Two-cardinal ideal operators and indescribability

A well-known version of Rowbottom's theorem for supercompactness ultrafilters leads naturally to notions of two-cardinal Ramseyness and corresponding normal ideals introduced herein. Generalizing results of Baumgartner [7, 8], Feng [22] and the first author [16, 17], we study the hierarchies associated with a particular version of two-cardinal Ramseyness and a strong version of two-cardinal ineffability, as well as the relationships between these hierarchies and a natural notion of transfinite two-cardinal indescribability.Comment: Fixed more typos in the first paragrap

Model Theory for a Compact Cardinal

We would like to develop model theory for T, a complete theory in L_{theta,theta}(tau) when theta is a compact cardinal. We already have bare bones stability theory and it seemed we can go no further. Dealing with ultrapowers (and ultraproducts) naturally we restrict ourselves to "D a theta-complete ultrafilter on I, probably (I,theta)-regular". The basic theorems of model theory work and can be generalized (like Los theorem), but can we generalize deeper parts of model theory? The first section tries to sort out what occurs to the notion of stable T for complete L_{theta,theta}-theories T. We generalize several properties of complete first order T, equivalent to being stable (see [Sh:c]) and find out which implications hold and which fail. In particular, can we generalize stability enough to generalize [Sh:c, Ch. VI]? Let us concentrate on saturation in the local sense (types consisting of instances of one formula). We prove that at least we can characterize the T's (of cardinality < theta for simplicity) which are minimal for appropriate cardinal lambda > 2^kappa +|T| in each of the following two senses. One is generalizing Keisler order which measures how saturated are ultrapowers. Another asks: Is there an L_{theta,theta}-theory T_1 supseteq T of cardinality |T| + 2^theta such that for every model M_1 of T_1 of cardinality > lambda, the tau(T)-reduct M of M_1 is lambda^+-saturated. Moreover, the two versions of stable used in the characterization are different

Topics in Topology and Homotopy Theory

This book is an account of certain topics in general and algebraic topology