Undefinability and Absolute Undefinability

I call a subset of the domain of a countable model absolutely undefinable if the set of its images under automorphisms of the model is uncountable. By the Kueker-Reyes theorem, all sets that are not absolutely undefinable are parametrically definable in the extension of first-order logic in which countably infinite conjunctions and disjunctions are allowed. I will survey classical results about first-order undefinability in the standard model of arithmetic, including Tarski's undefinability of truth theorem, and I will contrast them with some old and some new results about absolute undefinability in nonstandard models of Peano Arithmetic.