Predicativity through transfinite reflection
  Peano Arithmetic is known to be provably equivalent to reflection over Elementary Arithmetic. We prove a characterization of Predicative Analysis in the guise of ATR0 in terms of stronger reflection principles. 
Andr\'es Cordon Franco, David Fern\'andez Duque, Joost J. Joosten and   F\'elix Lara Mart\'in
math.LO