So far the strongest great cardinals hypothesis that has received a deep and
shared analysis is the existence of an elementary embedding j from
L(V_{lambda+1}) to itself, for some lambda>cp(j). There were various
attempts to define hypotheses stronger than I0, but Woodin's approach caught
my attention: since he found several similarities between L(V_{lambda+1})
under I0 and L(R) under AD, he continued to carry on the
comparison trying to find a hypothesis similar to AD_R,
constructing a sequence of E^0_alpha(V_{lambda+1}) such that
V_{lambda+1}subseteq E^0_alpha(V_{lambda+1})subseteq V_{lambda+2},
that imitates the construction of the minimum model of AD_R.
My attention is focused on the properties of the elementary embeddings from
L(E^0_alpha(V_{lambda+1})) to itself, and the first property that I analyzed
is PROPERNESS, i.e. the cofinality in Theta of L(E^0_alpha(V_{lambda+1}))
of the fixed points of the embedding, that it turns out is
quite important in preserving the similarity with determinacy.
The first original result is the existence of an alpha
and a j:L(E^0_alpha(V_{lambda+1})) < L(E^0_alpha(V_{lambda+1}))
that is not proper. This both validates the definition of proper elementary
embedding, since it states for the first time that the definition is not
trivial, and fills a gap in a Theorem by Woodin that is fundamental for this
new research.