Vincenzo Dimonte: Non-proper elementary embeddings beyond L(V_{lambda+1})

Week 1, Friday June 19, 17:05-17:30

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.