Provability, Computability and Reflection.

We show that Zilber’s conjecture that complex exponentiation is isomorphic to his pseudo-exponentiation follows from the a priori simpler conjecture that they are elementarily equivalent. An analysis of the first-order types in pseudo-exponentiation leads to a description of the elementary embeddings, and the result that pseudo-exponential fields are precisely the models of their common first-order theory which are atomic over exponential transcendence bases. We also show that the class of all pseudo-exponential fields is an example of a nonfinitary abstract elementary (...) class, answering a question of Kesälä and Baldwin. (shrink)

The language $L_A$ is formed by adding the quantifier $\Finv x$ , "few x", to the infinitary logic L A on an admissible set A. A complete axiomatization is obtained for models whose universe is the set of ordinals of A and where $\Finv x$ is interpreted as there exist A-finitely many x. For well-behaved A, every consistent sentence has a model with an A-recursive diagram. A principal tool is forcing for $L_A$.

We construct and study structures imitating the field of complex numbers with exponentiation. We give a natural, albeit non first-order, axiomatisation for the corresponding class of structures and prove that the class has a unique model in every uncountable cardinality. This gives grounds to conjecture that the unique model of cardinality continuum is isomorphic to the field of complex numbers with exponentiation.

The language $L_A(\Finv)$ is formed by adding the quantifier $\Finv x$ , "few x", to the infinitary logic L A on an admissible set A. A complete axiomatization is obtained for models whose universe is the set of ordinals of A and where $\Finv x$ is interpreted as there exist A-finitely many x. For well-behaved A, every consistent sentence has a model with an A-recursive diagram. A principal tool is forcing for $L_A(\Finv)$.