Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-22T15:57:38.400Z Has data issue: false hasContentIssue false
  • English
  • Français

ERDŐS AND SET THEORY

Published online by Cambridge University Press:  30 December 2014

AKIHIRO KANAMORI
AKIHIRO KANAMORI*
Affiliation:
BOSTON UNIVERSITY, 111 CUMMINGTON MALL, BOSTON, MASSACHUSETTS 02215, USAE-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Extract

Paul Erdős (26 March 1913—20 September 1996) was a mathematician par excellence whose results and initiatives have had a large impact and made a strong imprint on the doing of and thinking about mathematics. A mathematician of alacrity, detail, and collaboration, Erdős in his six decades of work moved and thought quickly, entertained increasingly many parameters, and wrote over 1500 articles, the majority with others. His modus operandi was to drive mathematics through cycles of problem, proof, and conjecture, ceaselessly progressing and ever reaching, and his modus vivendi was to be itinerant in the world, stimulating and interacting about mathematics at every port and capital.

Type
Articles
Information
Bulletin of Symbolic Logic , Volume 20 , Issue 4 , December 2014 , pp. 449 - 490
Copyright
Copyright © The Association for Symbolic Logic 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Abraham, Uri (1981). Free sets for nowhere-dense set mappings. Israel Journal of Mathematics, 39:167–176.Google Scholar
Aigner, Martin and Ziegler, Günter (2009). Proofs from THE BOOK. Springer, Berlin, 2013. Corrected printing of the 4th edition.Google Scholar
Bagemihl, Frederick (1973). The existence of an everywhere dense independent set. Michigan Mathematical Journal, 20:1–2.Google Scholar
Bateman, Paul T. (2002). Some personal memories of Paul Erdős. In Halász, Gábor, Lovász, Lázló, Simonovits, Miklós, and Sós, Vera T., editors, Paul Erdős and His Mathematics I, volume 11 of Bolyai Society Mathematical Studies, pages 9–12. Springer, Berlin.Google Scholar
Baumgartner, James E. (1970). Results and Independence Proofs in Combinatorial Set Theory. PhD thesis, University of California at Berkeley.Google Scholar
Baumgartner, James E., (1984). Generic graph construction. The Journal of Symbolic Logic, 49:234–240.CrossRefGoogle Scholar
Baumgartner, James E. and Galvin, Fred (1978). Generalized Erdős cardinals and 0 #. Annals of Mathematical Logic, 15:289–311.Google Scholar
Baumgartner, James E. and Prikry, Karel L. (1976). On a theorem of Silver. Discrete Mathematics, 14:17–21.Google Scholar
Bollobás, Béla (2013). Paul Erdős: Life and work. In Graham, Ronald L., Nešetřil, Jaroslav, and Butler, Steve, editors, The Mathematics of Paul Erdős I. Springer, Berlin.Google Scholar
de Bruijn, Nicolaas G. (2002). Remembering Paul Erdős. In Halász, Gábor, Lovász, Lázló, Simonovits, Miklós, and Sós, Vera T., editors, Paul Erdős and His Mathematics I, volume 11 of Bolyai Society Mathematical Studies, pages 13–16. Springer, Berlin.Google Scholar
Devlin, Keith J. and Paris, Jeffrey B. (1973). More on the free subset problem. Annals of Mathematical Logic, 5:327–336.CrossRefGoogle Scholar
Dushnik, Ben and Miller, Edwin W. (1941). Partially ordered sets. American Journal of Mathematics, 63:606–610.CrossRefGoogle Scholar
Ehenfeucht, Andrzej and Mostowski, Andrzej M. (1956). Models of axiomatic theories admitting automorphisms. Fundamenta Mathematicae, 43:50–68.Google Scholar
Erdős, Paul (1932). Beweis eines Satzes von Tschebyschef. Acta Universitatis Szegediensis, 5:194–198.Google Scholar
Erdős, Paul, (1963). On a combinatorial problem. Nordisk Matematisk Tidskrift, 11:5–10.Google Scholar
Erdős, Paul, (1964). On a combinatorial problem. II. Acta Mathematica Academiae Scientiarum Hungaricae, 15:445–447.Google Scholar
Erdős, Paul, (1969). On a combinatorial problem III. Canadian Mathematical Bulletin, 12:413–416.Google Scholar
Erdős, Paul and Lovász, Lázló (1975). Problems and results on 3-chromatic hypergraphs and some related questions. In Hajnal, András, Rado, Richard, and Sós, Vera T., editors, Infinite and Finite Sets, Keszthely (Hungary), 1973, volume II, volume 10 of Colloquia Mathematica Societatis János Bolyai, pages 609–627. North-Holland, Amsterdam.Google Scholar
Erdős, Paul and Szekeres, György (1935). A combinatorial problem in geometry. Compositio Mathematica, 2:463–470.Google Scholar
Foreman, Matthew (1998). An א1-dense ideal on א2. Israel Journal of Mathematics, 108:253–290.Google Scholar
Foreman, Matthew and Laver, Richard (1988). Some downwards transfer properties for א 1. Advances in Mathematics, 67:230–238.Google Scholar
Freiling, Chris (1986). Axioms of symmetry: Throwing darts at the real number line. The Journal of Symbolic Logic, 51:190–200.Google Scholar
Galvin, Fred and Hajnal, András (1975). Inequalities for cardinal powers. Annals of Mathematics, 101:491–498.Google Scholar
Graham, Ronald L., Nešetřil, Jaroslav, and Butler, Steve, editors (2013a). The Mathematics of Paul Erdős I. Springer, Berlin, 2013. Second, expanded edition; first edition 1997.Google Scholar
Graham, Ronald L., Nešetřil, Jaroslav, and Butler, Steve, editors, (2013b). The Mathematics of Paul Erdős II. Springer, Berlin, 2013. Second, expanded edition; first edition 1997.Google Scholar
Hajnal, András (1961). Proof of a conjecture of S. Ruziewicz. Fundamenta Mathematicae, 50:123–128.CrossRefGoogle Scholar
Hajnal, András, (1997). Paul Erdős’ set theory. In Graham, Ronald L. and Nešetřil, Jaroslav, editors, The Mathematics of Paul Erdős II, volume 14 of Algorithms and Combinatorics, pages 352–393. Springer, Berlin.Google Scholar
Hajnal, András, Juhász, István, and Shelah, Saharon (1986). Splitting strongly almost disjoint families. Transactions of the American Mathematical Society, 295:369–387.Google Scholar
Hajnal, András, Juhász, István, and Shelah, Saharon, (2000). Strongly almost disjoint famil[i]es, revisited. Fundamental Mathematicae, 163:13–23.Google Scholar
Hajnal, András and Larson, Jean A. (2010). Partition relations. In Foreman, Matthew and Kanamori, Akihiro, editors, Handbook of Set Theory, volume 1, pages 129–213. Springer, Dordrecht.Google Scholar
Halász, Gábor, Lovász, Lázló, Simonovits, Miklós, and Sós, Vera T., editors (2002a). Paul Erdős and His Mathematics I, volume 11 of Bolyai Society Mathematical Studies. Springer, Berlin.Google Scholar
Halász, Gábor, Lovász, Lázló, Simonovits, Miklós, and Sós, Vera T., editors, (2002b). Paul Erdős and His Mathematics II, volume 11 of Bolyai Society Mathematical Studies. Springer, Berlin.Google Scholar
Hallett, Michael (1984). Cantorian Set Theory and Limitation of Size, volume 10 of Oxford Logic Guides. Clarendon Press, Oxford.Google Scholar
Hausdorff, Felix (2005). Grundzüge einer Theorie der geordneten Mengen. Mathematische Annalen, 65:435–505, 1908. Translated with an introduction in Jacob Plotkin, editor, Hausdorff on Ordered Sets, American Mathematical Society, pages 181–258.Google Scholar
Hechler, Stephen H. (1972). Directed graphs over topological spaces: some set theoretical aspects. Israel Journal of Mathematics, 11:231–248.CrossRefGoogle Scholar
Kanamori, Akihiro (1983). Morasses in combinatorial set theory. In Mathias, A. R. D., editor, Surveys in Set Theory, volume 87 of London Mathematical Society Lecture Note Series, pages 167–197. Cambridge University Press, Cambridge.Google Scholar
Kanamori, Akihiro, (2009). The Higher Infinite. Large Cardinals in Set Theory from Their Beginnings. Springer, Berlin, 2009. Second edition; first edition, 1994.Google Scholar
Jerome Keisler, H. (1962). Some applications of the theory of models to set theory. In Ernest Nagel, Patrick Suppes, and Alfred Tarski, editors, Logic, Methodology and Philosophy of Science. Proceedings of the 1960 International Congress, pages 80–86. Stanford University, Stanford.Google Scholar
Jerome Keisler, H., (1976). Six classes of theories. Journal of the Australian Mathematical Society, 21:257–266.CrossRefGoogle Scholar
Koepke, Peter (1984). The consistency strength of the free-subset property for ωω. The Journal of Symbolic Logic, pages 1198–1204.Google Scholar
Kojman, Menachem (2015). Splitting families of sets in ZFC. Advances in Mathematics, Volume 269, 10 January 2015, Pages 707–725.Google Scholar
Kojman, Menachem, Kubiś, Wiesław, and Shelah, Saharon (2004). On two problems of Erdős and Hechler: New methods in singular madness. Proceedings of the American Mathematical Society, 132:3357–3365.Google Scholar
Komjáth, Péter (1991). The chromatic number of some uncountable graphs. In Halász, Gábor, editor, Sets, Graphs, and Numbers: A Birthday Salute to Vera T. Sós and András Hajnal, volume 60 of Colloquia Mathematica Societatis János Bolyai, pages 439–444. North-Holland, Amsterdam.Google Scholar
Komjáth, Péter, (2013). Erdős’s work on infinite graphs. In Lovász, Lázló, Ruzsa, Imre Z., and Sós, Vera T., editors, Erdős Centennial, volume 25 of Bolyai Society Mathematical Studies, pages 325–345. Springer, Berlin.Google Scholar
Kőnig, Dénes (1936). Theorie der endliche und unendlichen Graphen. Akademische Verlagsgesellschaft MBG, Leipzig.Google Scholar
Kunen, Kenneth (1971). Elementary embeddings and infinitary combinatorics. The Journal of Symbolic Logic, 36:407–413.CrossRefGoogle Scholar
Kunen, Kenneth, (2013). The impact of Paul Erdős on set theory. In Lovász, Lázló, Ruzsa, Imre Z., and Sós, Vera T., editors, Erdős Centennial, volume 25 of Bolyai Society Mathematical Studies, pages 347–363. Springer, Berlin.Google Scholar
Kurepa, Ðuro (1939). Sur la puissance des ensembles partiellement ordonnés. Sprawozdania Towarzystwo Naukowe Warszawa, Mat.-Fiz, 32:61–67.Google Scholar
Kurepa, Ðuro, (1953). On reflexive symmetric relations and graphs. Slovenska Akademija Znanosti in Umetinosti. Razred za Matematične, Fizikalne in Technične Vede. Serija A, 4:65–92.Google Scholar
Ðuro Kurepa, , (1996). Selected Papers of Duro Kurepa. Matematicki Institute SANU.Google Scholar
Larson, Jean A. (2012). Infinite combinatorics. In Kanamori, Akihiro, Woods, John H., and Gabbay, Dov, editors, Sets and Extensions in the Twentieth Century, volume 6 of Handbook of the History of Logic. Elsevier, Amsterdam.Google Scholar
Laver, Richard (1978). Strong saturation properties of ideals (abstract). The Journal of Symbolic Logic, 43:371.Google Scholar
Laver, Richard, (1982). An (א2,א2,א0)-saturated ideal on ω1. In Dalen, Dirk van and Daniel Lascaraqnd Smiley, Timothy J., editors, Logic Colloquium ’80, volume 108 of Studies in Logic and the Foundatiions of Mathematics, pages 173–180. North-Holland, Amsterdam.Google Scholar
Łoś, Jerzy (1959). Linear equations and pure subgroups. Bulletin de l’Académie Polonaise des Sciences, Série des Sciences Mathématiques, Astronomiques et Physiques, 7:13–18.Google Scholar
Lovász, Lázló, Ruzsa, Imre Z., and Sós, Vera T., editors (2013). Erdős Centennial, volume 25 of Bolyai Society Mathematical Studies. Springer, Berlin.Google Scholar
Miller, Edwin W. (1937). On a property of families of sets. Comptes Rendus des Séances de la Société des Sciences et des Lettres de Varsovie, 30:31–38.Google Scholar
Morley, Michael (1965a). Categoricity in power. Transactions of the American Mathematical Society, 144:514–538.Google Scholar
Morley, Michael, (1965b). Omitting classes of elements. In Addison, John W., Henkin, Leon, and Tarski, Alfred, editors, The Theory of Models. Proceedings of the 1963 International Symposium at Berkeley. North-Holland, Amsterdam.Google Scholar
Prikry, Karel L. (1972). On a problem of Erdős, Hajnal and Rado. Discrete Mathematics, 2:51–59.Google Scholar
Shelah, Saharon (1971). Stability, the f.c.p., and superstability; model theoretic properties of formulas in first order theory. Annals of Mathematical Logic, 3:271–362.Google Scholar
Shelah, Saharon, (1972). A combinatorial problem; stability and order for models and theories in infinitary languages. Pacific Journal of Mathematics, 41:247–261.CrossRefGoogle Scholar
Shelah, Saharon, (1974). Categoricity of uncountable theories. In Henkin, Leon, editor, Proceedings of the Tarski Symposium, volume 25 of Proceedings of Symposia in Pure Mathematics, pages 187–203. American Mathematical Society, Providence.Google Scholar
Shelah, Saharon, (1975). A compactness theorem for singular cardinals, free algebra, Whitehead problem and transversals. Israel Journal of Mathematics, 62:319–349.Google Scholar
Shelah, Saharon, (1975). Notes on partition calculus. In Hajnal, András, Rado, Richard, and Sós, Vera T., editors, Infinite and Finite Sets, Keszthely (Hungary), 1973, volume II, volume 10 of Colloquia Mathematica Societatis János Bolyai, pages 1257–1276. North-Holland, Amsterdam.Google Scholar
Shelah, Saharon, (1981). Canonization theorems and applications. The Journal of Symbolic Logic, 46:345–353.Google Scholar
Shelah, Saharon, (1988). Was Sierpiński right? I. Israel Journal of Mathematics, 62:355–380.Google Scholar
Shelah, Saharon, (1990). Classification Theory and the Number of Non-Isomorphic Models. North-Holland, Amsterdam, revised edition.Google Scholar
Shelah, Saharon, (1990). Incompactness for chromatic numbers of graphs. In Baker, Alan, Bollobás, Béla, and Hajnal, András, editors, A Tribute to Paul Erdős, pages 361–371. Cambridge University Press, Cambridge.Google Scholar
Shelah, Saharon, (1992). Strong partition relations below the power set: Consistency, Was Sierpiński right? II. In Halász, Gábor, editor, Sets, Graphs and Numbers: A Birthday Salute to Vera T. Sós and András Hajnal, volume 60 of Colloquia Mathematica Societatis János Bolyai, pages 637–668. North-Holland, Amsterdam.Google Scholar
Shelah, Saharon, (1994). Cardinal Arithmetic, volume 29 of Oxford Logic Guides. Clarendon Press, Oxford.Google Scholar
Shelah, Saharon, (2000). Was Sierpiński right? IV. The Journal of Symbolic Logic, 65:1031–1054.Google Scholar
Shelah, Saharon, (2013). On incompactness of chromatic number of graphs. Acta Mathematica Hungarica, 139:363–371.Google Scholar
Shelah, Saharon and Stanley, Lee J. (1987). A theorem and some consistency results in partition calculus. Annals of Pure and Applied Logic, 36:119–152.Google Scholar
Shelah, Saharon and Stanley, Lee J., (1993). More consistency results in partition calculus. Israel Journal of Mathematics, 81:97–110.Google Scholar
Sierpiński, Wacław (1933). Sur un problème de la théorie des relations. Annali della Scuola Normale Superiore di Pisa, 2(2):285–287.Google Scholar
Sierpiński, Wacław, (1934a). Hypothèse du Continu, volume 4 of Monograpfie Matematyczne. Seminarjum Matematyczne Uniwersytetu Warszawskiego, Warsaw. Second revised edition, Chelsea, New York, 1956.Google Scholar
Sierpiński, Wacław, (1934b). Sur la dualité entre la première catégorie et la mesure nulle.Fundamenta Mathematicae, pages 276–280.CrossRefGoogle Scholar
Sierpiński, Wacław and Tarski, Alfred (1930). Sur une propriété caractéristique des nombres inaccessibles. Fundamenta Mathematicae, 15:292–300.Google Scholar
Silver, Jack H. (1975). On the singular cardinals problem. In Proceedings of the International Congress of Mathematicians, pages 265–268. Canadian Mathematical Congress, Montreal.Google Scholar
Sós, Vera T. (2002). Turbulent years: Erdős in his correspondence with Turán. In Halász, Gábor, Lovász, Lázló, Simonovits, Miklós, and Sós, Vera T., editors, Paul Erdős and His Mathematics I, volume 11 of Bolyai Society Mathematical Studies, pages 85–146. Springer, Berlin.Google Scholar
Tarski, Alfred (1938). Über unerreichbare Kardinalzahlen. Fundamenta Mathematicae, 30:68–89.Google Scholar
Tarski, Alfred, (1962). Some problems and results relevant to the foundations of set theory. In Nagel, Ernest, Suppes, Patrick, and Tarski, Alfred, editors, Logic, Methodology and Philosophy of Science. Proceedings of the 1960 International Congress, pages 125–135. Stanford University, Stanford.Google Scholar
Taylor, Alan D. (1980). On saturated sets of ideals and Ulam’s problem. Fundamenta Mathematicae, 109:37–53.Google Scholar
Todorcevic, Stevo (1987). Partitioning pairs of countable ordinals. Acta Mathematica, 159:261–294.Google Scholar
Todorcevic, Stevo, (1994). Some partition of three-dimensional combinatorial cubes. Journal of Combiantorial Theory, 68:410–437.Google Scholar
Todorcevic, Stevo, (1997). Comparing the continuum with the first two uncountable cardinals. In Dalla Chiara, M.L., editor, Logic and Scientific Methods, volume 259 of Synthese Library, pages 145–155. Kluwer, Dordrecht.Google Scholar
Todorcevic, Stevo, (2007). Walks on Ordinals and Their Characteristics, volume 263 of Progress in Mathematics. Birkhäuser, Basel.Google Scholar
Turán, Paul (2002). Pál Erdős is fifty. In Halász, Gábor, Lovász, Lázló, Simonovits, Miklós, and Sós, Vera T., editors, Paul Erdős and His Mathematics I, volume 11 of Bolyai Society Mathematical Studies, pages 55–83. Springer, Berlin.Google Scholar
Ulam, Stanisław M. (1930). Zur Masstheorie in der allgemeinen Mengenlehre. Fundamenta Mathematicae, 16:140–150.CrossRefGoogle Scholar
Vazsonyi, Andrew (1996). Paul Erdös, the world’s most beloved mathematical genius “leaves”. Pure Mathematics and Applications, 7:1–12. Anthologized at www.emis.de/classics/Erdos/textpdf/vazsonyi/genius.pdf.Google Scholar