Ultrafilters Throughout Mathematics

Goldbring, Isaac

doi:10.1090/gsm/220

AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution

Ultrafilters Throughout Mathematics

About this Title

Isaac Goldbring, University of California, Irvine, Irvine, CA

Publication: Graduate Studies in Mathematics
Publication Year: 2022; Volume 220
ISBNs: 978-1-4704-6900-9 (print); 978-1-4704-6960-3 (online)
DOI: https://doi.org/10.1090/gsm/220

ePUB View full volume as ePUB

PDF View full volume as PDF

Read more about this volume

C. Anantharaman and S. Popa, An introduction to II$_1$ factors, book in preparation.
K. J. Arrow, A difficulty in the concept of social welfare, Journal of political economy, 58 (1950), pp. 328–346.
James Ax, The elementary theory of finite fields, Ann. of Math. (2) 88 (1968), 239–271. MR 229613, DOI 10.2307/1970573
James E. Baumgartner, A short proof of Hindman’s theorem, J. Combinatorial Theory Ser. A 17 (1974), 384–386. MR 354394, DOI 10.1016/0097-3165(74)90103-4
James E. Baumgartner and Richard Laver, Iterated perfect-set forcing, Ann. Math. Logic 17 (1979), no. 3, 271–288. MR 556894, DOI 10.1016/0003-4843(79)90010-X
Benjamin Baumslag, Residually free groups, Proc. London Math. Soc. (3) 17 (1967), 402–418. MR 215903, DOI 10.1112/plms/s3-17.3.402
Gilbert Baumslag, Alexei Myasnikov, and Vladimir Remeslennikov, Discriminating completions of hyperbolic groups, Geom. Dedicata 92 (2002), 115–143. Dedicated to John Stallings on the occasion of his 65th birthday. MR 1934015, DOI 10.1023/A:1019687202544
Mathias Beiglböck, An ultrafilter approach to Jin’s theorem, Israel J. Math. 185 (2011), 369–374. MR 2837141, DOI 10.1007/s11856-011-0114-5
Itaï Ben Yaacov, Alexander Berenstein, C. Ward Henson, and Alexander Usvyatsov, Model theory for metric structures, Model theory with applications to algebra and analysis. Vol. 2, London Math. Soc. Lecture Note Ser., vol. 350, Cambridge Univ. Press, Cambridge, 2008, pp. 315–427. MR 2436146, DOI 10.1017/CBO9780511735219.011
Itaï Ben Yaacov and Alexander Usvyatsov, Continuous first order logic and local stability, Trans. Amer. Math. Soc. 362 (2010), no. 10, 5213–5259. MR 2657678, DOI 10.1090/S0002-9947-10-04837-3
Vitaly Bergelson, Combinatorial and Diophantine applications of ergodic theory, Handbook of dynamical systems. Vol. 1B, Elsevier B. V., Amsterdam, 2006, pp. 745–869. Appendix A by A. Leibman and Appendix B by Anthony Quas and Máté Wierdl. MR 2186252, DOI 10.1016/S1874-575X(06)80037-8
Vitaly Bergelson, Ultrafilters, IP sets, dynamics, and combinatorial number theory, Ultrafilters across mathematics, Contemp. Math., vol. 530, Amer. Math. Soc., Providence, RI, 2010, pp. 23–47. MR 2757532, DOI 10.1090/conm/530/10439
Andreas Blass, A model without ultrafilters, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 25 (1977), no. 4, 329–331 (English, with Russian summary). MR 476510
Andreas Blass, Combinatorial cardinal characteristics of the continuum, in Handbook of Set Theory, Springer, 2010, pp. 395–489.
Andreas Blass and Saharon Shelah, Ultrafilters with small generating sets, Israel J. Math. 65 (1989), no. 3, 259–271. MR 1005010, DOI 10.1007/BF02764864
Andreas Raphael Blass, ORDERINGS OF ULTRAFILTERS, ProQuest LLC, Ann Arbor, MI, 1970. Thesis (Ph.D.)–Harvard University. MR 2939947
William Boos, Infinitary compactness without strong inaccessibility, J. Symbolic Logic 41 (1976), no. 1, 33–38. MR 409184, DOI 10.2307/2272942
David Booth, Ultrafilters on a countable set, Ann. Math. Logic 2 (1970/71), no. 1, 1–24. MR 277371, DOI 10.1016/0003-4843(70)90005-7
N. Bourbaki, Éléments de mathématique. Topologie algébrique. Chapitres 1 à 4, Springer, Heidelberg, 2016 (French). MR 3617167
Emmanuel Breuillard, Ben Green, and Terence Tao, The structure of approximate groups, Publ. Math. Inst. Hautes Études Sci. 116 (2012), 115–221. MR 3090256, DOI 10.1007/s10240-012-0043-9
J. W. Cannon, W. J. Floyd, and W. R. Parry, Introductory notes on Richard Thompson’s groups, Enseign. Math. (2) 42 (1996), no. 3-4, 215–256. MR 1426438
Valerio Capraro and Martino Lupini, Introduction to sofic and hyperlinear groups and Connes’ embedding conjecture, Lecture Notes in Mathematics, vol. 2136, Springer, Cham, 2015. With an appendix by Vladimir Pestov. MR 3408561, DOI 10.1007/978-3-319-19333-5
H. Cartan, Filtres et ultrafiltres, Comptes Rendus de l’Académie des Sciences, 205 (1937), pp. 777–779.
H. Cartan, Théorie des filtres, Comptes Rendus de l’Académie des Sciences, 205 (1937), pp. 595–598.
Eduard Čech, On bicompact spaces, Ann. of Math. (2) 38 (1937), no. 4, 823–844. MR 1503374, DOI 10.2307/1968839
Christophe Champetier and Vincent Guirardel, Limit groups as limits of free groups, Israel J. Math. 146 (2005), 1–75. MR 2151593, DOI 10.1007/BF02773526
Chen-chung Chang and H. Jerome Keisler, Continuous model theory, Annals of Mathematics Studies, No. 58, Princeton University Press, Princeton, NJ, 1966. MR 231708
Chen-chung Chang and H. Jerome Keisler, Model theory, Dover Publications, 1990.
Charles Chidume, Geometric properties of Banach spaces and nonlinear iterations, Lecture Notes in Mathematics, vol. 1965, Springer-Verlag London, Ltd., London, 2009. MR 2504478
Ian Chiswell, Introduction to $\Lambda$-trees, World Scientific Publishing Co., Inc., River Edge, NJ, 2001. MR 1851337, DOI 10.1142/4495
Matt Clay and Dan Margalit, Groups, Office hours with a geometric group theorist, Princeton Univ. Press, Princeton, NJ, 2017, pp. 3–20. MR 3587214
E. Coleman, Banach space ultraproducts, Irish Math. Soc. Bull. 18 (1987), 30–39. MR 885794
W. W. Comfort, Ultrafilters: some old and some new results, Bull. Amer. Math. Soc. 83 (1977), no. 4, 417–455. MR 454893, DOI 10.1090/S0002-9904-1977-14316-4
A. Connes, Classification of injective factors. Cases $II_{1},$ $II_{\infty },$ $III_{\lambda },$ $\lambda \not =1$, Ann. of Math. (2) 104 (1976), no. 1, 73–115. MR 454659, DOI 10.2307/1971057
John B. Conway, A course in functional analysis, 2nd ed., Graduate Texts in Mathematics, vol. 96, Springer-Verlag, New York, 1990. MR 1070713
Abraham Robinson, A result on consistency and its application to the theory of definition, Indag. Math. 18 (1956), 47–58. Nederl. Akad. Wetensch. Proc. Ser. A 59. MR 78307
V. B. Dem′yanov, On cubic forms in discretely normed fields, Doklady Akad. Nauk SSSR (N.S.) 74 (1950), 889–891 (Russian). MR 37836
Keith J. Devlin, Fundamentals of contemporary set theory, Universitext, Springer-Verlag, New York-Heidelberg, 1979. MR 541746
Mauro Di Nasso, Hypernatural numbers as ultrafilters, Nonstandard analysis for the working mathematician, Springer, Dordrecht, 2015, pp. 443–474. MR 3409522
Mauro Di Nasso, Iterated hyper-extensions and an idempotent ultrafilter proof of Rado’s Theorem, Proc. Amer. Math. Soc. 143 (2015), no. 4, 1749–1761. MR 3314087, DOI 10.1090/S0002-9939-2014-12342-2
Mauro Di Nasso and Marco Forti, Hausdorff ultrafilters, Proc. Amer. Math. Soc. 134 (2006), no. 6, 1809–1818. MR 2207497, DOI 10.1090/S0002-9939-06-08433-4
Mauro Di Nasso, Isaac Goldbring, and Martino Lupini, Nonstandard methods in Ramsey theory and combinatorial number theory, Lecture Notes in Mathematics, vol. 2239, Springer, Cham, 2019. MR 3931702, DOI 10.1007/978-3-030-17956-4
Mauro Di Nasso and Eleftherios Tachtsis, Idempotent ultrafilters without Zorn’s lemma, Proc. Amer. Math. Soc. 146 (2018), no. 1, 397–411. MR 3723149, DOI 10.1090/proc/13719
Seán Dineen, The second dual of a JB$^*$ triple system, Complex analysis, functional analysis and approximation theory (Campinas, 1984) North-Holland Math. Stud., vol. 125, North-Holland, Amsterdam, 1986, pp. 67–69. MR 893410
Hans-Dieter Donder, Regularity of ultrafilters and the core model, Israel J. Math. 63 (1988), no. 3, 289–322. MR 969944, DOI 10.1007/BF02778036
Alan Dow, Saturated Boolean algebras and their Stone spaces, Topology Appl. 21 (1985), no. 2, 193–207. MR 813288, DOI 10.1016/0166-8641(85)90104-X
Cornelia Druţu and Mark Sapir, Tree-graded spaces and asymptotic cones of groups, Topology 44 (2005), no. 5, 959–1058. With an appendix by Denis Osin and Mark Sapir. MR 2153979, DOI 10.1016/j.top.2005.03.003
Gábor Elek and Endre Szabó, On sofic groups, J. Group Theory 9 (2006), no. 2, 161–171. MR 2220572, DOI 10.1515/JGT.2006.011
Herbert B. Enderton, Elements of set theory, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1977. MR 439636
Herbert B. Enderton, A mathematical introduction to logic, Elsevier, 2001.
Herbert B. Enderton, Computability theory: An introduction to recursion theory, Academic Press, 2010.
P. Enflo, Uniform structures and square roots in topological groups. I, II, Israel J. Math. 8 (1970), 230–252; ibid. 8 (1970), 253–272. MR 263969, DOI 10.1007/bf02771561
Ilijas Farah, Bradd Hart, and David Sherman, Model theory of operator algebras I: stability, Bull. Lond. Math. Soc. 45 (2013), no. 4, 825–838. MR 3081550, DOI 10.1112/blms/bdt014
Ilijas Farah, Bradd Hart, and David Sherman, Model theory of operator algebras II: model theory, Israel J. Math. 201 (2014), no. 1, 477–505. MR 3265292, DOI 10.1007/s11856-014-1046-7
Ilijas Farah and Saharon Shelah, A dichotomy for the number of ultrapowers, J. Math. Log. 10 (2010), no. 1-2, 45–81. MR 2802082, DOI 10.1142/S0219061310000936
S. Feferman and R. L. Vaught, The first order properties of products of algebraic systems, Fund. Math. 47 (1959), 57–103. MR 108455, DOI 10.4064/fm-47-1-57-103
Ben Fine, Anthony M. Gaglione, Gerhard Rosenberger, and Dennis Spellman, On CT and CSA groups and related ideas, J. Group Theory 19 (2016), no. 5, 923–940. MR 3545911, DOI 10.1515/jgth-2016-0005
Edward R. Fisher, Abelian structures. I, Abelian group theory (Proc. Second New Mexico State Univ. Conf., Las Cruces, N.M., 1976) Lecture Notes in Math., Vol. 616, Springer, Berlin-New York, 1977, pp. 270–322. MR 540014
T. Frayne, A. C. Morel, and D. S. Scott, Reduced direct products, Fund. Math. 51 (1962/63), 195–228. MR 142459, DOI 10.4064/fm-51-3-195-228
Tobias Fritz, Tsirelson’s problem and Kirchberg’s conjecture, Rev. Math. Phys. 24 (2012), no. 5, 1250012, 67. MR 2928100, DOI 10.1142/S0129055X12500122
Zdeněk Frolík, Non-homogeneity of $\beta P-P$, Comment. Math. Univ. Carolinae 8 (1967), 705–709. MR 266160
Harry Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. Analyse Math. 31 (1977), 204–256. MR 498471, DOI 10.1007/BF02813304
Anthony M. Gaglione and Dennis Spellman, Even more model theory of free groups, Infinite groups and group rings (Tuscaloosa, AL, 1992) Ser. Algebra, vol. 1, World Sci. Publ., River Edge, NJ, 1993, pp. 37–40. MR 1377955
D. Galvin, Ultrafilters, with applications to analysis, social choice and combinatorics, unpublished notes, 2009.
Murray Gerstenhaber and Oscar S. Rothaus, The solution of sets of equations in groups, Proc. Nat. Acad. Sci. U.S.A. 48 (1962), 1531–1533. MR 166296, DOI 10.1073/pnas.48.9.1531
Saeed Ghasemi, Reduced products of metric structures: a metric Feferman-Vaught theorem, J. Symb. Log. 81 (2016), no. 3, 856–875. MR 3569108, DOI 10.1017/jsl.2016.20
Leonard Gillman and Meyer Jerison, Rings of continuous functions, Graduate Texts in Mathematics, No. 43, Springer-Verlag, New York-Heidelberg, 1976. Reprint of the 1960 edition. MR 407579
Robert Goldblatt, Lectures on the hyperreals, Graduate Texts in Mathematics, vol. 188, Springer-Verlag, New York, 1998. An introduction to nonstandard analysis. MR 1643950, DOI 10.1007/978-1-4612-0615-6
I. Goldbring and H. J. Keisler, Continuous sentences preserved under reduced products, to appear in Journal of Symbolic Logic.
R. I. Grigorchuk, Degrees of growth of finitely generated groups and the theory of invariant means, Izv. Akad. Nauk SSSR Ser. Mat. 48 (1984), no. 5, 939–985 (Russian). MR 764305
Mikhael Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math. 53 (1981), 53–73. MR 623534
M. Gromov, Endomorphisms of symbolic algebraic varieties, J. Eur. Math. Soc. (JEMS) 1 (1999), no. 2, 109–197. MR 1694588, DOI 10.1007/PL00011162
A. Grothendieck and J. A. Dieudonné, Éléments de géométrie algébrique. I, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 166, Springer-Verlag, Berlin, 1971 (French). MR 3075000
A. Grothendieck, Éléments de géométrie algébrique. II. Étude globale élémentaire de quelques classes de morphismes, Inst. Hautes Études Sci. Publ. Math. 8 (1961), 222 (French). MR 217084
A. Grothendieck, Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents. I, Inst. Hautes Études Sci. Publ. Math. 11 (1961), 167. MR 217085
R. Gurevič, On ultracoproducts of compact Hausdorff spaces, J. Symbolic Logic 53 (1988), no. 1, 294–300. MR 929393, DOI 10.2307/2274446
Eric J. Hall, Kyriakos Keremedis, and Eleftherios Tachtsis, The existence of free ultrafilters on $\omega$ does not imply the extension of filters on $\omega$ to ultrafilters, MLQ Math. Log. Q. 59 (2013), no. 4-5, 258–267. MR 3100753, DOI 10.1002/malq.201100092
C. Ward Henson, Foundations of nonstandard analysis: a gentle introduction to nonstandard extensions, Nonstandard analysis (Edinburgh, 1996) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 493, Kluwer Acad. Publ., Dordrecht, 1997, pp. 1–49. MR 1603228
Horst Herrlich, Kyriakos Keremedis, and Eleftherios Tachtsis, Remarks on the Stone spaces of the integers and the reals without $\textbf {AC}$, Bull. Pol. Acad. Sci. Math. 59 (2011), no. 2, 101–114. MR 2852866, DOI 10.4064/ba59-2-1
Edwin Hewitt, Rings of real-valued continuous functions. I, Trans. Amer. Math. Soc. 64 (1948), 45–99. MR 26239, DOI 10.1090/S0002-9947-1948-0026239-9
Neil Hindman, The existence of certain ultra-filters on $N$ and a conjecture of Graham and Rothschild, Proc. Amer. Math. Soc. 36 (1972), 341–346. MR 307926, DOI 10.1090/S0002-9939-1972-0307926-0
Neil Hindman, Finite sums from sequences within cells of a partition of $N$, J. Combinatorial Theory Ser. A 17 (1974), 1–11. MR 349574, DOI 10.1016/0097-3165(74)90023-5
Neil Hindman and Dona Strauss, Algebra in the Stone-Čech compactification, De Gruyter Textbook, Walter de Gruyter & Co., Berlin, 2012. Theory and applications; Second revised and extended edition [of MR1642231]. MR 2893605
Paul Howard and Jean E. Rubin, Consequences of the axiom of choice, Mathematical Surveys and Monographs, vol. 59, American Mathematical Society, Providence, RI, 1998. With 1 IBM-PC floppy disk (3.5 inch; WD). MR 1637107, DOI 10.1090/surv/059
Paul E. Howard, Łoś’ theorem and the Boolean prime ideal theorem imply the axiom of choice, Proc. Amer. Math. Soc. 49 (1975), 426–428. MR 384548, DOI 10.1090/S0002-9939-1975-0384548-X
Ehud Hrushovski, Stable group theory and approximate subgroups, J. Amer. Math. Soc. 25 (2012), no. 1, 189–243. MR 2833482, DOI 10.1090/S0894-0347-2011-00708-X
José Iovino, Applications of model theory to functional analysis, Dover Publications, Inc., Mineola, NY, 2014. Revised reprint of the 2002 original; With a new preface, notes, and an updated bibliography. MR 3362124
C. Ward Henson, José Iovino, Alexander S. Kechris, and Edward Odell, Analysis and logic, London Mathematical Society Lecture Note Series, vol. 262, Cambridge University Press, Cambridge, 2002. Lectures from the mini-courses offered at the International Conference held at the University of Mons-Hainaut, Mons, August 25–29, 1997; Edited by Catherine Finet and Christian Michaux. MR 1967837
T. Jech, Set theory, Springer Science & Business Media, 2002.
Thomas J. Jech, The axiom of choice, Studies in Logic and the Foundations of Mathematics, Vol. 75, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973. MR 396271
Z. Ji, A. Natarajan, T. Vidick, J. Wright, and H. Yuen, MIP*= RE, arXiv preprint arXiv:2001.04383, 2020.
Renling Jin, The sumset phenomenon, Proc. Amer. Math. Soc. 130 (2002), no. 3, 855–861. MR 1866042, DOI 10.1090/S0002-9939-01-06088-9
M. Junge, M. Navascues, C. Palazuelos, D. Perez-Garcia, V. B. Scholz, and R. F. Werner, Connes embedding problem and Tsirelson’s problem, J. Math. Phys. 52 (2011), no. 1, 012102, 12. MR 2790067, DOI 10.1063/1.3514538
Akihiro Kanamori, The higher infinite, 2nd ed., Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2009. Large cardinals in set theory from their beginnings; Paperback reprint of the 2003 edition. MR 2731169
A. Karrass and D. Solitar, On a theorem of Cohen and Lyndon about free bases for normal subgroups, Canadian J. Math. 24 (1972), 1086–1091. MR 320150, DOI 10.4153/CJM-1972-112-0
Alexander S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. MR 1321597, DOI 10.1007/978-1-4612-4190-4
H. Jerome Keisler, Ultraproducts and elementary classes, Indag. Math. 23 (1961), 477–495. Nederl. Akad. Wetensch. Proc. Ser. A 64. MR 140396
H. Jerome Keisler, Limit ultrapowers, Trans. Amer. Math. Soc. 107 (1963), 382–408. MR 148547, DOI 10.1090/S0002-9947-1963-0148547-1
H. Jerome Keisler, Good ideals in fields of sets, Ann. of Math. (2) 79 (1964), 338–359. MR 166105, DOI 10.2307/1970549
H. Jerome Keisler, Ultraproducts and saturated models, Indag. Math. 26 (1964), 178–186. Nederl. Akad. Wetensch. Proc. Ser. A 67. MR 168483
H. Jerome Keisler, Ultraproducts which are not saturated, J. Symbolic Logic 32 (1967), 23–46. MR 218224, DOI 10.2307/2271240
H. Jerome Keisler, The ultraproduct construction, Ultrafilters across mathematics, Contemp. Math., vol. 530, Amer. Math. Soc., Providence, RI, 2010, pp. 163–179. MR 2757537, DOI 10.1090/conm/530/10444
Juliette Kennedy, Saharon Shelah, and Jouko Väänänen, Regular ultrapowers at regular cardinals, Notre Dame J. Form. Log. 56 (2015), no. 3, 417–428. MR 3373611, DOI 10.1215/00294527-3132788
Kyriakos Keremedis, Tychonoff products of two-element sets and some weakenings of the Boolean prime ideal theorem, Bull. Pol. Acad. Sci. Math. 53 (2005), no. 4, 349–359. MR 2214925, DOI 10.4064/ba53-4-1
Jonathan Kirby, An invitation to model theory, Cambridge University Press, Cambridge, 2019. MR 3967730, DOI 10.1017/9781316683002
Eberhard Kirchberg, On nonsemisplit extensions, tensor products and exactness of group $C^*$-algebras, Invent. Math. 112 (1993), no. 3, 449–489. MR 1218321, DOI 10.1007/BF01232444
V. L. Klee Jr., Invariant metrics in groups (solution of a problem of Banach), Proc. Amer. Math. Soc. 3 (1952), 484–487. MR 47250, DOI 10.1090/S0002-9939-1952-0047250-4
Péter Komjáth and Vilmos Totik, Ultrafilters, Amer. Math. Monthly 115 (2008), no. 1, 33–44. MR 2375774, DOI 10.1080/00029890.2008.11920493
Linus Kramer, Saharon Shelah, Katrin Tent, and Simon Thomas, Asymptotic cones of finitely presented groups, Adv. Math. 193 (2005), no. 1, 142–173. MR 2132762, DOI 10.1016/j.aim.2004.04.012
J. L. Krivine, Sous-espaces et cônes convexes dans les espaces Lp, PhD thesis, Centre national de la recherche scientifique, 1967.
Kenneth Kunen, Ultrafilters and independent sets, Trans. Amer. Math. Soc. 172 (1972), 299–306. MR 314619, DOI 10.1090/S0002-9947-1972-0314619-7
Kenneth Kunen, Weak P-points in $\mathbb {N}^*$, Colloquia Mathematica Societatis Janos Bolyai, 23 (1978), pp. 741–749.
Jerzy Łoś, Quelques remarques, théorèmes et problèmes sur les classes définissables d’algèbres, Mathematical interpretation of formal systems, North-Holland, Amsterdam, 1955, pp. 98–113 (French). MR 75156
D. J. Lewis, Cubic homogeneous polynomials over $p$-adic number fields, Ann. of Math. (2) 56 (1952), 473–478. MR 49947, DOI 10.2307/1969655
W. A. J. Luxemburg, A general theory of monads, Applications of Model Theory to Algebra, Analysis, and Probability (Internat. Sympos., Pasadena, Calif., 1967) Holt, Rinehart and Winston, New York-Montreal, Que.-London, 1969, pp. 18–86. MR 244931
Saunders Mac Lane, Categories for the working mathematician, 2nd ed., Graduate Texts in Mathematics, vol. 5, Springer-Verlag, New York, 1998. MR 1712872
Dugald Macpherson, Model theory of finite and pseudofinite groups, Arch. Math. Logic 57 (2018), no. 1-2, 159–184. MR 3749405, DOI 10.1007/s00153-017-0584-1
Anatoliĭ Ivanovič Mal′cev, The metamathematics of algebraic systems. Collected papers: 1936–1967, Studies in Logic and the Foundations of Mathematics, Vol. 66, North-Holland Publishing Co., Amsterdam-London, 1971. Translated, edited, and provided with supplementary notes by Benjamin Franklin Wells, III. MR 349383
Maryanthe Elizabeth Malliaris, Persistence and regularity in unstable model theory, ProQuest LLC, Ann Arbor, MI, 2009. Thesis (Ph.D.)–University of California, Berkeley. MR 2713926
M. E. Malliaris, Hypergraph sequences as a tool for saturation of ultrapowers, J. Symbolic Logic 77 (2012), no. 1, 195–223. MR 2951637, DOI 10.2178/jsl/1327068699
M. Malliaris and S. Shelah, Cofinality spectrum theorems in model theory, set theory, and general topology, J. Amer. Math. Soc. 29 (2016), no. 1, 237–297. MR 3402699, DOI 10.1090/jams830
M. Malliaris and S. Shelah, An example of a new simple theory, arXiv preprint arXiv:1804.03254, 2018.
Maryanthe Malliaris and Saharon Shelah, Keisler’s order has infinitely many classes, Israel J. Math. 224 (2018), no. 1, 189–230. MR 3799754, DOI 10.1007/s11856-018-1647-7
M. Malliaris and S. Shelah, Keisler’s order is not simple (and simple theories may not be either), arXiv preprint arXiv:1906.10241, 2019.
M. Malliaris and S. Shelah, A new look at interpretability and saturation, Ann. Pure Appl. Logic 170 (2019), no. 5, 642–671. MR 3926500, DOI 10.1016/j.apal.2019.01.001
David Marker, Model theory, Graduate Texts in Mathematics, vol. 217, Springer-Verlag, New York, 2002. An introduction. MR 1924282
Donald A. Martin, The axiom of determinateness and reduction principles in the analytical hierarchy, Bull. Amer. Math. Soc. 74 (1968), 687–689. MR 227022, DOI 10.1090/S0002-9904-1968-11995-0
Donald A. Martin, Measurable cardinals and analytic games, in Mathematical Logic in the 20th Century, World Scientific, 2003, pp. 264–268.
D. A. Martin and R. M. Solovay, Internal Cohen extensions, Ann. Math. Logic 2 (1970), no. 2, 143–178. MR 270904, DOI 10.1016/0003-4843(70)90009-4
Donald A. Martin and John R. Steel, Projective determinacy, Proc. Nat. Acad. Sci. U.S.A. 85 (1988), no. 18, 6582–6586. MR 959109, DOI 10.1073/pnas.85.18.6582
Antonio Martínez-Abejón, An elementary proof of the principle of local reflexivity, Proc. Amer. Math. Soc. 127 (1999), no. 5, 1397–1398. MR 1476378, DOI 10.1090/S0002-9939-99-04687-0
A. R. D. Mathias, Solution of problems of Choquet and Puritz, Conference in Mathematical Logic—London ’70 (Proc. Conf., Bedford Coll., London, 1970) Lecture Notes in Math., Vol. 255, Springer, Berlin-New York, 1972, pp. 204–210. MR 363911
Hideyuki Matsumura, Commutative ring theory, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1989. Translated from the Japanese by M. Reid. MR 1011461
Dusa McDuff, Central sequences and the hyperfinite factor, Proc. London Math. Soc. (3) 21 (1970), 443–461. MR 281018, DOI 10.1112/plms/s3-21.3.443
Arnold W. Miller, There are no $Q$-points in Laver’s model for the Borel conjecture, Proc. Amer. Math. Soc. 78 (1980), no. 1, 103–106. MR 548093, DOI 10.1090/S0002-9939-1980-0548093-2
John Milnor, Growth of finitely generated solvable groups, J. Differential Geometry 2 (1968), 447–449. MR 244899
Deane Montgomery and Leo Zippin, Topological transformation groups, Interscience Publishers, New York-London, 1955. MR 73104
Itay Neeman, Ultrafilters and large cardinals, Ultrafilters across mathematics, Contemp. Math., vol. 530, Amer. Math. Soc., Providence, RI, 2010, pp. 181–200. MR 2757538, DOI 10.1090/conm/530/10445
Alexander Yu. Ol′shanskii and Mark V. Sapir, A finitely presented group with two non-homeomorphic asymptotic cones, Internat. J. Algebra Comput. 17 (2007), no. 2, 421–426. MR 2310154, DOI 10.1142/S021819670700369X
D. Osin and A. O. Houcine, Finitely presented groups with infinitely many non-homeomorphic asymptotic cones, to appear in Algebraic and Geometric Topology.
Narutaka Ozawa, Tsirelson’s problem and asymptotically commuting unitary matrices, J. Math. Phys. 54 (2013), no. 3, 032202, 8. MR 3059438, DOI 10.1063/1.4795391
Pierre Pansu, Croissance des boules et des géodésiques fermées dans les nilvariétés, Ergodic Theory Dynam. Systems 3 (1983), no. 3, 415–445 (French, with English summary). MR 741395, DOI 10.1017/S0143385700002054
Vladimir G. Pestov, Hyperlinear and sofic groups: a brief guide, Bull. Symbolic Logic 14 (2008), no. 4, 449–480. MR 2460675, DOI 10.2178/bsl/1231081461
Bedřich Pospíšil, Remark on bicompact spaces, Ann. of Math. (2) 38 (1937), no. 4, 845–846. MR 1503375, DOI 10.2307/1968840
Florin Rădulescu, The von Neumann algebra of the non-residually finite Baumslag group $\langle a,b|ab^3a^{-1}=b^2\rangle$ embeds into $R^\omega$, Hot topics in operator theory, Theta Ser. Adv. Math., vol. 9, Theta, Bucharest, 2008, pp. 173–185. MR 2436761
F. P. Ramsey, On a Problem of Formal Logic, Proc. London Math. Soc. (2) 30 (1929), no. 4, 264–286. MR 1576401, DOI 10.1112/plms/s2-30.1.264
V. N. Remeslennikov, $\exists$-free groups, Sibirsk. Mat. Zh. 30 (1989), no. 6, 193–197 (Russian); English transl., Siberian Math. J. 30 (1989), no. 6, 998–1001 (1990). MR 1043446, DOI 10.1007/BF00970922
Abraham Robinson, Non-standard analysis, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1996. Reprint of the second (1974) edition; With a foreword by Wilhelmus A. J. Luxemburg. MR 1373196, DOI 10.1515/9781400884223
Walter Roelcke and Susanne Dierolf, Uniform structures on topological groups and their quotients, Advanced Book Program, McGraw-Hill International Book Co., New York, 1981. MR 644485
Mary Ellen Rudin, Partial orders on the types in $\beta N$, Trans. Amer. Math. Soc. 155 (1971), 353–362. MR 273581, DOI 10.1090/S0002-9947-1971-0273581-5
Volker Runde, Lectures on amenability, Lecture Notes in Mathematics, vol. 1774, Springer-Verlag, Berlin, 2002. MR 1874893, DOI 10.1007/b82937
Mark Sapir, On groups with locally compact asymptotic cones, Internat. J. Algebra Comput. 25 (2015), no. 1-2, 37–40. MR 3325875, DOI 10.1142/S0218196715400020
Hans Schoutens, The use of ultraproducts in commutative algebra, Lecture Notes in Mathematics, vol. 1999, Springer-Verlag, Berlin, 2010. MR 2676525, DOI 10.1007/978-3-642-13368-8
D. Scott, Measurable cardinals and constructible sets, in Mathematical Logic in the 20th Century, World Scientific, 2003, pp. 407–410.
Zlil Sela, Diophantine geometry over groups. I. Makanin-Razborov diagrams, Publ. Math. Inst. Hautes Études Sci. 93 (2001), 31–105. MR 1863735, DOI 10.1007/s10240-001-8188-y
Zlil Sela, Diophantine geometry over groups VIII: Stability, Annals of Mathematics, 177 (2013), pp. 787–868.
Saharon Shelah, Every two elementarily equivalent models have isomorphic ultrapowers, Israel J. Math. 10 (1971), 224–233. MR 297554, DOI 10.1007/BF02771574
Saharon Shelah, Saturation of ultrapowers and Keisler’s order, Ann. Math. Logic 4 (1972), 75–114. MR 294113, DOI 10.1016/0003-4843(72)90012-5
Saharon Shelah, Proper forcing, in Proper Forcing, Springer, 1982, pp. 73–113.
Saharon Shelah, Vive la différence I: Nonisomorphism of ultrapowers of countable models, in Set Theory of the Continuum, Springer, 1992, pp. 357–405.
Saharon Shelah, Proper and improper forcing, vol. 5 of Perspectives in Logic, Cambridge University Press, 2017.
S. Shelah and M. E. Rudin, Unordered types of ultrafilters, Topology Proc. 3 (1978), no. 1, 199–204 (1979). MR 540490
W. Sierpiński, Fonctions additives non complètement additives et fonctions non mesurables, Fundamenta Mathematicae, 1 (1938), pp. 96–99.
T. Skolem, Über die Nicht-charakterisierbarkeit der Zahlenreihe mittels endlich oder abzählbar unendlich vieler Aussagen mit ausschliesslich Zahlenvariablen, Fundamenta mathematicae, 23 (1934), pp. 150–161.
R. Solovay, Measurable cardinals and the axiom of determinateness, Lecture Notes of UCLA Summer Institute on Axiomatic Set Theory, Los Angeles, (1967).
R. M. Solovay, $2^{\aleph _0}$ can be anything it ought to be, in The Theory of Models, Proceedings of the 1963 International Symposium at Berkeley, Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1965, p. 435.
R. M. Solovay, On the cardinality of $\sigma _2^1$ sets of reals, in Foundations of Mathematics, Springer, 1969, pp. 58–73.
Robert M. Solovay, A model of set-theory in which every set of reals is Lebesgue measurable, Ann. of Math. (2) 92 (1970), 1–56. MR 265151, DOI 10.2307/1970696
M. H. Stone, The theory of representations for Boolean algebras, Trans. Amer. Math. Soc. 40 (1936), no. 1, 37–111. MR 1501865, DOI 10.1090/S0002-9947-1936-1501865-8
M. H. Stone, Applications of the theory of Boolean rings to general topology, Trans. Amer. Math. Soc. 41 (1937), no. 3, 375–481. MR 1501905, DOI 10.1090/S0002-9947-1937-1501905-7
E. Szemerédi, On sets of integers containing no $k$ elements in arithmetic progression, Acta Arith. 27 (1975), 199–245. MR 369312, DOI 10.4064/aa-27-1-199-245
Terence Tao, Hilbert’s fifth problem and related topics, Graduate Studies in Mathematics, vol. 153, American Mathematical Society, Providence, RI, 2014. MR 3237440, DOI 10.1090/gsm/153
A. Tarski, Une contribution à la théorie de la mesure, Fundamenta Mathematicae, 15 (1930), pp. 42–50.
Katrin Tent and Martin Ziegler, A course in model theory, Lecture Notes in Logic, vol. 40, Association for Symbolic Logic, La Jolla, CA; Cambridge University Press, Cambridge, 2012. MR 2908005, DOI 10.1017/CBO9781139015417
Guy Terjanian, Un contre-exemple à une conjecture d’Artin, C. R. Acad. Sci. Paris Sér. A-B 262 (1966), A612 (French). MR 197450
Simon Thomas and Boban Velickovic, Asymptotic cones of finitely generated groups, Bull. London Math. Soc. 32 (2000), no. 2, 203–208. MR 1734187, DOI 10.1112/S0024609399006621
Stevo Todorcevic, Introduction to Ramsey spaces, Annals of Mathematics Studies, vol. 174, Princeton University Press, Princeton, NJ, 2010. MR 2603812, DOI 10.1515/9781400835409
Douglas Ulrich, Keisler’s order is not linear, assuming a supercompact, J. Symb. Log. 83 (2018), no. 2, 634–641. MR 3835081, DOI 10.1017/jsl.2018.1
Lou van den Dries, Lectures on the model theory of valued fields, Model theory in algebra, analysis and arithmetic, Lecture Notes in Math., vol. 2111, Springer, Heidelberg, 2014, pp. 55–157. MR 3330198, DOI 10.1007/978-3-642-54936-6_{4}
L. van den Dries and K. Schmidt, Bounds in the theory of polynomial rings over fields. A nonstandard approach, Invent. Math. 76 (1984), no. 1, 77–91. MR 739626, DOI 10.1007/BF01388493
L. van den Dries and A. J. Wilkie, Gromov’s theorem on groups of polynomial growth and elementary logic, J. Algebra 89 (1984), no. 2, 349–374. MR 751150, DOI 10.1016/0021-8693(84)90223-0
Nik Weaver, Forcing for mathematicians, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2014. MR 3184751, DOI 10.1142/8962
Benjamin Weiss, Sofic groups and dynamical systems, Sankhyā Ser. A 62 (2000), no. 3, 350–359. Ergodic theory and harmonic analysis (Mumbai, 1999). MR 1803462

AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution

Ultrafilters Throughout Mathematics

About this Title

Table of Contents

References [Enhancements On Off] (What's this?)