The Theory of Ultrafilters

1. Front Matter

   Pages I-X

   PDF

2. Set Theory

   • W. Wistar Comfort, Stylianos Negrepontis

   Pages 1-20

3. Topology and Boolean Algebras

   • W. Wistar Comfort, Stylianos Negrepontis

   Pages 21-60

4. Intersection Systems and Families of Large Oscillation

   • W. Wistar Comfort, Stylianos Negrepontis

   Pages 61-81

5. The General Theory of Jónsson Classes

   • W. Wistar Comfort, Stylianos Negrepontis

   Pages 82-100

6. The Jónsson Class of Ordered Sets

   • W. Wistar Comfort, Stylianos Negrepontis

   Pages 101-115

7. The Jónsson Class of Boolean Algebras

   • W. Wistar Comfort, Stylianos Negrepontis

   Pages 116-141

8. Basic Facts on Ultrafilters

   • W. Wistar Comfort, Stylianos Negrepontis

   Pages 142-163

9. Large Cardinals

   • W. Wistar Comfort, Stylianos Negrepontis

   Pages 164-203

10. The Rudin-Keisler Order on Types of Ultrafilters

    • W. Wistar Comfort, Stylianos Negrepontis

    Pages 204-232

11. Good Ultrafilters

    • W. Wistar Comfort, Stylianos Negrepontis

    Pages 233-261

12. Elementary Types

    • W. Wistar Comfort, Stylianos Negrepontis

    Pages 262-285

13. Families of Almost Disjoint Sets

    • W. Wistar Comfort, Stylianos Negrepontis

    Pages 286-310

14. Saturation of Ultraproducts

    • W. Wistar Comfort, Stylianos Negrepontis

    Pages 311-340

15. Topology of Spaces of Ultrafilters

    • W. Wistar Comfort, Stylianos Negrepontis

    Pages 341-380

16. Spaces Homeomorphic to (2^α) _α

    • W. Wistar Comfort, Stylianos Negrepontis

    Pages 381-409

17. Ultrafilters on ω

    • W. Wistar Comfort, Stylianos Negrepontis

    Pages 410-452

18. Back Matter

    Pages 453-484

    PDF

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.

• Compact space
• Mathematica
• Ultrafilter
• algebra
• arithmetic
• compactness
• duality
• function
• logic
• ordinal
• set theory
• theorem
• topology

• Department of Mathematics, Wesleyan University, Middletown, USA

  W. Wistar Comfort

• Department of Mathematics, Athens University, Athens, Greece

  Stylianos Negrepontis

• Department of Mathematics, McGill University, Montréal, Canada

  Stylianos Negrepontis

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