Open Access
2015 More on Generic Dimension Groups
Philip Scowcroft
Notre Dame J. Formal Logic 56(4): 511-553 (2015). DOI: 10.1215/00294527-3153570


While finitely generic (f.g.) dimension groups are known to admit no proper self-embeddings, these groups also have no automorphisms other than scalar multiplications, and every countable infinitely generic (i.g.) dimension group admits proper self-embeddings and has automorphisms other than scalar multiplications. The finite-forcing companion of the theory of dimension groups is recursively isomorphic to first-order arithmetic, the infinite-forcing companion of the theory of dimension groups is recursively isomorphic to second-order arithmetic, and the first-order theory of existentially closed (e.c.) dimension groups is a complete Π11-set. While many special properties of f.g. dimension groups may be realized in recursive e.c. dimension groups, and many special properties of i.g. dimension groups may be realized in hyperarithmetic e.c. dimension groups, no f.g. dimension group is arithmetic and no i.g. dimension group is analytical. Yet there is an f.g. dimension group recursive in first-order arithmetic, and (modulo a set-theoretic hypothesis) there is an i.g. dimension group recursive in second-order arithmetic.


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Philip Scowcroft. "More on Generic Dimension Groups." Notre Dame J. Formal Logic 56 (4) 511 - 553, 2015.


Received: 15 March 2012; Accepted: 28 June 2013; Published: 2015
First available in Project Euclid: 30 September 2015

zbMATH: 1372.03081
MathSciNet: MR3403090
Digital Object Identifier: 10.1215/00294527-3153570

Primary: 03C60 , 06F20
Secondary: 03C25

Keywords: dimension group , existentially closed , finitely generic , infinitely generic

Rights: Copyright © 2015 University of Notre Dame

Vol.56 • No. 4 • 2015
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