Definability and decidability in infinite algebraic extensions

We use a generalization of a construction by Ziegler to show that for any field F and any countable collection of countable subsets Ai⊆F, i∈I⊂Z>0 there exist infinitely many fields K of arbitrary greater than one transcendence degree over F and of infinite algebraic degree such that each Ai 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.