We apply set-theoretic methods to study projective modules and their generalizations over transfinite extensions of simple artinian rings R. prove that if R is small, then the Weak Diamond implies projectivity an arbitrary module can be tested at layer epimorphisms

Sets with transfinite cardinality can be said to have countably infinite cardinality or uncountably infinite cardinality. The cardinality of the set v is said to be countably infinite if there exists an injection between v and the set of natural numbers. It is represented by the symbol א0, read ”aleph.” An uncountably infinite set has a greater cardinality than that of the set of integers; ther...

In recent years there has emerged the study of discrete computational models which are allowed to act transfinitely . By ‘discrete’ we mean that the machine models considered are not analogue machines, but compute by means of distinct stages or in units of time. The paradigm of such models is, of course, Turing’s original machine model. If we concentrate on this for a moment, the machine is con...

We describe various computational models based initially, but not exclusively, on that of the Turing machine, that are generalized to allow for transfinitely many computational steps. Variants of such machines are considered that have longer tapes than the standard model, or that work on ordinals rather than numbers. We outline the connections between such models and the older theories of recur...