Feferman proved in 1962 that any arithmetical theorem is
a consequence of a suitable transfinite iteration of full uniform reflection of PA. This result is commonly known as Feferman's completeness
theorem. The purpose of this talk is to give two new proofs of Feferman's completeness theorem that, we hope, shed new light on this mysterious and often overlooked
result. Moreover, one of the proofs
furnishes sharp bounds on the order
types of well-orders necessary to attain completeness. 

joint work with Fedor Pakhomov and Dino Rossegger