A dilator D is a functor from ON to itself commuting with direct limits and pull-backs. A dilator D is a flower iff D(x) is continuous. A flower F is regular iff F(x) is strictly increasing and F(f)(F(z)) = F(f(z)) (for f ϵ ON(x,y), z ϵ X).

Equalization is the following axiom: if F, G ϵ Flr (class of regular flowers), then there is an H ϵ Flr such that F ° H = G ° H. From this we can deduce that if ℱ is a set ⊆ Flr, then there is an H ϵ Flr which is the smallest equalizer of ℱ (it can be said that H equalizes ℱ iff for every F, G ϵ ℱ we have F ° H = G ° H). Equalization is not provable in set theory because equalization for denumerable flowers is equivalent to -determinacy (see a forthcoming paper by Girard and Kechris).

Therefore it is interesting to effectively find, by elementary means, equalizers even in the simplest cases. The aim of this paper is to prove Girard and Kechris's conjecture: “ is the (smallest) equalizer for Flr < ω” (where Flr < ω denotes the set of finite regular flowers). We will verify that is an equalizer of Flr < ω; we will sketch the proof that it is the smallest one at the end of the paper. We will denote by H.

The object of this note is to extend to Post's m-valued propositional calculus1 the axiomatization by Anderson and Belnap2 of the 2-valued propositional calculus.

Notation and definitions. We shall use the symbols ∼ and v for the primitives of Post and shall otherwise, except as stated below, use the same notation and definitions as in the previous paper.

We study the degree of (weak) (Q-pointness, and that of (weak) P-pointness, of ideals on a regular infinite cardinal.

Let T be a recursively enumerable theory extending Elementary Arithmetic EA. L. D. Beklemishev proved that the Σ2 local reflection principle for T, (T), is conservative over the Σ1 local reflection principle, (T), with respect to boolean combinations of Σ1-sentences; and asked whether this result is best possible. In this work we answer Beklemishev's question by showing that Π2-sentences are not conserved for T = EA + “f is total,” where f is any nondecreasing computable function with elementary graph. We also discuss how this result generalizes to n > 0 and obtain as an application that for n > 0, is conservative over IΣn with respect to Πn+2-sentences.

We will show that there is a nontrivial strongly minimal expansion of (ω, s), the natural numbers with successor. Pillay and Steinhorn [1] proved that there is no -minimal expansion of (ω, ≤). This result provides an interesting contrast.

The strongly minimal expansion of (ω, s) is very easy to describe. Consider the order-two permutation of ω, π recursively defined by

Let T be Th(ω, s, π, 0).

Let p be a set. A function Φ is uniformly Σ 1(p) in every admissible set if there is a Σ 1 formula ϕ in the parameter p so that ϕ defines Φ in every Σ 1-admissible set which includes p. A theorem of Van de Wiele states that if Φ is a total function from sets to sets then Φ is uniformly Σ 1 in every admissible set if and only if it is E-recursive. A function is ESp -recursive if it can be generated from the schemes for E-recursion together with a selection scheme over the transitive closure of p. The selection scheme is exactly what is needed to insure that the ESP -recursively enumerable predicates are closed under existential quantification over the transitive closure of p. Two theorems are established: a) If the transitive closure of p is countable then a total function on sets is ESp -recursive if and only if it is uniformly Σ 1(p) in every admissible set. b) For any p, if Φ is a function on the ordinal numbers then Φ is ESP -recursive if and only if it is uniformly Σ 1(p) in every admissible set.

The global function on finite systems is a new concept defined by Gurevich in [1] and discussed in [2] and [3]. In the last ten years this concept has become more and more useful in computer science and logic. Gurevich also pointed out the importance of global functionals on finite systems. In this paper we will give a brief introduction to the concepts of global functions and global functionals on finite systems.

In studying the natural number system N = 〈N, +,0〉 we often refer to its functions and functionals. There are a lot of books and papers in this area. Kleene in [4] gave a detailed introduction to the recursive functions of N. The functionals of N are normally very difficult to compute because here we need to tell the machine what the input function is, which is not very easy to do. In developing the theory of finite systems the functions and functionals are also very useful. For computing the functionals in finite systems we can take the entire graph of a function as the input, which is not possible in N. We will discuss recursive functions and functionals for finite systems. The definitions of recursive functions are very similar to the case in N, but we will have a very different situation. In N the number of elements is infinite. The number of all possible functions from N to N is the continuum. In a finite system the number of all possible functions is finite. It seems that there is no necessity to define the global functions.

We give several characterizations of weakly minimal abelian structures. In two special cases, dual in a sense to be made explicit below, we give precise structure theorems:

1. when the only finite 0-definable subgroup is {0}, or equivalently 0 is the only algebraic element (the co-strongly minimal case);

2. when the theory of the structure is strongly minimal.

In the first case, we identify the abelian structure as a “near-subspace” A of a vector space V over a division ring D with its induced structure, with possibly some collection of distinguished subgroups of A of finite index in A and (up to acl(∅)) no further structure. In the second, the structure is that of V/A for a vector space and near-subspace as above, with the only further possible structure some collection of distinguished points. Here a near-subspace of V is a subgroup A such that for any nonzero d ∈ D. the index of A ∩ dA, in A is finite. We also show that any weakly minimal abelian structure is a reduct of a weakly minimal module.

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].

We characterize some large cardinal properties, such as μ-measurability and P2(κ)-measurability, in terms of ultrafilters, and then explore the Rudin-Keisler (RK) relations between these ultrafilters and supercompact measures on Pκ(2κ). This leads to the characterization of 2κ-supercompactness in terms of a measure on measure sequences, and also to the study of a certain natural subset, Fullκ, of Pκ(2κ) whose elements code measures on cardinals less than κ. The hypothesis that Fullκ is stationary (a weaker assumption than 2κ-supercompactness) is equivalent to a higher order Löwenheim-Skolem property, and settles a question about directed versus chain-type unions on Pκλ.