ATRo and Some Related Theories

The main point of interest of this dissertation is to study theories related to the theory ATRo in the realm of second order arithmetic. It is divided into two parts. In Part I, the equivalence of several axiom schemas to (ATR) over ACAo is proven. In particular, so-called reduction principles – also known as separation principles – are discussed. Part I is then concluded with an analysis of set-parameter free variants of ATRo and related systems. In Part II we are interested in set-theoretic analogues of questions that were treated in Part I. To this end, a range of basic set theories featuring the natural numbers as urelements and induction principles on sets and the natural numbers of various strengths are introduced. To interpret set-theoretic objects within second order arithmetic, we adapt the method of representation trees introduced by Jäger and Simpson. Making use of representation trees, the effect on proof-theoretic strength when adding reduction principles to our basic set theories is discussed. Finally, the effect of adding Axiom Beta is examined