Laver property

The Laver property is a combinatorial property related to embeddings of the Laver tree and its connection to large cardinal axioms in set theory. It is a property of an elementary embedding j: V → M, where V represents the set-theoretic universe and M is a transitive inner model of V, asserting a strong form of elementarity related to the preservation of certain structures under the embedding.

More precisely, an elementary embedding j: V → M is said to have the Laver property if there exists an a in M such that j(a) = {j ↑ a : a ∈ V}. This condition essentially states that the restriction of j to a is preserved under the embedding in a specific way. It relates the embedding j to a sequence of its restrictions, providing a way to "build" the embedding from its initial segments.

The Laver property is intimately connected to the existence of Laver sequences and the indestructibility of supercompact cardinals under certain forcing extensions. A Laver sequence, generally denoted as <gα : α < κ>, for a cardinal κ, is a sequence of κ-complete ultrafilters on κ such that for all α < κ, gα is a V[G] - ultrafilter, where G is a generic filter for a forcing notion which adds a Laver sequence. This is related to the idea that the ultrafilter measure is preserved under iteration.

The Laver property plays a crucial role in constructing models where supercompact cardinals remain supercompact after forcing, addressing a fundamental problem in inner model theory and the consistency strength of large cardinal axioms. Specifically, it allows one to control the behavior of embeddings under forcing, ensuring that the desirable properties of the large cardinal are not destroyed by forcing extensions.

The existence of elementary embeddings with the Laver property is a consequence of the existence of supercompact cardinals. Conversely, it is often used in showing the consistency strength of certain large cardinal axioms by constructing models with embeddings possessing this property.

The study of the Laver property involves sophisticated techniques from set theory, including forcing, inner model theory, and the theory of large cardinals. It provides a deeper understanding of the structure of the set-theoretic universe and the relationships between different large cardinal axioms.