UC Berkeley

We explore the technique of inverse limits, a tool first introduced by Laver in the context of rank-into-rank embeddings. We extend reflection results of Laver using inverse limits up to and including embeddings of the form $L(V_{lambda+1}) to L(V_{lambda+1})$. Isolating this technique as `inverse limit reflection,' we use it to prove structural results of $L(V_{lambda+1})$ very similar to properties of $L(R)$ under $AD^{L(R)}$. We then define a representation, first introduced by Woodin, for subsets of $V_{lambda+1}$, and prove basic closure properties of this representation. Employing the technique of inverse limit reflection we then prove an important property called the Tower Condition and analyze certain measures generated from fixed points of embeddings to broaden the extent of these representations in $L(V_{lambda+1})$.