Submission date: 16 May 2011.

Abstract:

We use a generalization of a construction by Ziegler to show that for any field F and any countable collection of countable subsets A_i \subseteq F, i \in I \subset Z_{>0} there exist infinitely many fields K of arbitrary positive transcendence degree over F and of infinite algebraic degree such that each A_i is first-order definable over K. We also use the construction to show that many infinitely axiomatizable theories of fields which are not compatible with the theory of algebraically closed fields are finitely hereditarily undecidable.

Mathematics Subject Classification: 03C07, 03C20

Keywords and phrases:

Full text arXiv 1105.2792: pdf, ps.