This note is written in reply to López-Escobar's paper [L-E] where a “sequence” of intermediate propositional systems NLCn (n ≥ 1) and corresponding implicative propositional systems NLICn (n ≥ 1) is given. We will show that the “sequence” NLCn contains three different systems only. These are the classical propositional calculus NLC1, Dummett's system NLC2 and the system NLC3. Accordingly (see [C], [Hs2], [Hs3], [B 1], [B2], [Hs4], [L-E]), the problem posed in the paper [L-E] can be formulated as follows: is NLC3a conservative extension of NLIC3? Having in mind investigations of intermediate propositional calculi that give more general results of this type (see V. I. Homič [H1], [H2], C. G. McKay [Mc], T. Hosoi [Hs 1]), in this note, using a result of Homič (Theorem 2, [H1]), we will give a positive solution to this problem.

NLICnand NLCn. If X and Y are propositional logical systems, by X ⊆ Y we mean that the set of all provable formulas of X is included in that of Y. And X = Y means that X ⊆ Y and Y ⊆ X. A(P1/B1, …, Pn/Bn) is the formula (or the sequent) obtained from the formula (or the sequent) A by substituting simultaneously B1, …, Bn for the distinct propositional variables P1, …, Pn in A.

Let Cn(n ≥ 1) be the string of the following sequents:

Having in mind that the calculi of sequents can be understood as meta-calculi for the deducibility relation in the corresponding systems of natural deduction (see [P]), the systems of natural deductions NLCn and NLICn (n ≥ 1), introduced in [L-E], can be identified with the calculi of sequents obtained by adding the sequents Cn as axioms to a sequential formulation of the Heyting propositional calculus and to a system of positive implication, respectively (see [C], [Ch], [K], [P]).

This paper is the first of several works: the matter is the systematic rewriting of usual results of ordinal notations, using Π½-logic. The purpose of these articles is already realized in Π½-logic, in a relatively abstract form; also, they are examples for Π½-logic. The first and second articles in this series are complete versions of unpublished results that one of authors had announced in 1976.

Veblen introduced his hierarchy in 1908. The hierarchy has since been widely used in proof theory; in fact, what is actually needed is the more natural hierarchy: , with (not necessarily a Cantor normal form) and K the Veblen hierarchy.

We shall show:

i) φ is in fact a two-variables functor (denoted by V in the text) commuting with direct limits and with pull-backs.

ii) The property of commutation with pull-backs permits one to give a representation of each ordinal x < φα(0) by means of a finite sequence of ordinals < α. This representation is independent of α it is ⊆ x, and can be used in the building up of notations systems. This representation is the “blowing-up” of the Cantor normal form, and it corresponds exactly to the denotation of ordinals < φα(0) by means of the functions (φβ)β<α.

In a recent series of papers Kanamori ([4], [5], and [6]) defines generalizations of several combinatorial principles known to follow from the existence of morasses. Kanamori proves the consistency of his generalizations by forcing arguments which come close to satisfying the hypotheses of the Martin's Axiom-type characterizations of morasses developed independently by Shelah and Stanley [9] and the author [12]. A similar “almost application” of morasses appears in [11], in which Todorčević uses forcing to prove the consistency of the existence of Kurepa trees with no Aronszajn or Cantor subtrees. In all cases the attempted proofs using morasses fail for the same reason: the partial orders involved do not have strong enough closure properties.

In an attempt to solve this problem Shelah and Stanley strengthened their characterization of morasses to allow applications to what they called “good canonical limit” partial orders. However, for rather subtle reasons even this strengthened forcing axiom is not good enough for the proposed applications. The problem this time is that Shelah and Stanley's “weak commutativity of Lim and restriction” requirement (see [9, 3.9(iv)]) is not satisfied. Furthermore, there is reason to believe that an ordinary morass is just not good enough for these applications, since in L morasses exist at all regular uncountable cardinals, but even a weak form of Todorčević's conclusion cannot hold at ineffable cardinals (see the end of §4).

A possible solution to this problem is suggested by the fact that □κ is equivalent to a forcing axiom which applies to partial orders satisfying precisely the kind of weak closure conditions involved in the examples described above (see [13]). What is needed to make the proposed morass applications work is something which will do for morass constructions what □κ does for ordinary transfinite recursion constructions. In this paper we show how extra structure can be built into a morass to accomplish this goal.

We prove the elimination of Magidor-Malitz quantifiers for R-modules relative to certain -core sentences and positive primitive formulas. For complete extensions of the elementary theory of R-modules it follows that all Ramsey quantifiers (ℵ0-interpretation) are eliminable. By a result of Baldwin and Kueker [1] this implies that there is no R-module having the finite cover property.

In [1] two interesting invariance notions were introduced: the notions of a set of godel numbers being invariant to automorphisms of the structures (ω, ·) and (ω, E) respectively. Here, · and E are defined by n · m ≃ φn (m) and nEm if and only if n Є Wm, where {φn} and {Wn} are acceptable enumerations of the partial recursive functions and r.e. sets respectively. In this paper we continue the study of the invariant sets, and especially the invariant r.e. sets, of gödel numbers.

We start off with an easy result which characterizes the Turing degrees containing invariant sets. We then take a closer look at r.e. sets invariant with respect to automorphisms of (ω,E). Using the characterization [1, Theorem 4.2] of such sets, we will derive a somewhat different characterization (which was stated, but not proved, in [1, Proposition 4.4]) and, using it as a tool for constructing invariant sets, prove that the r.e. sets invariant with respect to automorphisms of (ω, E) cannot be effectively enumerated.

We will next discuss representations of r.e. sets invariant with respect to automorphisms of (ω, ·). Although these sets do not have as nice a characterization as the r.e. sets invariant with respect to automorphisms of (ω, E) do, the techniques of [1] can still profitably be used to investigate their structure. In particular, if f is a partial recursive function whose graph is invariant with respect to automorphisms of (ω, ·), then for every a in the domain of f, there is a term t(a) built up from a and · only such that f(a) ≃ t(a). This is an analog to [1, Corollary 4.3]. We will also prove an analog to a result mentioned in the previous paragraph: the r.e. sets invariant with respect to automorphisms of (ω, ·) cannot be effectively enumerated.

This work results from an attempt to give the vague notion of relevance a concrete semantical interpretation. The idea is that propositions may be divided into different “domains of relevance”. Each “domain” has its own “T” and “F” values, and propositions “belonging” to one domain can never entail propositions “belonging” to another, unconnected one.

The semantics we have developed were found to correspond to an already known system, which we call here RMI⥲. Its axioms are the implication-negation axioms of the system RM ([1, Chapter 5]). However, as Meyer has shown, RM is not a conservative extension of RMI⥲, since RMI⥲ has the sharing-of-variables property ([5], and [1, pp. 148–149[), which the implication-negation fragment of RM has not.

RMI⥲ has four advantages in comparison to its more famous sister R⥲ (the pure intentional fragment of the system R; see [1]):

a) It has a very natural (from a relevance point of view) many-valued semantics, the simple form of which we describe here.

b) RMI⥲, ⊢ A1 → [A2 → (… → (An → A) …)] iff there is a proof of A from the set {A1, …, An} that actually uses all the members of this set. In R⥲, this holds only if we talk about “sequences” instead of “sets”. This is somewhat less intuitive (see [1, pp. 394–395]).

c) RMI⥲ is a maximal “natural” relevance logic, in the sense that every proper extension of it limits the number of “domains of relevance” (§III).

Assume G.C.H. We prove that for singular λ, □λ implies the diamonds hold for many S ⊆ λ+ (including S ⊆ {δ:δ ∈ λ+, cf δ = cf λ}.) We also have complementary consistency results.

We present a formal theory of propositions and combinator terms, and in this theory we give an interpretation of Martin-Löf's type theory. The construction of the interpretation is inspired by the semantics for type theory, but it can also be viewed as a formalized realizability interpretation.

§1. A counterexample and preservation of “proper + X”.

Theorem. Suppose V satisfies, , and for some A ⊆ ω1, every B ⊆ ω1, belongs to L[A].

Then we can define a countable support iterationsuch that the following conditions hold:

a) EachQiis proper and ⊩Pi “Qi, has power ℵ1”.

b) Each Qi is -complete for some simple ℵ1-completeness system.

c) Forcing with Pα = Lim adds reals.

Proof. We shall define Qi by induction on i so that conditions a) and b) are satisfied, and Ci, is a Qi-name of a closed unbounded subset of ω1. Let : ξ < ω1› ∈ L[A] be a list of all functions f which are from δ to δ for some δ < ω1 and let h: ω1 → ω1, h ∈ L[A], be defined by h(α) = Min{β: β > α and Lβ[A]⊨ “∣α∣ = ℵ0”}.

Suppose we have defined Qj for every j < i; then Pi is defined, is proper (as each Qj, j < i, is proper, and by III 3.2) and has a dense subset of power ℵ (by III 4.1). Let Gi ⊆ Pi be generic so clearly there is B ⊆ ω1, such that in V[Gi] every subset of ω1 belongs to L[A, B], The following now follows:

Fact. In V[Gi], every countableN ⥽(H(ℵ2), ∈, A, B) is isomorphic toLβ[A ∩ δ, B ∩ δ] for some β < h(δ), where δ = δ(N) = ω1, ∩ N.

The universe of sets, V, is usually seen as an entity structured in successive levels, each level being made up of objects and collections of objects belonging to the previous levels. This process of obtaining sets and axioms for set theory can be seen in Scott [74] and Shoenfield [77].

The approach we want to take differs from the previous one very strongly: the seeds from which we want to generate our universe of classes are to be the one-variable predicates (given by one-free-variable formulas) of the formal language we shall be using. In other words, any one-variable predicate of the language is to be represented as a class in our universe. In this sense, we view our theory as being about a self-referential language, a language whose predicates refer to objects which are predicates of the language itself.

We want, in short, a system such that: (i) any predicate may be represented by an object to be studied by the theory itself; (ii) the axioms for the theory may be derived from the general principle that we are dealing with a language that aims at describing its own predicates; and (iii) the theory should be strong enough to derive ZFC and suggest answers to the existence of large cardinals and to the continuum hypothesis.

An objection to such a project arises immediately: in view of the Russell-Zermelo paradox, how is it possible to have all predicates of the language as elements of the universe? This objection will be easy to deal with: we shall provide our language with a type structure to avoid paradox.

Several stepping up lemmas are proved which are then used to investigate the connection between definable partition relations and admissible ordinals.

Abstract.

Banach introduced the following two-person, perfect information, infinite game on the real numbers and asked the question: For which sets A ⊆ R is the game determined?

Rules: The two players alternate moves starting with player I. Each move an is legal iff it is a real number and 0 < an, and for n > 1, an < an−1. The first player to make an illegal move loses. Otherwise all moves are legal and I wins iff exists and .

We will look at this game and some variations of it, called Banach games. In each case we attempt to find the relationship between Banach determinacy and the determinacy of other well-known and much-studied games.

We consider extensions of Peano arithmetic suitable for doing some of nonstandard analysis, in which there is a predicate N(x) for an elementary initial segment, along with axiom schemes approximating ω1-saturation. We prove that such systems have the same proof-theoretic strength as their natural analogues in second order arithmetic. We close by presenting an even stronger extension of Peano arithmetic, which is equivalent to ZF for arithmetic statements.

Solovay has shown that if F: [ω]ω → 2 is a clopen partition with recursive code, then there is an infinite homogeneous hyperarithmetic set for the partition (a basis result). Simpson has shown that for every 0α, where α is a recursive ordinal, there is a clopen partition F: [ω]ω → 2 such that every infinite homogeneous set is Turing above 0α (an anti-basis result). Here we refine these results, by associating the “order type” of a clopen set with the Turing complexity of the infinite homogeneous sets. We also consider the Nash-Williams barrier theorem and its relation to the clopen Ramsey theorem.

Let A and B be subsets of the space 2N of sets of natural numbers. A is said to be Wadge reducible to B if there is a continuous map Φ from 2N into 2N such that A = Φ−1 (B); A is said to be monotone reducible to B if in addition the map Φ is monotone, that is, a ⊂ b implies Φ(a) ⊂ Φ(b). The set A is said to be monotone if a ∈ A and a ⊂ b imply b ∈ A. For monotone sets, it is shown that, as for Wadge reducibility, sets low in the arithmetical hierarchy are nicely ordered. The sets are all reducible to the ( but not ) sets, which are in turn all reducible to the strictly sets, which are all in turn reducible to the strictly sets. In addition, the nontrivial sets all have the same degree for n ≤ 2. For Wadge reducibility, these results extend throughout the Borel hierarchy. In contrast, we give two natural strictly monotone sets which have different monotone degrees. We show that every monotone set is actually positive. We also consider reducibility for subsets of the space of compact subsets of 2N. This leads to the result that the finitely iterated Cantor-Bendixson derivative Dn is a Borel map of class exactly 2n, which answers a question of Kuratowski.

This is the third and last part of an investigation in several topics of first order model theory of modules which I began in Some model theory of modules. I. On total transcendence of modules (this Journal, vol. 48 (1983), pp. 570–574) and continued in Some model theory of modules. II. On stability and categoricity of flat modules (this Journal, vol. 48 (1983), pp. 970–985). Throughout I refer to these papers as “Part I” and “Part II”.

Although these parts are only loosely connected, I will tacitly use the notation and preliminaries already introduced in the preceding ones. Further details are given in §0. Concerning Part II, the reader is assumed to be familiar with most of §1, a very small part of §4, and the criterion for elementary equivalence of modules over regular rings given in §3 (Lemma 20) of that part.

Using those tools in this paper I consider completeness of the (elementary) theory of all modules (and of some of its extensions) (§2), and eliminability of cardinality quantifiers in the elementary theories of modules (§3).

§2 is almost entirely devoted to a new short proof based on the technique developed in Part II of a theorem of Tukavkin which seems to be the first relevant result concerning the completeness of the theory of all modules (after the simple observation that any theory of infinite vector spaces is complete).