New axioms in set theory

Giorgio Venturi; Matteo Viale

Matematica, Cultura e Società. Rivista dell'Unione Matematica Italiana (2018)

  • Volume: 3, Issue: 3, page 211-236
  • ISSN: 2499-751X

Abstract

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In this article we review the present situation in the foundations of set theory, discussing two programs meant to overcome the undecidability results, such as the independence of the continuum hypothesis; these programs are centered, respectively, on forcing axioms and Woodin's V = Ultimate-L conjecture. While doing so, we briefly introduce the key notions of set theory.

How to cite

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Venturi, Giorgio, and Viale, Matteo. "New axioms in set theory." Matematica, Cultura e Società. Rivista dell'Unione Matematica Italiana 3.3 (2018): 211-236. <http://eudml.org/doc/294070>.

@article{Venturi2018,
abstract = {In this article we review the present situation in the foundations of set theory, discussing two programs meant to overcome the undecidability results, such as the independence of the continuum hypothesis; these programs are centered, respectively, on forcing axioms and Woodin's V = Ultimate-L conjecture. While doing so, we briefly introduce the key notions of set theory.},
author = {Venturi, Giorgio, Viale, Matteo},
journal = {Matematica, Cultura e Società. Rivista dell'Unione Matematica Italiana},
language = {eng},
month = {12},
number = {3},
pages = {211-236},
publisher = {Unione Matematica Italiana},
title = {New axioms in set theory},
url = {http://eudml.org/doc/294070},
volume = {3},
year = {2018},
}

TY - JOUR
AU - Venturi, Giorgio
AU - Viale, Matteo
TI - New axioms in set theory
JO - Matematica, Cultura e Società. Rivista dell'Unione Matematica Italiana
DA - 2018/12//
PB - Unione Matematica Italiana
VL - 3
IS - 3
SP - 211
EP - 236
AB - In this article we review the present situation in the foundations of set theory, discussing two programs meant to overcome the undecidability results, such as the independence of the continuum hypothesis; these programs are centered, respectively, on forcing axioms and Woodin's V = Ultimate-L conjecture. While doing so, we briefly introduce the key notions of set theory.
LA - eng
UR - http://eudml.org/doc/294070
ER -

References

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  1. AUDRITO, G. and VIALE, M.. Absoluteness via resurrection. Journal of Mathematical Logic, 17(2):1750005, 36, 2017. Zbl06815211MR3730561DOI10.1142/S0219061317500052
  2. BARTON, N., TERNULLO, C., and VENTURI, G.. On forms of justification in set theory. Preprint, 2018. 
  3. BELL, J.. Set Theory. Boolean Valued Models and Independence Proofs. Oxford Science Pubblications, 2005. Zbl1065.03034MR2257858DOI10.1093/acprof:oso/9780198568520.001.0001
  4. BERNAYS, P.. Sur le platonism dans les mathématiques. L'enseignement mathématique, (34):52-69, 1935. English transl.: On mathematical platonism, in P. Benacerraf and H. Putnam, editors, Philosophy of mathematics: selected readings, Cambridge University Press, pp. 258-271, 1983. MR742474
  5. BOOLOS, G.. The iterative conception of set. Journal of Philosophy, 68(8):215-231, 1971. 
  6. CALDERONI, F. and THOMAS, S.. The bi-embeddability relation for countable abelian groups. Transaction of the American Mathematical Society, To appear. Zbl06999078MR3894051DOI10.1090/tran/7513
  7. CANTOR, G.. Grundlagen einer allgemeinen Mannigfaltigkeitslehre. Ein mathematich-philosophischer Versuch in der Lehre des Unendichen. Teubner, 1883. English transl.: Foundations of a general theory of manifolds: a mathematico-philosophical investigation into the theory of the infinite, in W. Ewald, editor, From Kant to Hilbert: A Source Book in the Foundations of Mathematics, vol. II, Clarendon Press, Oxford, pp. 878-919, 2008. 
  8. CANTOR, G.. Beiträge zur Begründung der transfiniten Mengenlehre. Mathematische Annalen, 46(4):481-512, 1895. Italian transl.: in G. Cantor, La Formazione della Teoria degli Insiemi (Scritti 1872-1899), G. Rigamonti, editor, Mimesis, 2012. MR1510964DOI10.1007/BF01444205
  9. COHEN, J. P.. The independence of the continuum hypothesis. Proceedings of the National Academy of Sciences of the United States of America, 50(6):1143-1148, 1963. Zbl0192.04401MR157890DOI10.1073/pnas.50.6.1143
  10. COHEN, J. P.. Skolem and pessimism about proof in mathematics. Philosophical Transaction of the Royal Society A, 363:2407-2418, 2005. Zbl1128.03002MR2197657DOI10.1098/rsta.2005.1661
  11. DALES, H. G. and WOODIN, W. H.. An Introduction to Independence for Analysts, volume 115 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 1987. Zbl0629.03030MR942216DOI10.1017/CBO9780511662256
  12. FARAH, I.. All automorphisms of the Calkin algebra are inner. Annals of Mathematics, 173(2):619-661, 2010. Zbl1250.03094MR2776359DOI10.4007/annals.2011.173.2.1
  13. FEFERMAN, S.. Does mathematics need new axioms?American Mathematical Monthly, 106:106-111, 1999. Zbl1076.00501MR1671849DOI10.2307/2589047
  14. FEFERMAN, S., FRIEDMAN, H., MADDY, P., and STEEL, J.. Does mathematics need new axioms?Bulletin of Symbolic Logic, 6(4):401-446, 2000. Zbl0977.03002MR1814122DOI10.2307/420965
  15. FENG, Q., MAGIDOR, M., and WOODIN, H.. Universally Baire sets of reals. In H. Judah, W. Just, and H. Woodin, editors, Set Theory of the Continuum, pages 203-242. Springer, 1992. Zbl0781.03034MR1233821DOI10.1007/978-1-4613-9754-0_15
  16. FERREIRÓS, J.. Labyrinth of Thought. A History of Set Theory and its Role in Modern Mathematics. Birkhäuser, 1999. MR1726552DOI10.1007/978-3-0348-5049-0
  17. FERREIRÓS, J.. On arbitrary sets and ZFC. Bullettin of Symbolic logic, 17(3):361-393, 2011. MR2856078DOI10.2178/bsl/1309952318
  18. FINE, K. and TENNANT, N.. A Defence of Arbitrary Objects. Proceedings of the Aristotelian Society, Supplementary Volumes, 57:55-77 + 79-89, 1983. 
  19. FOREMAN, M., MAGIDOR, M. and SHELAH, S.. Martin's Maximum, saturated ideals and nonregular ultrafilters. Annals of Mathematics (2), 127(1):1-47, 1988. Zbl0645.03028MR924672DOI10.2307/1971415
  20. GALE, D. and STEWART, F.. Infinite games with perfect information. In H. W. Kuhn and A. W. Tucker, editors, Contributions to the Theory of Games, pages 245-266. Princeton University press, 1953. Zbl0050.14305MR54922
  21. GIVANT, S. and HALMOS, P.. Introduction to Boolean Algebras. Undergraduate Texts in Mathematics. Springer, New York, 2009. Zbl1168.06001MR2466574DOI10.1007/978-0-387-68436-9
  22. GÖDEL, K.. The consistency of the axiom of choice and of the generalized continuum-hypothesis. Proceedings of the National Academy of Sciences of the United States of America. National Academy of Sciences, 24(12):556-557, 1938. 
  23. GÖDEL, K.. What is Cantor's continuum problem?American Mathematical Monthly, (54):515-525, 1947; errata, 55, p. 151. Italian trasl.: Cos'è il problema del continuo di Cantor?, in E. Ballo and G. Lolli and C. Mangione, editors, Opere. Volume 2, 1938-1974, Bollati Boringhieri, pp. 180-192, 2002. MR23780DOI10.2307/2304666
  24. GOLDBLATT, R.. On the role of the Baire category theorem and dependent choice in the foundation of logic. The Journal of Symbolic Logic, 50(2):412-422, 1985. Zbl0567.03023MR793122DOI10.2307/2274230
  25. GRAY, G.. Plato's Ghost. The Modernist Trasnformation of Mathematics. Princeton University Press, 2008. Zbl1166.00005MR2452344
  26. HAMKINS, J. D. and SEABOLD, D.. Well-founded boolean ultrapowers as large cardinal embeddings. 40 pages, 2012. 
  27. HILBERT, D.. Mathematical problems. Bulletin of the American Mathematical Society, 37(4):407-436, 2000. Italian translation in: V. Abrusci (editor), D. Hilbert, Ricerche sui fondamenti della matematica, Bibliopolis, 1978, pp. 145-162. Zbl0979.01028MR1779412DOI10.1090/S0273-0979-00-00881-8
  28. HRBACEK, K. and JECH, T.. Introduction to set theory, volume 220 of Monographs and Textbooks in Pure and Applied Mathematics. Marcel Dekker, Inc., New York, third edition, 1999. MR1697766
  29. JENSEN, R. B.. The fine structure of the constructible hierarchy. Annals of Pure and Applied Logic, 4:229-308; erratum, ibid. 4 (1972), 443, 1972. With a section by Jack Silver. Zbl0257.02035MR309729DOI10.1016/0003-4843(72)90001-0
  30. KANAMORI, A.. The Higher Infinite. Large Cardinals in Set Theory from their Beginnings. Springer-Verlag, 1994. Zbl0813.03034MR1321144
  31. KECHRIS, A.. Classical Descriptive Set Theory. Springer Verlag, 1994. Zbl0805.54035MR1321597DOI10.1007/978-1-4612-4190-4
  32. KOELLNER, P.. On the question of absolute undecidability. Philosophia Mathematica, 14:153-188, 2006. Zbl1113.03011MR2245398DOI10.1093/philmat/nkj009
  33. KOELLNER, P.. On reflection principles. Annals of Pure and Applied Logic, 157(2)(4):206-219, 2009. Zbl1162.03030MR2499709DOI10.1016/j.apal.2008.09.007
  34. KUNEN, K.. Set Theory. An Introduction to Independence Proofs. North-Holland, 1980. Zbl0443.03021MR597342
  35. LARSON, P. B.. The Stationary Tower: Notes on a Course by W. Hugh Woodin. AMS, 2004. Zbl1072.03031MR2069032DOI10.1090/ulect/032
  36. LARSON, P. B.. A brief history of determinacy. In Sets and Extensions in the Twentieth Century, 2012. Zbl1255.03010MR3409863DOI10.1016/B978-0-444-51621-3.50006-2
  37. MADDY, P.. Naturalism in Mathematics. Clarendon Press, 1997. Zbl0931.03003MR1699270
  38. MADDY, P.. Defending the Axioms: On the Philosophical Foundations of Set Theory. Oxford University Press, 2011. Zbl1219.00014MR2779203DOI10.1093/acprof:oso/9780199596188.001.0001
  39. MARTIN, D.. Measurable cardinals and analytic games. Fundamenta Mathematicae, 66:287-291, 1970. Zbl0216.01401MR258637DOI10.4064/fm-66-3-287-291
  40. MARTIN, D.. Borel determinacy. Annals of Mathematics, 102:363-371, 1975. MR403976DOI10.2307/1971035
  41. MARTIN, D. A. and STEEL, J. R.. A proof of projective determinacy. Journal of the American Mathematical Society, 2(1):71-125, 1989. Zbl0668.03021MR955605DOI10.2307/1990913
  42. MAYBERRY, J. P.. The Foundations of Mathematics in the Theory of Sets. Cambridge University Press, 2000. Zbl0972.03001MR1826603
  43. MOORE, G. H.. Zermelo's Axiom of Choice. Dover, 1982. Zbl0497.01005MR679315DOI10.1007/978-1-4613-9478-5
  44. MOORE, J. T.. The Proper Forcing Axiom. In R. Bhatia, editor, Proceedings of the International Congress of Mathematicians. Volume 2, pages 1-25. World Scientific, 2010. MR2827783
  45. MOORE, J. T.. What makes the continuum N2. In Foundations of Mathematics, volume 690 of Contemp. Math., pages 259-287. Amer. Math. Soc., Providence, RI, 2017. Zbl06767151MR3656315
  46. MYCIELSKI, J. and STEINHAUS, H.. A mathematical axiom contradicting the axiom of choice. Bulletin de l'Académie Polonaise des Sciences, 10:1-3, 1962. Zbl0106.00804MR140430
  47. PHILLIPS, N. and WEAVER, N.. The calkin algebra has outer automorphisms. Duke Mathematical Journal, 139(1):185-202, 2007. Zbl1220.46040MR2322680DOI10.1215/S0012-7094-07-13915-2
  48. SCOTT, D.. Measurable cardinals and constructible sets. Bulletin de l'Académie Polonaise des Sciences, Série des sciences mathématiques, astronomiques et physiques, 9:521-524, 1961. Zbl0154.00702MR143710
  49. SHELAH, S.. Whitehead groups may not be free, even assuming CH. I. Israel Journal of Mathematics, 28(3):193-203, 1977. Zbl0369.02035MR469757DOI10.1007/BF02759809
  50. SHELAH, S.. Whitehead groups may not be free, even assuming CH. II. Israel Journal of Mathematics, 35(4):257-285, 1980. Zbl0467.03049MR594332DOI10.1007/BF02760652
  51. SHULMAN, M. A.. Set theory for category theory. arXiv:0810.1279v2, 2008. 
  52. SOLOVAY, R.. A model of set-theory in which every set of reals is Lebesgue measurable. Annals of Mathematics. Second Series, 92(1):1-56, 1970. Zbl0207.00905MR265151DOI10.2307/1970696
  53. STEEL, J.. What is a Woodin cardinal?Notices of the American Mathematical Society, 54(9):1146-1147, 2007. Zbl1153.03315MR2349534
  54. TODORCÆEVIĆ, S.. Partition Problems in Topology, volume 84 of Contemporary Mathematics. American Mathematical Society, Providence, RI, 1989. MR980949DOI10.1090/conm/084
  55. VACCARO, A. and VIALE, M.. Generic absoluteness and boolean names for elements of a Polish space. Boll. Unione Mat. Ital., 10(3):293-319, 2017. Zbl06807740MR3691801DOI10.1007/s40574-017-0124-2
  56. VENTURI, G.. Forcing, multiverse and realism. In F. Boccuni and A. Sereni, editors, Objectivity, Knowledge and Proof. FIlMat Studies in the Philosophy of Mathematics, 211-241. Springer, 2016. MR3618488
  57. VENTURI, G.. Genericity and arbitrariness, Logique et Analyse. To appear. 
  58. VENTURI, G.. On generic arbitrary models of set theory. Preprint, 2017. 
  59. VIALE, M.. Notes on forcing. Zbl1150.03015
  60. VIALE, M.. Category forcings, MM+++ and generic absoluteness for the theory of strong forcing axioms. Journal of the American Mathematical Society, 29(3):675-728, 2016. Zbl1403.03108MR3486170DOI10.1090/jams/844
  61. VIALE, M.. Forcing the truth of a weak form of Schanuel's conjecture. Confluentes Math., 8(2):59-83, 2016. Zbl06754758MR3633221DOI10.5802/cml.33
  62. VIALE, M.. Useful axioms. Ifcolog Journal of Logics and their Applications, 4(10):3427-3462, 2017. 
  63. WOODIN, H.. The Continuum Hypothesis. Part I. Notices of the American Mathematical Society, 48(6):567-576, 2001. Zbl0992.03063MR1834351
  64. ZERMELO, E.. Untersuchungen über die Grundlagen der Mengenlehre I. Mathematische Annalen, 65:261-281, 1908. English transl.: Investigations in the foundations of set theory I, in J. van Heijenoort, editor, From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931, Source Books in the History of the Sciences, Harvard Univ. Press, pp. 199-215, 1967. Zbl39.0097.03MR1511466DOI10.1007/BF01449999
  65. ZERMELO, E.. Über Grenzzahlen und Mengenbereiche: Neue Untersuchungen über die Grundlagen der Mengenlehre. Fundamenta Mathematicae, 16:29-47, 1930. English transl.: On boundary numbers and domains of sets: New investigations in the foundations of set theory, in W. Ewald, editor, From Kant to Hilbert: A Source Book in the Foundations of Mathematics, vol. II, Clarendon Press, Oxford, pp. 1219-1233, 2008; Italian transl.: in C. Cellucci, editor, Il Paradiso di Cantor, Bibliopolis, pp. 178-195, 1978. MR270860

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