Generic Large Cardinals and Systems of Filters

We introduce the notion of $\mathcal{C}$-system of filters, generalizing the standard definitions of both extenders and towers of normal ideals. This provides a framework to develop the theory of extenders and towers in a more general and concise way. In this framework we investigate the topic of definability of generic large cardinals properties.Comment: 36 page

Two-cardinal ideal operators and indescribability

A well-known version of Rowbottom's theorem for supercompactness ultrafilters leads naturally to notions of two-cardinal Ramseyness and corresponding normal ideals introduced herein. Generalizing results of Baumgartner [7, 8], Feng [22] and the first author [16, 17], we study the hierarchies associated with a particular version of two-cardinal Ramseyness and a strong version of two-cardinal ineffability, as well as the relationships between these hierarchies and a natural notion of transfinite two-cardinal indescribability.Comment: Fixed more typos in the first paragrap

Implementation of classical communication in a quantum world

Observations of quantum systems carried out by finite observers who subsequently communicate their results using classical data structures can be described as "local operations, classical communication" (LOCC) observations. The implementation of LOCC observations by the Hamiltonian dynamics prescribed by minimal quantum mechanics is investigated. It is shown that LOCC observations cannot be described using decoherence considerations alone, but rather require the \textit{a priori} stipulation of a positive operator-valued measure (POVM) about which communicating observers agree. It is also shown that the transfer of classical information from system to observer can be described in terms of system-observer entanglement, raising the possibility that an apparatus implementing an appropriate POVM can reveal the entangled system-observer states that implement LOCC observations.Comment: 17 pages, 2 figures; final versio

2013 (Spring)

Abstracts of the talks given at the 2013 Spring Colloquium

Category forcings, MM+++MM^{+++}, and generic absoluteness for the theory of strong forcing axioms

We introduce a category whose objects are stationary set preserving complete boolean algebras and whose arrows are complete homomorphisms with a stationary set preserving quotient. We show that the cut of this category at a rank initial segment of the universe of height a super compact which is a limit of super compact cardinals is a stationary set preserving partial order which forces $MM^{++}$ and collapses its size to become the second uncountable cardinal. Next we argue that any of the known methods to produce a model of $MM^{++}$ collapsing a superhuge cardinal to become the second uncountable cardinal produces a model in which the cutoff of the category of stationary set preserving forcings at any rank initial segment of the universe of large enough height is forcing equivalent to a presaturated tower of normal filters. We let $MM^{+++}$ denote this statement and we prove that the theory of $L(Ord^{\omega_1})$ with parameters in $P(\omega_1)$ is generically invariant for stationary set preserving forcings that preserve $MM^{+++}$. Finally we argue that the work of Larson and Asper\'o shows that this is a next to optimal generalization to the Chang model $L(Ord^{\omega_1})$ of Woodin's generic absoluteness results for the Chang model $L(Ord^{\omega})$. It remains open whether $MM^{+++}$ and $MM^{++}$ are equivalent axioms modulo large cardinals and whether $MM^{++}$ suffices to prove the same generic absoluteness results for the Chang model $L(Ord^{\omega_1})$.Comment: - to appear on the Journal of the American Mathemtical Societ