if there is a simpler proof of his Theorem I.

In [1], Chang proves also another interpolation theorem (Theorem II), which is not an extension of his Theorem I, but extends Craig's interpolation theorem to Lα+,ω languages with interpolant in Lα+,α where α is a strong limit cardinal of cofinality ω.

In remark k) in [1], Chang asks if there is a generalization of both Theorems I and II in [1], or at least a generalization of both Theorem I in [1] and Lopez-Escobar's interpolation theorem in [7].

Maehara and Takeuti, in [8], show that there is a completely different proof of Chang's interpolation Theorem I as a consequence of their interpolation theorems. The proofs of these theorems of Maehara and Takeuti are proof theoretical in character, involving the notion of cut-free natural deduction, and it uses devices as simple as those needed for the usual Craig's interpolation theorem. Hence this can be considered as a positive answer to Chang's question in remark e) in [1].

Ever since Post [4] the structure of recursively enumerable sets and their classification has been an important area in recursion theory. It is also intimately connected with the study of the lattices and of r.e. sets and r.e. sets modulo finite sets respectively. (This lattice theoretic viewpoint was introduced by Myhill [3].) Key roles in both areas have been played by the lattice of r.e. supersets, , of an r.e. set A (along with the corresponding modulo finite sets) and more recently by the group of automorphisms of and . Thus for example we have Lachlan's deep result [1] that Post's notion of A being hyperhypersimple is equivalent to (or ) being a Boolean algebra. Indeed Lachlan even tells us which Boolean algebras appear as —precisely those with Σ3 representations. There are also many other simpler but still illuminating connections between the older typology of r.e. sets and their roles in the lattice . (r-maximal sets for example are just those with completely uncomplemented.) On the other hand, work on automorphisms by Martin and by Soare [8], [9] has shown that most other Post type conditions on r.e. sets such as hypersimplicity or creativeness which are not obviously lattice theoretic are in fact not invariant properties of .

In general the program of analyzing and classifying r.e. sets has been directed at the simple sets. Thus the subtypes of simple sets studied abound — between ten and fifteen are mentioned in [5] and there are others — but there seems to be much less known about the nonsimple sets. The typologies introduced for the nonsimple sets begin with Post's notion of creativeness and add on a few variations. (See [5, §8.7] and the related exercises for some examples.) Although there is a classification scheme for r.e. sets along the simple to creative line (see [5, §8.7]) it is admitted to be somewhat artificial and arbitrary. Moreover there does not seem to have been much recent work on the nonsimple sets.