Topics of Ultrafilters
Keywords:
ultrafilters, ultrafilter applications, nonprincipal ultrafilter on N
Abstract
Ultrafilters are very important mathematical objects in mathematical research [6, 22, 23]. There are a wide variety of classical theorems in various branches of mathematics where ultrafilters are applied in their proof, and other classical theorems that deal directly with ultrafilters. The objective of this article is to contribute (in a divulgative way) to ultrafilter research by describing the demonstrations of some such theorems related (uniquely or in
combination) to topology, measurement theory, algebra, combinatorial infinite, set theory and first-order logic, also formulating some updated open problems of set theory that refer to non-main ultrafilters on N, the Mathias’s model and the Solovay’s model.
References
Betz, C., Introducción a la teorı́a de la medida e integración. Univesidad Central de Venezuela. 1992.
Bourbaki, N., Topologie générale, Actualités Sci. Ind. 858 (1940), 916 (1942), 1029 (1947), 1045 (1948), 1084 (1949), Paris.
Cartan, H., Théorie des filtres, C.R. Acad. Sci. Paris (1937), 595–598.
Cartan, H., Filtres et ultrafiltres, C. R. Acad. Sci. Paris (1937), 777–779.
Chang, C. and Keisler, H., Model Theory. Dover Publications. 2012.
Confort, W. and Negropontis, S., The Theory of Ultrafilters. Springer-Verlag. 1974.
Corbillón, M., Análisis real no estándar. Tesis de licenciatura en Matemáticas. Tutor: Dr. Josep Maria Font Llovet. Facultat de Matemátiques. Universitat de Barcelona. 2015.
Di Prisco, C. and Henle, H., Doughnuts, Floating Ordinals, Square Brackets, and Ultraflitters. Journal of Symbolic Logic 65(2000), 462–473.
Di Prisco, C. and Henle, H., Partitions of the reals and choice. En “Models, algebras and proofs”. X. Caicedo y C. M. Montenegro. Eds. Lecture Notes in Pure and Appl. Math, 203, Marcel Dekker, 1999.
Di Prisco, C., Teorı́a de conjuntos. Universidad Central de Venezuela. 2009.
Di Prisco, C. y Uzcátegui, C., Una introducción a la teorı́a descriptiva de conjuntos. 2019. (En prensa).
Dudley, R., Real Analysis and Probabiblity. Cambridge University Press. 2002.
Enderton, H., Elements of Set Theory. Academic Press. New York. 1977.
Galindo, F., Álgebras booleanas, órdenes parciales y el axioma de elección. Divulgaciones Matemáticas, 18(1) (2017), 34–54.
Halmos, P., Measure Theory. Springer-Verlag. 1970.
Halpern, J. and Lévy, A., The Boolean prime ideal theorem does not imply the axiom of choice. En “ Axiomatic Set Theory” (D. S. Scoott, ed), Proc. Sympos. Pure Math., Vol. XIII, Part I, Univ. California, Los Angeles, 1967.
Herrlich, H., Axiom of Choice. Springer. 2006.
Howard, P. and Rubin, J., Consecuences of the Axiom of Choice. American Mathematical Society. 1998.
Jech, T., Set Theory. 1978
Jech, T., Set Theory. Springer. 2000.
Jech, T., The Axiom Choice. North-Holland Publishing Company. 1973.
Kelley, J., General Topology. Springer. 1991.
Kunen, K., Set Theory. Elsevier. 2006.
Manzano, M., Teorı́a de Modelos. Alianza. Madrid. 1989.
D. Mathias, A. R., The Order Extension Principle. Procedings of Symposia in Pura Mathematics, Vol. 13, Part II, 1974.
Mendelson, E., Introduction to Mathematical Logic. Chapman and Hall/CRL. U.S.A. 2009.
Royden, H., Real Analysis. Pearson. 2010.
Sikorski, R., Boolean Algebras. Springer-Verlag. 1960.
Stone, M., The theory of representations for Booleam algebras. Trans. Amer. Math. Soc. 40 (1936), 37–111. Zbl.014.34002
https://math.stackexchange.com/questions/1130615/non-measurability-of-ultrafilter-on-omega.
Mathematics: Why is compactness in log called compactness? https://math. stackexchange.com/questions/842/why-is-compactness-in-logic-called-
compactness.
Bourbaki, N., Topologie générale, Actualités Sci. Ind. 858 (1940), 916 (1942), 1029 (1947), 1045 (1948), 1084 (1949), Paris.
Cartan, H., Théorie des filtres, C.R. Acad. Sci. Paris (1937), 595–598.
Cartan, H., Filtres et ultrafiltres, C. R. Acad. Sci. Paris (1937), 777–779.
Chang, C. and Keisler, H., Model Theory. Dover Publications. 2012.
Confort, W. and Negropontis, S., The Theory of Ultrafilters. Springer-Verlag. 1974.
Corbillón, M., Análisis real no estándar. Tesis de licenciatura en Matemáticas. Tutor: Dr. Josep Maria Font Llovet. Facultat de Matemátiques. Universitat de Barcelona. 2015.
Di Prisco, C. and Henle, H., Doughnuts, Floating Ordinals, Square Brackets, and Ultraflitters. Journal of Symbolic Logic 65(2000), 462–473.
Di Prisco, C. and Henle, H., Partitions of the reals and choice. En “Models, algebras and proofs”. X. Caicedo y C. M. Montenegro. Eds. Lecture Notes in Pure and Appl. Math, 203, Marcel Dekker, 1999.
Di Prisco, C., Teorı́a de conjuntos. Universidad Central de Venezuela. 2009.
Di Prisco, C. y Uzcátegui, C., Una introducción a la teorı́a descriptiva de conjuntos. 2019. (En prensa).
Dudley, R., Real Analysis and Probabiblity. Cambridge University Press. 2002.
Enderton, H., Elements of Set Theory. Academic Press. New York. 1977.
Galindo, F., Álgebras booleanas, órdenes parciales y el axioma de elección. Divulgaciones Matemáticas, 18(1) (2017), 34–54.
Halmos, P., Measure Theory. Springer-Verlag. 1970.
Halpern, J. and Lévy, A., The Boolean prime ideal theorem does not imply the axiom of choice. En “ Axiomatic Set Theory” (D. S. Scoott, ed), Proc. Sympos. Pure Math., Vol. XIII, Part I, Univ. California, Los Angeles, 1967.
Herrlich, H., Axiom of Choice. Springer. 2006.
Howard, P. and Rubin, J., Consecuences of the Axiom of Choice. American Mathematical Society. 1998.
Jech, T., Set Theory. 1978
Jech, T., Set Theory. Springer. 2000.
Jech, T., The Axiom Choice. North-Holland Publishing Company. 1973.
Kelley, J., General Topology. Springer. 1991.
Kunen, K., Set Theory. Elsevier. 2006.
Manzano, M., Teorı́a de Modelos. Alianza. Madrid. 1989.
D. Mathias, A. R., The Order Extension Principle. Procedings of Symposia in Pura Mathematics, Vol. 13, Part II, 1974.
Mendelson, E., Introduction to Mathematical Logic. Chapman and Hall/CRL. U.S.A. 2009.
Royden, H., Real Analysis. Pearson. 2010.
Sikorski, R., Boolean Algebras. Springer-Verlag. 1960.
Stone, M., The theory of representations for Booleam algebras. Trans. Amer. Math. Soc. 40 (1936), 37–111. Zbl.014.34002
https://math.stackexchange.com/questions/1130615/non-measurability-of-ultrafilter-on-omega.
Mathematics: Why is compactness in log called compactness? https://math. stackexchange.com/questions/842/why-is-compactness-in-logic-called-
compactness.
Published
2020-12-27
How to Cite
Galindo, F. (2020). Topics of Ultrafilters. Divulgaciones Matemáticas, 21(1-2), 54-77. Retrieved from https://www.produccioncientificaluz.org/index.php/divulgaciones/article/view/36605
Section
Expository and historical papers
