Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Gross spaces


Authors: Saharon Shelah and Otmar Spinas
Journal: Trans. Amer. Math. Soc. 348 (1996), 4257-4277
MSC (1991): Primary 11E04, 03E35; Secondary 12L99, 15A36
DOI: https://doi.org/10.1090/S0002-9947-96-01658-3
MathSciNet review: 1357403
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A Gross space is a vector space $E$ of infinite dimension over some field $F$, which is endowed with a symmetric bilinear form $\Phi :E^{2} \rightarrow F$ and has the property that every infinite dimensional subspace $U\subseteq E$ satisfies dim $U^{\perp }<$ dim $E$. Gross spaces over uncountable fields exist (in certain dimensions) (see [H. Gross and E. Ogg, Quadratic spaces with few isometries, Comment. Math. Helv. 48 (1973), 511-519]). The existence of a Gross space over countable or finite fields (in a fixed dimension not above the continuum) is independent of the axioms of ZFC. Here we continue the investigation of Gross spaces. Among other things, we show that if the cardinal invariant b equals $\omega _{1}$, a Gross space in dimension $\omega _{1}$ exists over every infinite field, and that it is consistent that Gross spaces exist over every infinite field but not over any finite field. We also generalize the notion of a Gross space and construct generalized Gross spaces in ZFC.


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

  • B. Balcar and P. Simon, Cardinal invariants in Boolean spaces, General topology and its relations to modern analysis and algebra, V (Prague, 1981) Sigma Ser. Pure Math., vol. 3, Heldermann, Berlin, 1983, pp. 39–47. MR 698388
  • James E. Baumgartner, Iterated forcing, Surveys in set theory, London Math. Soc. Lecture Note Ser., vol. 87, Cambridge Univ. Press, Cambridge, 1983, pp. 1–59. MR 823775, DOI https://doi.org/10.1017/CBO9780511758867.002
  • James E. Baumgartner and Otmar Spinas, Independence and consistency proofs in quadratic form theory, J. Symbolic Logic 56 (1991), no. 4, 1195–1211. MR 1136450, DOI https://doi.org/10.2307/2275468
  • Walter Baur and Herbert Gross, Strange inner product spaces, Comment. Math. Helv. 52 (1977), no. 4, 491–495. MR 457475, DOI https://doi.org/10.1007/BF02567381
  • Andreas Blass and Saharon Shelah, There may be simple $P_{\aleph _1}$- and $P_{\aleph _2}$-points and the Rudin-Keisler ordering may be downward directed, Ann. Pure Appl. Logic 33 (1987), no. 3, 213–243. MR 879489, DOI https://doi.org/10.1016/0168-0072%2887%2990082-0
  • Eric K. van Douwen, The integers and topology, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 111–167. MR 776622
  • D. H. Fremlin, Cichoń’s diagram, Initiation à l’Analyse, Université Piere et Marie Curie, Paris, 1985.
  • Herbert Gross, Quadratic forms in infinite-dimensional vector spaces, Progress in Mathematics, vol. 1, Birkhäuser, Boston, Mass., 1979. MR 537283
  • Herbert Gross and Erwin Ogg, Quadratic forms and linear topologies. VI. Quadratic spaces with few isometries, Comment. Math. Helv. 48 (1973), 511–519. MR 370131, DOI https://doi.org/10.1007/BF02566137
  • Martin Goldstern, Tools for your forcing construction, Set theory of the reals (Ramat Gan, 1991) Israel Math. Conf. Proc., vol. 6, Bar-Ilan Univ., Ramat Gan, 1993, pp. 305–360. MR 1234283
  • Thomas Jech, Set theory, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. Pure and Applied Mathematics. MR 506523
  • Jaime I. Ihoda and Saharon Shelah, Souslin forcing, J. Symbolic Logic 53 (1988), no. 4, 1188–1207. MR 973109, DOI https://doi.org/10.2307/2274613
  • Kenneth Kunen, Set theory, Studies in Logic and the Foundations of Mathematics, vol. 102, North-Holland Publishing Co., Amsterdam-New York, 1980. An introduction to independence proofs. MR 597342
  • S. Shelah, Proper forcing, Lecture Notes in Mathematics, vol. 942, Springer, 1982.
  • S. Shelah, On cardinal invariants of the continuum, Proceedings of the 6/83 Boulder conference in set theory (J. Baumgartner D. Martin and S. Shelah, eds.), Contemporary mathematics, vol. 31, AMS, 1984, pp. 183-207.
  • Saharon Shelah, Vive la différence. I. Nonisomorphism of ultrapowers of countable models, Set theory of the continuum (Berkeley, CA, 1989) Math. Sci. Res. Inst. Publ., vol. 26, Springer, New York, 1992, pp. 357–405. MR 1233826, DOI https://doi.org/10.1007/978-1-4613-9754-0_20
  • S. Shelah, Strong negative partition relations below the continuum, Acta Math. Hungar. 58 (1991), no. 1-2, 95–100. MR 1152830, DOI https://doi.org/10.1007/BF01903551
  • S. Shelah, There are Jońsson algebras in many inaccessible cardinals, Cardinal Arithmetic, Oxford University Press, 1994.
  • S. Shelah, Further cardinal arithmetic, in press ([Sh430] in Shelah’s list of publications), Israel Journal of Mathematics.
  • S. Shelah, Colouring and $\aleph _{2}$-c.c. not productive, in preparation ([Sh572] in Shelah’s list of publications).
  • O. Spinas, Konsistenz- und Unabhängigkeitsresultate in der Theorie der quadratischen Formen, Dissertation, University of Zürich, 1989.
  • O. Spinas, Iterated forcing in quadratic form theory, Israel J. Math. 79 (1992), no. 2-3, 297–315. MR 1248920, DOI https://doi.org/10.1007/BF02808222
  • O. Spinas, An undecidability result in lattice theory, Abstracts of papers presented to the AMS 11, no. 2. (March 1990), 161.
  • Otmar Spinas, Cardinal invariants and quadratic forms, Set theory of the reals (Ramat Gan, 1991) Israel Math. Conf. Proc., vol. 6, Bar-Ilan Univ., Ramat Gan, 1993, pp. 563–581. MR 1234289

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 11E04, 03E35, 12L99, 15A36

Retrieve articles in all journals with MSC (1991): 11E04, 03E35, 12L99, 15A36


Additional Information

Saharon Shelah
Affiliation: Department of Mathematics, The Hebrew University of Jerusalem, Givat Ram, Jerusalem, Israel
MR Author ID: 160185
ORCID: 0000-0003-0462-3152
Email: shelah@math.huji.ac.il

Otmar Spinas
Affiliation: Department of Mathematics, University of California, Irvine, California 92717
Email: ospinas@math.uci.edu

Received by editor(s): August 1, 1995
Additional Notes: The authors are supported by the Basic Research Foundation of the Israel Academy of Science.
Article copyright: © Copyright 1996 American Mathematical Society