According to Sobociński's recollection, the consistency of Mereology was proved by Leśniewski by means of an appropriate interpretation within the framework of the theory of real numbers. His proof was never published, but in a recent paper R. E. Clay has succeeded in reconstructing a version of it.1 Clay's result amounts to showing that if Lesniewski's Ontology expanded by the addition of the axioms for the real numbers is consistent then Mereology is consistent. Without casting any doubts on the validity of the proof one can hardly fail to note that here we have a case where the consistency of a conceptually simple theory is made to depend on the consistency of a theory which from the point of view of intuition is far from being obvious.

A formula α of modal LPC is a de re modality iff it contains a free individual variable within the scope of a modal operator; otherwise α is de dicto.1 We shall say that de re modalities are eliminable in a system S of modal LPC iff there is a transformation R on wffs of S such that R(α) is de dicto and ├sα ≡ R(α). We shall show that where S is LPC + S5 + Pr2 then de re modalities are not eliminable in S.

Cylindric algebras were introduced by Alfred Tarski about 1952 to provide an algebraic analysis of (first-order) predicate logic. With each cylindric algebra one can, in fact, associate a certain, in general infinitary, predicate logic; for locally finite cylindric algebras of infinite dimension the associated predicate logics are finitary. As with Boolean algebras and sentential logic, the algebraic counterpart of completeness is representability. Tarski proved the fundamental result that every locally finite cylindric algebra of infinite dimension is representable.

The notion of polyadic algebra was introduced by Halmos to reflect algebraically the predicate logic without equality. Later Halmos enriched the study with the introduction of the notion of equality. These algebras are very closely related to the cylindric algebras of Tarski. The notion of diagonal free cylindric algebra predates that of cylindric algebra and is also due to Tarski. The theory of diagonal free algebras forms an important fragment of the theories of polyadic and cylindric algebras.

This paper compares the subsystem of analysis based on the Π11-comprehension axiom (Π11-CA) with classical Bar Induction (BI). An extension of Bar Induction using infinitary formulae is also considered, which was suggested to us by G. Kreisel (oral communication).

To describe these formal systems, we first describe the language, L, appropriate for their formalization. L contains the language of ENT (1st order arithmetic) plus function variables ƒi, gi, hi, etc. Number-theoretic terms are built up from +, × , ′ , 0, and application.

We use the word “number” for “nonnegative integer” and “set” for collection of numbers. The set of all numbers is denoted by ε and the empty set by ο. We write ⊂ for the relation of inclusion and ⊆ for that of proper inclusion. If ƒ is a function from a subset of ε into ε, its domain and range are denoted by δƒ and ρƒ respectively. For any collection Γ of entities, card Γ or card (Γ) stands for the cardinality of Γ. The cardinality of the continuum is denoted by c.

Since Cohen [1] showed the independence of the axiom of choice from the other axioms of ZF, a number of people have used his forcing technique to show how badly choice can fail while certain vestiges of it remain. Many of these arguments involve the introduction of a generic sequence or generic set of generic sets of ordinals into a countable standard model of ZF + V = L. The desired results then follow by the use of clever permutation and forcing arguments. (See Cohen [2, pp. 136–142].)

It is known that the theorems of the intuitionist predicate calculus are exactly those formulas which are valid in every Heyting algebra (that is, pseudo-Boolean algebra). The simplest kind of Heyting algebra is a linearly ordered set. This paper concerns the question of determining all formulas which are valid in every linearly ordered Heyting algebra. The question is of interest because it is a particularly simple case intermediate between the intuitionist and classical logics. Also the interpretation of implication is such that in general there exists no nondiscrete Hausdorff topology for which this operation is continuous.

Consider the predicate of natural numbers defined by:

where R is recursive. If, as usual, the variable ƒ ranges over ωω (the set of functions from natural numbers to natural numbers) then this is just the usual normal form for Π11 sets. If, however, ƒ ranges over 2ω (the set of functions from ω into {0, 1}) then every such predicate is recursively enumerable.3 Thus the second type of formula is generally ignored. However, the reduction just mentioned requires proof, and the proof uses some form of the Brower-König Infinity Lemma.

The results of the present paper were announced in [1]. The work is divided into our parts. In the first part we define relations (relations between relational struc tures) and we show their connection with the equivalence of the languages LΚ,λ (Theorem 1). The relations generalize the games Gn (n < ω) of Ehrenfeucht (see [2]) and the conditions (i)–(ii) which were used by Karp in [5].

In this paper we prove infinitary analogues of model-theoretic results known for finitary logic. The infinitary language we deal with is Lω1ω which is roughly described by saying that, in addition to the usual formation rules of the lower predicate calculus with identity, also the formation of the conjunction and disjunction of countably many formulas is allowed.

By γ, we mean the rule, “From ├ A and ├ Ā V B, infer ├ B”.1 This rule has played an important and a controversial role in a set of relevant logics free of certain well-known paradoxes of implication, like AĀ-→B and A-→(B-→B). Among these logics we count the pioneering systems of strenge Implikation presented by Ackermann in [1],2 as well as the Anderson-Belnap systems E of entail-ment and R of relevant implication.3

Dummett's LC [1] is a system which characterizes all formulas of the propositional calculus which are valid in every chain (for definitions and notation see the first section of [2]). An L-algebra is a Heyting algebra in which (x → y) + (y → x) = 1 for all x, y. L-algebras bear the same relation to LC as Boolean algebras to the classical propositional calculus and Heyting algebras to the intuitionist propositional calculus.

In 1956 W. Ackermann proposed a new axiomatic set theory, that has received some attention in recent years. (See references [3], [4], [6], [7], and [9].) This theory distinguishes between sets and classes. In this paper we study mainly the natural models of this theory. We show, among other results, that the set-theoretical fragment of these models are also models of Zermelo-Fraenkel set theory. This result gives a partial answer to the question, raised by A. Levy, of the relative strength of Ackermann's set theory with respect of Zermelo-Fraenkel's.2

In [5, Corollary 3.1] D. A. Martin characterized the degrees of the hyperhypersimple sets as those r.e. degrees a satisfying a′ = 0″. In the present paper we investigate the degrees of the hyperhyperimmune (h.h.i.) sets. We do not achieve a characterization of these degrees, but we do show that the condition a′ ≥ 0″ is sufficient, and the condition a′ ≥ 0′ is necessary, for a degree a to contain an h.h.i. set.