This paper uses an atomistic ontology of universals, individuals, and facts to provide a semantics for ramified type theory. It is shown that with some natural constraints on the sort of universals and facts admitted into a model, the axiom of reducibility is made valid.

Let A be a complex, commutative unital Banach algebra. We introduce two notions of exponential reducibility of Banach algebra tuples and present an analogue to the Corach-Suárez result on the connection between reducibility in A and in C(M(A)). Our methods are of an analytical nature. Necessary and sufficient geometric/topological conditions are given for reducibility (respectively reducibility...

A c.e. real x is Solovay reducible (S-reducible) to another c.e. real y if y is at least as difficult to be approximated as x. In this case, y is at least as random as x. Thus, the S-reducibility classifies relative randomness of c.e. reals such that the c.e. random reals are complete in the class of c.e. reals under the S-reducibility. In this paper we investigate extensions of the S-reducibil...

We show that if A and B form a nontrivial K-pair, then there is a semi-computable set C such that A ≤e C and B ≤e C. As a consequence, we obtain a definition of the total enumeration degrees: a nonzero enumeration degree is total if and only if it is the join of a nontrivial maximal K-pair. This answers a long-standing question of Hartley Rogers, Jr. We also obtain a definition of the “c.e. in”...

Wadge reducibility in the Baire and Cantor spaces is very important in descriptive set theory. We consider Wadge reducibility in so called φ-spaces which are topological counterpart of the algebraic directed-complete partial orderings. It turns out that in many spaces the Wadge reducibility behaves worse than in the classical case but there exist also interesting examples of spaces with a bette...

We show the decidability of the existential theory of the recursively enumerable degrees in the language of Turing reducibility, Turing reducibility of the Turing jumps, and least and greatest element. The Existential Theory of the Poset of R.E. Degrees with a Predicate for Single Jump Reducibility

We introduce the notion of finitary computable reducibility on equivalence relations on the domain ω. This is a weakening of the usual notion of computable reducibility, and we show it to be distinct in several ways. In particular, whereas no equivalence relation can be Π n+2-complete under computable reducibility, we show that, for every n, there does exist a natural equivalence relation which...