Freiling's axioms of symmetry in a general setting and some applications

We formulate C. Freiling’s axioms of symmetry for general second-order structures with respect to a certain ideal of small sets contained in them and find several equivalent formulations of the principles. Then we focus on particular models, namely saturated and recursively saturated ones, and show that they are symmetric with respect to appropriate classes of small sets when their second-order part consists of definable sets. Some asymmetric models are also exhibited as well as partial asymmetric ones constructed by forcing.