Zermelo in his remarkable 1930a offered his final axiomatization of set theory as well as a striking, synthetic view of a procession of natural models that would have a modern resonance. Appearing only six articles after Skolem 1930 in Fundamenta mathematicae, Zermelo’s article seemed strategically placed as a response, an aspect that we will discuss below, but its dramatically new picture of s...

If the power set operation is considered as a definite operation, but the universe of all sets is regarded as an indefinite totality, we are led to systems of set theory having Power Set as an axiom but only Bounded Separation axioms and intuitionistic logic for reasoning about the universe at large. The study of subsystems of ZF formulated in intuitionistic logic with Bounded Separation but co...

separation logics are a family of extensions of Hoare logic for reasoning about programs that manipulate resources such as memory locations. These logics are “abstract” because they are independent of any particular concrete resource model. Their assertion languages, called propositional abstract separation logics (PASLs), extend the logic of (Boolean) Bunched Implications (BBI) in various ways...

There is a gap in the proof of Lemma VII.4 in [1]. We present an alternative proof of Theorem B (C 1 Ω-stable vector fields satisfy Axiom A) in [1]. The novel and essential part in the proof of the stability and Ω-stability conjectures for flows is the connecting lemma introduced in [1]. A mistake in the proof of the last conjecture was pointed out to me by Toyoshiba [5], who later also provide...

In the present paper, we introduce soft generalized closed sets in soft topological spaces which are defined over an initial universe with a fixed set of parameters. A sufficient condition for a soft g-closed set to be a soft closed is also presented. Moreover, the union and intersection of two soft g-closed sets are discussed. Finally, the new soft separation axiom, namely soft 1 2 T space is ...

The power and usefulness of cleavability (also known as splittability) have been well established within the framework of topology by A.V. Arhangel’skii and his associates. The concept was transferred to partially ordered sets by D.J. Marron and T.B.M. McMaster. The connections between topology and order are here exploited as we examine the properties enjoyed by a topological space that is spli...

We define a constructive topos to be a locally cartesian closed pretopos. The terminology is supported by the fact that constructive toposes enjoy a relationship with constructive set theory similar to the relationship between elementary toposes and (impredicative) intuitionistic set theory. This paper elaborates upon one aspect of the relationship between constructive toposes and constructive ...