Talk held by Vincenzo Dimonte (University of Udine, Italy) at the KGRC seminar on 2018-05-03.

**Abstract:** In the momentous years when the community of set theorists was reaching the definite answer for the problem of the consistency of the Axiom of Determinacy, Martin wrote a small paper in the Proceedings of the International Congress of Mathematicians, 1978, in which he proved that the iterable version of I3, a very large cardinal, implied the determinacy of $\Pi^1_2$ sets of reals. Later it was proved that AD had much lower consistency, and iterable I3 fell into oblivion. In the last decade interest on I3 re-emerged, but iterable I3 is still elusive, and the small paper by Martin is not helpful, as it is terse and full of gaps. Even the definition of iterable I3 is not convincing. In this seminar we will bring back to life this abandoned hypothesis, clean it up to modern standards, and reveal the existence of a new hierarchy of axioms that was previously overlooked.