The theory of ultrafilters
Author(s)
Bibliographic Information
The theory of ultrafilters
（Die Grundlehren der mathematischen Wissenschaften, Bd. 211）
Springer-Verlag, 1974
- : gw
- : us
- : pbk
Available at 80 libraries
Note
Bibliography: p. [453]-471
Includes indexes
Description and Table of Contents
- Volume
-
: us ISBN 9780387066042
Description
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.
- Volume
-
: gw ISBN 9783540066040
Description
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.
Table of Contents
- 1. Set Theory.- Ordinals.- Cardinal Arithmetic.- Notes for 1.- 2. Topology and Boolean Algebras.- Topology.- Finitary Properties of Boolean Algebras.- Stone's Duality.- The Completion of a Boolean Algebra and the Gleason Space of a Compact Space.- Notes for 2.- 3. Intersection Systems and Families of Large Oscillation.- Intersection Systems and the Souslin Number.- Families of Large Oscillation.- Notes for 3.- 4. The General Theory of Jonsson Classes.- Notes for 4.- 5. The Jonsson Class of Ordered Sets.- Notes for 5.- 6. The Jonsson Class of Boolean Algebras.- The Stone Space of the Homogeneous-Universal Boolean Algebras.- Properties of the Space S?.- Notes for 6.- 7. Basic Facts on Ultrafilters.- Notes for 7.- 8. Large Cardinals.- Weakly Compact Cardinals: Combinatorial Equivalences.- Weakly Compact Cardinals: Boolean-Algebraic and Topological Equivalences.- Measurable Cardinals.- Descendingly Incomplete Ultrafilters.- Notes for 8.- 9. The Rudin-Keisler Order on Types of Ultrafilters.- The Rudin-Keisler Order.- Rudin-Keisler Minimal Types in ?[?(?)\?].- Good Ultrafilters.- Notes for 9..- 10. Good Ultrafilters.- Families of Large Oscillation Modulo Filters
- the Fundamental Existence Theorem of Good Ultrafilters.- Additional Existence Results.- Directedness Properties of the Rudin-Keisler Order.- Adequate Ultrafilters on Special Boolean Algebras.- Notes for 10.- 11. Elementary Types.- Notes for 11.- 12. Families of Almost Disjoint Sets.- Cardinalities of Families of Almost Disjoint Sets.- The Balcar-Vop?nka Theorem.- Cardinalities of Ultraproducts.- Notes for 12.- 13. Saturation of Ultraproducts.- Ultraproducts Modulo Regular Ultrafilters.- Ultraproducts Modulo Good Ultrafilters.- Shelah's Characterisation of Elementary Equivalence.- Characterisation of the Rudin-Keisler Order.- Notes for 13.- 14. Topology of Spaces of Ultrafilters.- Certain Properties of F?-Spaces.- The Space of ?-Uniform Ultrafilters on ?.- Spaces of Uniform Ultrafilters and Homogeneous-Universal Boolean Algebras.- The Space of Sub-Uniform Ultrafilters.- Relations to Measurable Cardinals.- Notes for 14.- 15. Spaces Homeomorphic to (2?)?.- The Topological Characterization of (2?)?.- The Baire Category Properties of (2?)?.- Spaces of Ultrafilters Homeomorphic to Spaces (2?)?.- Applications to the Growth Spaces ?X\X.- Notes for 15.- 16. Ultrafilters on ?.- The Rudin-Keisler Order on ?(?) and the Canonical Function ?:?(? x ?) ? ?(?) x ?(?).- The Rudin-Frolik Order.- Non-Homogeneity of Certain Spaces.- Notes for 16.- Index of Symbols.
- Volume
-
: pbk ISBN 9783642657825
Description
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.
Table of Contents
- 1. Set Theory.- Ordinals.- Cardinal Arithmetic.- Notes for 1.- 2. Topology and Boolean Algebras.- Topology.- Finitary Properties of Boolean Algebras.- Stone's Duality.- The Completion of a Boolean Algebra and the Gleason Space of a Compact Space.- Notes for 2.- 3. Intersection Systems and Families of Large Oscillation.- Intersection Systems and the Souslin Number.- Families of Large Oscillation.- Notes for 3.- 4. The General Theory of Joensson Classes.- Notes for 4.- 5. The Jonsson Class of Ordered Sets.- Notes for 5.- 6. The Jonsson Class of Boolean Algebras.- The Stone Space of the Homogeneous-Universal Boolean Algebras.- Properties of the Space S?.- Notes for 6.- 7. Basic Facts on Ultrafilters.- Notes for 7.- 8. Large Cardinals.- Weakly Compact Cardinals: Combinatorial Equivalences.- Weakly Compact Cardinals: Boolean-Algebraic and Topological Equivalences.- Measurable Cardinals.- Descendingly Incomplete Ultrafilters.- Notes for 8.- 9. The Rudin-Keisler Order on Types of Ultrafilters.- The Rudin-Keisler Order.- Rudin-Keisler Minimal Types in ?[?(?)\?].- Good Ultrafilters.- Notes for 9..- 10. Good Ultrafilters.- Families of Large Oscillation Modulo Filters
- the Fundamental Existence Theorem of Good Ultrafilters.- Additional Existence Results.- Directedness Properties of the Rudin-Keisler Order.- Adequate Ultrafilters on Special Boolean Algebras.- Notes for 10.- 11. Elementary Types.- Notes for 11.- 12. Families of Almost Disjoint Sets.- Cardinalities of Families of Almost Disjoint Sets.- The Balcar-Vop?nka Theorem.- Cardinalities of Ultraproducts.- Notes for 12.- 13. Saturation of Ultraproducts.- Ultraproducts Modulo Regular Ultrafilters.- Ultraproducts Modulo Good Ultrafilters.- Shelah's Characterisation of Elementary Equivalence.- Characterisation of the Rudin-Keisler Order.- Notes for 13.- 14. Topology of Spaces of Ultrafilters.- Certain Properties of F?-Spaces.- The Space of ?-Uniform Ultrafilters on ?.- Spaces of Uniform Ultrafilters and Homogeneous-Universal Boolean Algebras.- The Space of Sub-Uniform Ultrafilters.- Relations to Measurable Cardinals.- Notes for 14.- 15. Spaces Homeomorphic to (2?)?.- The Topological Characterization of (2?)?.- The Baire Category Properties of (2?)?.- Spaces of Ultrafilters Homeomorphic to Spaces (2?)?.- Applications to the Growth Spaces ?X\X.- Notes for 15.- 16. Ultrafilters on ?.- The Rudin-Keisler Order on ?(?) and the Canonical Function ?:?(? x ?) ? ?(?) x ?(?).- The Rudin-Frolik Order.- Non-Homogeneity of Certain Spaces.- Notes for 16.- Index of Symbols.
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