The Theory of Ultra Filters (Grundlehren Der Mathematischen Wissenschaften Series, Vol 211)

An ultrafilter is a truth-value assignment to the family of subsets of a set, and a method of convergence to infinity. From the first (logical) property arises its connection with two-valued logic and model theory; from the second (convergence) property arises its connection with topology and set theory. Both these descriptions of an ultrafilter are connected with compactness. The model-theoretic property finds its expression in the construction of the ultraproduct and the compactness type of theorem of Los (implying the compactness theorem of first-order logic); and the convergence property leads to the process of completion by the adjunction of an ideal element for every ultrafilter-i. e. , to the Stone-Cech com­ pactification process (implying the Tychonoff theorem on the compact­ ness of products). Since these are two ways of describing the same mathematical object, it is reasonable to expect that a study of ultrafilters from these points of view will yield results and methods which can be fruitfully crossbred. This unifying aspect is indeed what we have attempted to emphasize in the present work. An ultrafilter is a truth-value assignment to the family of subsets of a set, and a method of convergence to infinity. From the first (logical) property arises its connection with two-valued logic and model theory; from the second (convergence) property arises its connection with topology and set theory. Both these descriptions of an ultrafilter are connected with compactness. The model-theoretic property finds its expression in the construction of the ultraproduct and the compactness type of theorem of Los (implying the compactness theorem of first-order logic); and the convergence property leads to the process of completion by the adjunction of an ideal element for every ultrafilter-i. e., to the Stone-Cech comƯ pactification process (implying the Tychonoff theorem on the compactƯ ness of products). Since these are two ways of describing the same mathematical object, it is reasonable to expect that a study of ultrafilters from these points of view will yield results and methods which can be fruitfully crossbred. This unifying aspect is indeed what we have attempted to emphasize in the present work Front Matter....Pages I-X Set Theory....Pages 1-20 Topology and Boolean Algebras....Pages 21-60 Intersection Systems and Families of Large Oscillation....Pages 61-81 The General Theory of Jónsson Classes....Pages 82-100 The Jónsson Class of Ordered Sets....Pages 101-115 The Jónsson Class of Boolean Algebras....Pages 116-141 Basic Facts on Ultrafilters....Pages 142-163 Large Cardinals....Pages 164-203 The Rudin-Keisler Order on Types of Ultrafilters....Pages 204-232 Good Ultrafilters....Pages 233-261 Elementary Types....Pages 262-285 Families of Almost Disjoint Sets....Pages 286-310 Saturation of Ultraproducts....Pages 311-340 Topology of Spaces of Ultrafilters....Pages 341-380 Spaces Homeomorphic to (2 α ) α ....Pages 381-409 Ultrafilters on ω....Pages 410-452 Back Matter....Pages 453-484