No abstract

In this paper we solve the satisfiability problem for the quantifier-free fragment of set theory MLSSPF involving in addition to the basic Boolean set operators of union, intersection, and difference, also the powerset and singleton operators, and a finiteness predicate. \ud The more restricted fragment obtained by dropping the finiteness predicate has been shown to have a solvable satisfiability problem in a previous paper, by establishing for it a small model property. \ud We exploit the latter decision result for dealing also with the finiteness predicate (and therefore with the infiniteness predicate too) and prove a small witness-model property for MLSSPF, asserting that any model for a satisfiable formula Phi with m distinct variables of the fragment of our interest admits a finite representation bounded by c(m), where c is a suitable computable function. Since such candidate representations are finitely many, their number does not exceed a known bound, and it can be recognized algorithmically whether they indeed represent a(n infinite) model for the input formula, the decidability of the satisfiability problem for MLSSPF follows

The satisfiability problem for multilevel syllogistic extended with the Cartesian product operator (MLS×) is a long-standing open problem in computable set theory. For long, it was not excluded that such a problem were undecidable, due to its remarkable resemblance with the well-celebrated Hilbert's tenth problem, as it was deemed reasonable that union of disjoint sets and Cartesian product might somehow play the roles of integer addition and multiplication.To dispense with nonessential technical difficulties, we report here about a positive solution to the satisfiability problem for a slight simplified variant of MLS×, yet fully representative of the combinatorial complications due to the presence of the Cartesian product, in which membership is not present and the Cartesian product operator is replaced with its unordered variant.We are very confident that such decidability result can be generalized to full MLS×, though at the cost of considerable technicalities. * We gratefully acknowledge partial support from the projects STORAGE and MEGABIT -

Abstract. Given a Boolean algebra B and an embedding e : B → P(N)/fin we consider the possibility of extending each or some automorphism of B to the whole P(N)/fin. Among other things, we show, assuming CH, that for a wide class of Boolean algebras there are embeddings for which no non-trivial automorphism can be extended.

We investigate the phenomenon of concentration of measure from a "phenomenological" point of view, by working on specific examples. In particular, we will get some speculative hints about extreme amenability of certain examples of infinite dimensional Lie groups.