An arithmetical interpretation of dynamic propositional logic (DPL) is a mapping ƒ satisfying the following: (1) ƒ associates with each formula A of DPL a sentence ƒ(A) of Peano arithmetic (PA) and with each program α a formula ƒ(α) of PA with one free variable describing formally a supertheory of PA; (2) ƒ commutes with logical connectives; (3) ƒ([α]A) is the sentence saying that ƒ(A) is provable in the theory ƒ(α); (4) for each axiom A of DPL, ƒ(A) is provable in PA (and consequently, for each A provable in DPL, ƒ(A) is provable in PA). The arithmetical completeness theorem is proved saying that a formula A of DPL is provable in DPL iff for each arithmetical interpretation ƒ, ƒ(A) is provable in PA. Various modifications of this result are considered.

In this paper we consider the special case of the Banach-Mazur game played on a topological space when the space also has an underlying Boolean Algebra structure. This case was first studied by Jech [2]. The version of the Banach-Mazur game we will play is the following game played on the Boolean algebra:

Players I and II alternate moves playing a descending sequence of elements of a Boolean algebra ℬ.

Player II wins the game iff Πi∈ωbi ≠ 0. Jech first considered these games and showed:

Theorem (Jech [2]). ℬ is (ω1, ∞)-distributive iff player I does not have a winning strategy in the game played on ℬ.

If ℬ has a dense ω-closed subset then it is easy to see that player II has a winning strategy in this game. This paper establishes a partial converse to this, namely it gives cardinality conditions on ℬ under which II having a winning strategy implies ω-closure.

In the course of proving the converse, we consider games of length > ω and generalize Jech's theorem to these games. Finally we present an example due to C. Gray that stands in counterpoint to the theorems in this paper.

In this section we give a few basis definitions and explain our notation. These definitions are all standard.

In Forcing and reducibilities, I [20] we reviewed various forcing techniques in the context of arithmetic. This second part deals with the same topics in the context of analysis. The numbering of sections is the continuation of the numbering of the first part. For the reader's convenience, we collect in an independent bibliography the papers which are referred to in this part.

In spite of the philosophical significance of higher-order modal logic, the modal logician's main concern has been with sentential logic. In this paper we do not intend to go into philosophical details, but we only remark that higher-order modal logic has a close relationship with Montague's well-known idea of “universal grammar”, which is an ambitious attempt to build a logical theory of natural languages with exact syntax and semantics, comparable with the artificial languages of mathematical logic. For this matter, the reader can consult, e.g., Montague [8], [9] and Gallin [2].

We find necessary and sufficient conditions for compatibility of the Myhill and the Nerode extensions of a combinatorial function to the isols.

In this paper we will investigate closure properties of the class of immune sets and of other, more restricted classes of sets of natural numbers under union and similar operations. Our main theorem states that there exists a hyperhyperimmune set A such that A join A is not hyperhyperimmune. We will prove this result in §3. The remaining questions about closure properties are all much easier and will be answered in §2.

We consider the closure properties which are given in the following

Definition. Let bea. class of subsets of N.

(i) is closed under join, if A join B ∈ whenever A ∈ and B ∈ .

(ii) is closed under self-join, if A join A ∈ whenever A ∈ .

(iii) is closed under union, if A ⋃ B ∈ whenever A ∈ and B ∈ .

(iv) is closed under cartesian product, if A × B ∈ whenever A ∈ and B ∈ .

(v) is closed under cartesian self-product, if A × A ∈ whenever A ∈ .

The almost-finiteness classes we consider are the following:

1 = {X∣X is immune};

2 = {X∣ simple};

3 = {X∣X is hyperimmune};

4 = {X∣ is hypersimple};

5 = {X∣X is strongly hyperimmune};

6 = {X∣ is strongly hypersimple};

7 = {X∣X is hyperhyperimmune};

8 = {X∣X is strongly hyperhyperimmune};

9 = {X∣ is hyperhypersimple};

10 = {X∣X is dense immune};

11 = {X∣ is dense simple}.

The definitions are all contained in Rogers [4, Chapter 12] and/or in Robinson [3]. In addition we could consider the class of -strongly hyperimmune sets and classes defined by lim-properties; see Rogers [4, pp. 243–244]. For other classes, however, such as the class of cohesive sets, the questions become trivial.

Our aim is to establish the table on the next page.

This paper deals with forcing in arithmetic (as first introduced by Feferman [2]) and its connections with recursive function theory. We define for each n ≥ 1 the class of weakly n-generic sets. We prove that these classes merge with the classes of n-generic sets to form the hierarchy suggested by the terminology. Our notation is the same as that of Jockusch [5].

A large number of formal systems based on combinatory logic or λ-calculus have been extended to include first order predicate calculus. Several of these however have been shown to be inconsistent, all, as far as the author knows, in the strong sense that all well formed formulas (which here include all strings of symbols) are provable. We will call the corresponding consistency notion—an arbitrary wff ⊥ is provable—weak consistency. We will say that a system is strongly consistent if no formula and its negation are provable.

Now for some systems, such as that of Kuzichev [11], the strong and weak consistency notions are equivalent, but in the systems of [5] and [6], which we will be considering, they are not. Each of these systems is strong enough to have all of ZF set theory, except Grounding and Choice, interpretable in it, and the system of [5] can also encompass first order arithmetic (see [7]). It therefore seems unlikely that a strong consistency result could be proved for these systems using elementary methods. In this paper however, we prove the weak consistency of both these systems by means that could be formulated, at least within the theory of [5]. The method also applies to the typed systems of Curry, Hindley and Seldin [10] and to Seldin's generalised types [12].

We begin with a simple observation and a simple question. If we fix Th(x), some reasonable formulation of “x is the Gödel number of a theorem of Peano Arithmetic”, then for any sentence σ, Peano Arithmetic proves σ → Th(⌈σ⌉). (Here ⌈σ⌉ is the canonical term denoting the Gödel number of σ.) This observation is crucial to the proof of the Second Incompleteness Theorem. Call ψ a self-prover (with respect to Th(x)) if ψ → Th(⌈ψ⌉) is a theorem of Peano Arithmetic (from now on, PA). Our simple question is this: Does the observation have a converse—must every self-prover be provably equivalent to a sentence? Whatever φ happens to be, the formula φ ∧ Th(⌈φ⌉) is a self-prover. This makes it seem clear that there must exist self-provers which are not provably : We need only use a suitably complicated φ.

Deciding what sort of complication is “suitable” and finding such a φ is surprisingly annoying. Here is a quick example: One might guess that any φ which is unprovable and would work. In that case we could take φ to be CON—that is, ¬Th(⌈0 = 1⌉); but CON ∧ Th(⌈CON⌉) is refutable in PA, so is provably equivalent to the quantifier-free formula 0 = 1.

In his thesis [2] Sy Friedman showed that for any inadmissible ordinal β there is an easily definable β-r.e. degree, 01/2, with β-degree strictly between 0 and 0′. In this paper we define, for some weakly admissible β, a β-r.e. degree 03/4 intermediate between 01/2 and 0′. This definition is given using the machinery developed by Maass [6], [7] to study the β-r.e. degrees.

A weak jump for β-recursion theory, the ½-jump, was first defined by Friedman [4] and has been studied by Sacks and Homer [5]. The degree 03/4 is of interest because of the way it interacts with the half-jump. In particular 03/4 forms the boundary between the half-jump of “hyperregular” and “nonhyperregular” β-r.e. degrees less than 01/2. One consequence of these results is that there are weakly inadmissible β for which either the jump theorem or the density theorem fails.

Much of this paper is based on earlier work in this area by Sy Friedman and Wolfgang Maass. I would like to thank Gerald Sacks for many helpful conversations and comments concerning this work.

The modern theory of computability is based on the works of Church, Markov and Turing who, starting from quite different models of computation, arrived at the same class of computable functions. The purpose of this paper is the show how the main results of the Church-Markov-Turing theory of computable functions may quickly be derived and understood without recourse to the largely irrelevant theories of recursive functions, Markov algorithms, or Turing machines. We do this by ignoring the problem of what constitutes a computable function and concentrating on the central feature of the Church-Markov-Turing theory: that the set of computable partial functions can be effectively enumerated. In this manner we are led directly to the heart of the theory of computability without having to fuss about what a computable function is.

The spirit of this approach is similar to that of [RGRS]. A major difference is that we operate in the context of constructive mathematics in the sense of Bishop [BSH1], so all functions are computable by definition, and the phrase “you can find” implies “by a finite calculation.” In particular if P is some property, then the statement “for each m there is n such that P(m, n)” means that we can construct a (computable) function θ such that P(m, θ(m)) for all m. Church's thesis has a different flavor in an environment like this where the notion of a computable function is primitive.

One point of such a treatment of Church's thesis is to make available to Bishopstyle constructivists the Markovian counterexamples of Russian constructivism and recursive function theory. The lack of serious candidates for computable functions other than recursive functions makes it quite implausible that a Bishopstyle constructivist could refute Church's thesis, or any consequence of Church's thesis. Hence counterexamples such as Specker's bounded increasing sequence of rational numbers that is eventually bounded away from any given real number [SPEC] may be used, as Brouwerian counterexamples are, as evidence of the unprovability of certain assertions.

If C1, …, Ck are members of a certain class of suitable categories (which contains those arising from models with dimension), C = C1 × … × Ck, C′ is a suitable category, F: C → C′ is a partial recursive combinatorial functor satisfying a certain property (which, if C = C1, is that F is nonconstant) and ∈ C, then (1) if F is regressive so is as is each i, and (2) if F is Dedekind then each i is Dedekind.

Under a weak set-theoretic assumption we interpret second-order logic in the monadic theory of order.

A many-one degree is functional if it contains the index set of some class of partial recursive functions but does not contain an index set of a class of r.e. sets. We give a natural embedding of the r.e. m-degrees into the functional degrees of 0′. There are many functional degrees in 0′ in the sense that every partial-order can be embedded. By generalizing, we show there are many functional degrees in every complete Turing degree.

There is a natural tie between the studies of index sets and partial-many-one reducibility. Every partial-many-one degree contains one or two index sets.

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.

In a recent work on -recursivity ([M] in which the principal result of [G.2] is obtained by more direct methods) I have given a very simple lemma on the structure of the so-called rungs [G.1] which enables us to describe precisely the relation they have with Jervell's tree [J] in the wellfounded case (R.O. Gandy has also worked out this relation in the wellfounded case, but with techniques very different from those introduced here).

In the present work I treat the general case in §§I and II. The results are summarized in §III. These results are then exploited by means of the operator Σ*, which is the principal tool of Lemma (4.1) (i.e. the generalized version of the previous structural lemma). A systematical use of Σ* is also the basis of the study of ladders in the last section.

Recall that the notions of rungs and ladders were introduced by J.-Y. Girard [G, 0] and [G.1] as a basic tool for ordinal analysis of Π½-logic (i.e. roughly speaking, logic with β-rule) and the introduction of trees and homogeneous trees by H.R. Jervell is a first attempt to give to these notions a geometrical interpretation without any explicit mention of functors and categorical limits. In fact, as it is evident in the last section of the present paper, limits play a modest role in the study of ladders, but their properties with respect to limits are of fundamental interest in their applications (a domain which is out of the scope of this basic work. Readers interested in applications to ordinal analysis are invited to consult the references). Note that my trees are not exactly those introduced by H.R. Jervell: his extra symbols λ and ρ are respectively changed into 0 and the type of the concerned tree. More important is the fact that we consider only the extremal nodes of the trees. The objects so obtained are then as close as possible to Girard's dendroids.

I am indebted to J.-Y. Girard for his advice about my work.