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MADNESS IN VECTOR SPACES

Published online by Cambridge University Press:  29 August 2019

IIAN B. SMYTHE
IIAN B. SMYTHE*
Affiliation:
DEPARTMENT OF MATHEMATICS RUTGERS UNIVERSITY – NEW BRUNSWICK 110 FRELINGHUYSEN ROAD PISCATAWAY, NJ08854, USAE-mail: [email protected]: www.iiansmythe.com
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Abstract

We consider maximal almost disjoint families of block subspaces of countable vector spaces, focusing on questions of their size and definability. We prove that the minimum infinite cardinality of such a family cannot be decided in ZFC and that the “spectrum” of cardinalities of mad families of subspaces can be made arbitrarily large, in analogy to results for mad families on ω. We apply the author’s local Ramsey theory for vector spaces [32] to give partial results concerning their definability.

Keywords

Primary 03E1703E15Secondary 15A03mad familiesinfinite-dimensionalvector spacesblock sequencescardinal invariantsforcingdefinabilityRamsey theory
Type
Articles
Information
The Journal of Symbolic Logic , Volume 84 , Issue 4 , December 2019 , pp. 1590 - 1611
Copyright
Copyright © The Association for Symbolic Logic 2019 

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References

REFERENCES

Baumgartner, J. E., Sacks forcing and the total failure of Martin’s axiom. Topology and its Applications, vol. 19 (1985), no. 3, pp. 211–225.CrossRefGoogle Scholar
Baumgartner, J. E. and Laver, R., Iterated perfect-set forcing. Annals of Mathematical Logic, vol. 17 (1979), no. 3, pp. 271–288.CrossRefGoogle Scholar
Bell, M. G., On the combinatorial principle $P\left( \mathfrak{c} \right)$. Fundamenta Mathematicae, vol. 114 (1981), no. 2, pp. 149–157.CrossRefGoogle Scholar
Bice, T. M., MAD families of projections on l2 and real-valued functions on ω. Archive for Mathematical Logic, vol. 50 (2011), no. 7–8, pp. 791–801.CrossRefGoogle Scholar
Blass, A., Combinatorial cardinal characteristics of the continuum, Handbook of Set Theory (Foreman, M. and Kanamori, A., editors), vol. 1, Springer, Dordrecht, 2010, pp. 395–489.CrossRefGoogle Scholar
Brendle, J., The almost-disjointness number may have countable cofinality. Transactions of the American Mathematical Society, vol. 355 (2003), no. 7, pp. 2633–2649.CrossRefGoogle Scholar
Brendle, J. and Ávila, L. M. G., Forcing-theoretic aspects of Hindman’s Theorem. Journal of the Mathematical Society of Japan, vol. 69 (2017), no. 3, pp. 1247–1280.CrossRefGoogle Scholar
Prisco, C. D., Mijares, J. G., and Nieto, J., Local Ramsey theory: An abstract approach. Mathematical Logic Quarterly, vol. 63 (2017), no. 5, pp. 384–396.Google Scholar
Farah, I., Semiselective coideals. Mathematika, vol. 45 (1998), no. 1, pp. 79–103.CrossRefGoogle Scholar
Farah, I. and Wofsey, E., Set theory and operator algebras, Appalachian Set Theory 2006–2012 (Cummings, J. and Schimmerling, E., editors), London Mathematical Society Lecture Note Series, vol. 406, Cambridge University Press, Cambridge, 2013, pp. 63–120.Google Scholar
Gowers, W. T., An infinite Ramsey theorem and some Banach-space dichotomies. Annals of Mathematics (2), vol. 156 (2002), no. 3, pp. 797–833.CrossRefGoogle Scholar
Guzmán-González, O., There is a +-Ramsey MAD family. Israel Journal of Mathematics, vol. 229 (2019), no. 1, pp. 393–414.CrossRefGoogle Scholar
Guzmán-González, O., Hrušák, M., Martínez-Ranero, C. A., and Ramos-García, U. A., Generic existence of MAD families, this Journal, vol. 82 (2017), no. 1, pp. 303–316.Google Scholar
Hechler, S. H., Short complete nested sequences in βN \ N and small maximal almost-disjoint families. Topology and its Applications, vol. 2 (1972), pp. 139–149.CrossRefGoogle Scholar
Hindman, N., Finite sums from sequences within cells of a partition of $\mathbb{N}$. Journal of Combinatorial Theory, Series A, vol. 17 (1974), pp. 1–11.CrossRefGoogle Scholar
Horowitz, H. and Shelah, S., Can you take Törnquist’s inaccessible away? preprint, 2016.Google Scholar
Horowitz, H. and Shelah, S., On the definability of mad families of vector spaces, preprint, 2018.Google Scholar
Hrušák, M., Selectivity of almost disjoint families. Acta Universitatis Carolinae. Mathematica et Physica, vol. 41 (2000), no. 2, pp. 13–21.Google Scholar
Hrušák, M., Life in the Sacks model. Acta Universitatis Carolinae. Mathematica et Physica, vol. 42 (2001), no. 2, pp. 43–58. 29th Winter School on Abstract Analysis (Lhota nad Rohanovem/Zahrádky u České Lípy, 2001).Google Scholar
Kolman, O., Almost disjoint families: An application to linear algebra. Electronic Journal of Linear Algebra, vol. 7 (2000), pp. 41–52.CrossRefGoogle Scholar
Kunen, K., Set Theory, Studies in Logic and the Foundations of Mathematics, vol. 102, North-Holland, Amsterdam, 1980.Google Scholar
Martin, D. A. and Solovay, R. M., Internal Cohen extensions. Annals of Mathematical Logic, vol. 2 (1970), no. 2, pp. 143–178.CrossRefGoogle Scholar
Mathias, A. R. D., Happy families. Annals of Mathematical Logic, vol. 12 (1977), no. 1, pp. 59–111.CrossRefGoogle Scholar
Miller, A. W., Infinite combinatorics and definability. Annals of Pure and Applied Logic, vol. 41 (1989), no. 2, pp. 179–203.CrossRefGoogle Scholar
Moore, J. T., The proper forcing axiom, Proceedings of the International Congress of Mathematicians (Bhatia, R., Pal, A., Rangarajan, G., Srinivas, V., and Vanninathan, M., editors), vol. II, Hindustan Book Agency, New Delhi, 2010, pp. 3–29.Google Scholar
Moore, J. T., Hrušák, M., and Džamonja, M., Parametrized ◊ principles. Transactions of the American Mathematical Society, vol. 356 (2004), no. 6, pp. 2281–2306.CrossRefGoogle Scholar
Rosendal, C., Infinite asymptotic games. Annales de l’institut Fourier (Grenoble), vol. 59 (2009), no. 4, pp. 1359–1384.CrossRefGoogle Scholar
Rosendal, C., An exact Ramsey principle for block sequences. Collectanea Mathematica, vol. 61 (2010), no. 1, pp. 25–36.CrossRefGoogle Scholar
Shelah, S., Two cardinal invariants of the continuum $\left( {\mathfrak{d} < \mathfrak{a}} \right)$ and FS linearly ordered iterated forcing . Acta Mathematica, vol. 192 (2004), no. 2, pp. 187–223.CrossRefGoogle Scholar
Shelah, S. and Spinas, O., Mad spectra, this Journal, vol. 80 (2015), no. 3, pp. 901–916.Google Scholar
Smythe, I. B., Set theory in infinite-dimensional vector spaces, Ph.D. thesis, Cornell University, 2017.Google Scholar
Smythe, I. B., A local Ramsey theory for block sequences. Transactions of the American Mathematical Society, vol. 370 (2018), no. 12, pp. 8859–8893.CrossRefGoogle Scholar
Todorcevic, S., Topics in Topology, Lecture Notes in Mathematics, vol. 1652, Springer-Verlag, Berlin, 1997.CrossRefGoogle Scholar
Törnquist, A., ${\text{\Sigma }}_2^1 $ and ${\text{\Pi }}_1^1 $ mad families, this Journal, vol. 78 (2013), no. 4, pp. 1181–1182.Google Scholar
Törnquist, A., Definability and almost disjoint families. Advances in Mathematics, vol. 330 (2018), pp. 61–73.CrossRefGoogle Scholar
van Douwen, E. K., The integers and topology, Handbook of Set-Theoretic Topology (Kunen, K. and Vaughan, J. E., editors), North-Holland, Amsterdam, 1984, pp. 111–167.CrossRefGoogle Scholar
Wofsey, E., P(ω) / fin and projections in the Calkin algebra. Proceedings of the American Mathematical Society, vol. 136 (2008), no. 2, pp. 719–726.CrossRefGoogle Scholar
Zapletal, J., Isolating cardinal invariants, Journal of Mathematical Logic. vol. 3 (2003), no. 1, pp. 143–162.CrossRefGoogle Scholar