The purpose of this paper is to suggest two alternatives to Quine's definition of closure. These new definitions have two advantages over Quine's definition, and they probably are the simplest definitions having both advantages. The two advantages are:

(1). Principle *101 becomes superfluous and may be dropped from Quine's set of principles for quantification. (In the case of my second definition, however, the dropping of *101 must be balanced by a slight change in *104.)

(2). Closure is made independent of the alphabetical order of variables.

The second of these advantages turns on the fact that the “alphabetical order” possessed by variables in virtue of their respective positions in the alphabet (or arbitrarily assigned to them) is a mere convention and not of genuine logical significance. It seems therefore desirable to consider some alternatives to Quine's definition of closure, since according to his definition the closure of a given formula will be one statement or another, depending upon whether or not one letter of the alphabet is alphabetically prior to a certain other letter. It is interesting that the removal of this minor artificiality also enables us to dispense with *101.

According to Quine, the closure of a formula containing n free variables is obtained by prefixing to it in alphabetical order the n universal quantifiers formed from these variables by enclosing each in a pair of parentheses. (If n = 0 the formula is its own closure and is a “statement” rather than a “matrix.”) Thus the statement ‘(x)(y)(z)(xϵy ▪ yϵz)’ would be the closure of ‘xϵy ▪ yϵz’, but ‘(z)(x)(y)(xϵy ▪ yϵz)’ would not be its closure. Now there is no reason why ‘(z)(x)(y)(xϵy ▪ yϵz)’ or (y)(z)(x)(xϵy ▪ yϵz) and so on, could not just as well be regarded as “the” closure of ‘(xϵy ▪ yϵz’ as ‘(x)(y)(z)(xϵy ▪ yϵz)’. I therefore propose to allow to each formula not merely one closure, but as many closures as can be obtained by permuting in various ways the n prefixed universal quantifiers. In this way alphabetical order becomes irrelevant and no preference is given to one order of prefixed quantifiers in contrast to other orders which seem equally good. This constitutes my first redefinition of closure.

We improve the upper bound for the consistency strength of stationary reflection at successors of singular cardinals.

Let X and Y be uncountable Polish spaces. We show in ZF that there is a coanalytic subset P of X × Y with countable sections which cannot be expressed as the union of countably many partial coanalytic, or even PCA = , graphs. If X = Y = ωω, P may be taken to be . Assuming stronger set theoretic axioms, we identify the least pointclass such that any such coanalytic P can be expressed as the union of countably many graphs in this pointclass. This last result is extended (under suitable hypotheses) to all levels of the projective hierarchy.

An example is given of a complete theory with minimal models of arbitrarily large minimality rank.

A basic goal in model-theoretic algebra is to obtain the classification of the complete extensions of a given (first-order) algebraic theory.

Results of this type, for the theory of totally ordered abelian groups, were obtained first by A. Robinson and E. Zakon [5] in 1960, later extended by Yu. Gurevich [4] in 1964, and further clarified by P. Schmitt in [6].

Within this circle of ideas, we give in this paper an axiomatization of the first-order theory of the class of all direct products of totally ordered abelian groups, construed as lattice-ordered groups (l-groups)—see the theorem below. We think of this result as constituing a first step—undoubtedly only a small one—towards the more general goal of classifying the first-order theory of abelian l-groups.

We write groups for abelian l-groups construed as structures in the language 〈 ∨, ∧, +, −, 0〉 (“−” is an unary operation). For unproved statements and unexplicated definitions, the reader is referred to [1].

We study the following open question in computable model theory: does there exist a structure of computable dimension two which is the prime model of its first-order theory? We construct an example of such a structure by coding a certain family of c.e. sets with exactly two one-to-one computable enumerations into a directed graph. We also show that there are examples of such structures in the classes of undirected graphs, partial orders, lattices, and integral domains.