Abstract
This chapter presents certain weak formal systems that are only strong enough to formalize certain aspects of mathematical practice. All of the systems considered in the chapter use classical logic. ATR 0 means the formal system of arithmetical transfinite recursion with quantifier free induction on the natural numbers. This is an interesting finitely axiomatizable subsystem of second order arithmetic. Although ATR 0 is a subsystem of second order arithmetic, ATR 0 is examined in the chapter from a set theoretic viewpoint. The chapter studies a certain finitely axiomatizable system of set theory ATR 0 s whose key axiom asserts that every well ordering is isomorphic to a von Neumann ordinal.
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