publication . Preprint . 2017

The Class of Countable Projective Planes is Borel Complete

Paolini, Gianluca;
Open Access English
  • Published: 16 Nov 2017
Abstract
We observe that Hall's free projective extension $P \mapsto F(P)$ of partial planes is a Borel map, and use a modification of the construction introduced in [9] to conclude that the class of countable non-Desarguesian projective planes is Borel complete. In the process, we also rediscover the main result of [7] on the realizability of every group as the group of collineations of some projective plane. Finally, we use classical results of projective geometry to prove that the class of countable Pappian projective planes is Borel complete.
Subjects
arXiv: Mathematics::LogicMathematics::General Topology
free text keywords: Mathematics - Logic, 51A35, 03E15, 54H05, 05B35, 22F50
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[1] Riccardo Camerlo and Su Gao. The Completeness of the Isomorphism Relation for Countable Boolean Algebras. Trans. Amer. Math. Soc. 353 (2001), no. 2, 491-518.

[2] Harvey Friedman and Lee Stanley. A Borel Reducibility Theory for Classes of Countable Structures. J. Symbolic Logic 54 (1989), no. 3, 894-914.

[3] Su Gao. Invariant Descriptive Set Theory. Pure and Applied Mathematics (Boca Raton), 293. CRC Press, Boca Raton, FL, 2009.

[4] Marshall Hall. Projective Planes. Trans. Amer. Math. Soc. 54 (1943), 229-277.

[5] Greg Hjorth and Alexander S. Kechris. Analytic Equivalence Relations and Ulm-Type Classifications. J. Symbolic Logic 60 (1995), no. 4, 1273-1300.

[6] Daniel R. Hughes and Fred C. Piper. Projective Planes. Graduate Texts in Mathematics, Vol. 6. Springer-Verlag, New York-Berlin, 1973.

[7] E. Mendelsohn. Every Group is the Collineation Group of some Projective Plane. J. Geometry 2 (1972), 97-106.

[8] Gianluca Paolini and Tapani Hyttinen. Beyond Abstract Elementary Classes: On The Model Theory of Geometric Lattices. Ann. Pure Appl. Logic, to appear.

[9] Gianluca Paolini. The Class of Countable Locally Finite Planes is Borel Complete. Submitted.

[10] Fredrick W. Stevenson. Projective Planes. W. H. Freeman and Co., San Francisco, Calif., 1972.

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