Let (X, T) be a topological space and *X a nonstandard extension of X. Sets of the form *G. where G ∈ T, form a base for the “standard” topology ST on *X. The topological space (*X, ST) will be used to study compactifications of (X, T) in a systematic way.

The logical system P-W is an implicational non-commutative intuitionistic logic denned by axiom schemes B − (b → c) → (a → b) → a → c. B′ = (a → b) → (b → c) → a → c. I - a → a with the rules of modus ponens and substitution. The P-W problem is a problem asking whether α - β holds if α → β and β → α are both provable in P-W. The answer is affirmative. The first to prove this was E. P. Martin by a semantical method.

In this paper, we give the first proof of Martin's theorem based on the theory of simply typed λ-calculus. This proof is obtained as a corollary to the main theorem of this paper, shown without using Martin's Theorem, that any closed hereditary right-maximal linear (HRML) λ-term of type α → α is βη-reducible to λxx. Here the HRML λ-terms correspond, via the Curry-Howard isomorphism, to the P-W proofs in natural deduction style.

A bracket abstraction algorithm is a means of translating λ-terms into combinators. Broda and Damas, in [1], introduce a new, rather natural set of combinators and a new form of bracket abstraction which introduces at most one combinator for each λ-abstraction. This leads to particularly compact combinatory terms. A disadvantage of their abstraction process is that it includes the whole Schönfinkel [4] algorithm plus two mappings which convert the Schönfinkel abstract into the new abstract. This paper shows how the new abstraction can be done more directly, in fact, using only 2n − 1 algorithm steps if there are n occurrences of the variable to be abstracted in the term. Some properties of the Broda-Damas combinators are also considered.

We characterize the 3-stratiflable theorems of NF as a 3-stratifiable extension of NF3: and show that NF is equiconsistent with TT plus raising type axioms for sentences asserting the existence of some predicate over an atomic Boolean algebra.

We exhibit a construction which produces for every Turing machine T with two halting states μ0 and μ−1, an algebra B(T) (finite and of finite type) with the property that the variety generated by B(T) is residually large if T halts in state μ−1, while if T halts in state μ0 then this variety is residually bounded by a finite cardinal.

We show that each non-compact Polish group admits a continuous action on a Polish space with non-smooth orbit equivalence relation. We actually construct a free such action. Thus for a Polish group compactness is equivalent to all continuous free actions of this group being smooth. This answers a question of Kechris. We also establish results relating local compactness of the group with its inability to induce orbit equivalence relations not reducible to countable Borel equivalence relations. Generalizing a result of Hjorth, we prove that each non-locally compact, that is, infinite dimensional, separable Banach space has a continuous action on a Polish space with non-Borel orbit equivalence relation, thus showing that this property characterizes non-local compactness among Banach spaces.

Combining techniques of the first author and Shelah with ideas of Magidor, we show how to get a model in which, for fixed but arbitrary finite n, the first n strongly compact cardinals k1..…kn are so that ki; for i = 1..…n is both the ith measurable cardinal and supercompact. This generalizes an unpublished theorem of Magidor and answers a question of Apter and Shelah.

The idea of the positive theory is to avoid the Russell's paradox by postulating an axiom scheme of comprehension for formulas without “too much” negations. In this paper, we show that the axiom of choice is inconsistent with the positive theory .