22 research outputs found
Quotients of Strongly Proper Forcings and Guessing Models
We prove that a wide class of strongly proper forcing posets have quotients
with strong properties. Specifically, we prove that quotients of forcing posets
which have simple universal strongly generic conditions on a stationary set of
models by certain nice regular suborders satisfy the -approximation
property. We prove that the existence of stationarily many -guessing
models in , for sufficiently large cardinals ,
is consistent with the continuum being arbitrarily large, solving a problem of
Viale and Weiss
Set Theory
This stimulating workshop exposed some of the most exciting recent develops in set theory, including major new results about the proper forcing axiom, stationary reflection, gaps in P(ω)/Fin, iterated forcing, the tree property, ideals and colouring numbers, as well as important new applications of set theory to C*-algebras, Ramsey theory, measure theory, representation theory, group theory and Banach spaces
On Indestructible Strongly Guessing Models
In \cite{MV} we defined and proved the consistency of the principle which implies that many consequences of strong
forcing axioms hold simultaneously at and . In this paper
we formulate a strengthening of that we call
. We also prove, modulo the consistency of two
supercompact cardinals, that is consistent
with ZFC. In addition to all the consequences of , the principle ,
together with some mild cardinal arithmetic assumptions that hold in our model,
implies that any forcing that adds a new subset of either adds a
real or collapses some cardinal. This gives a partial answer to a question of
Abraham \cite{AvrahamPhD} and extends a previous result of Todor\v{c}evi\'{c}
\cite{Todorcevic82} in this direction
On what I do not understand (and have something to say): Part I
This is a non-standard paper, containing some problems in set theory I have
in various degrees been interested in. Sometimes with a discussion on what I
have to say; sometimes, of what makes them interesting to me, sometimes the
problems are presented with a discussion of how I have tried to solve them, and
sometimes with failed tries, anecdote and opinion. So the discussion is quite
personal, in other words, egocentric and somewhat accidental. As we discuss
many problems, history and side references are erratic, usually kept at a
minimum (``see ... '' means: see the references there and possibly the paper
itself).
The base were lectures in Rutgers Fall'97 and reflect my knowledge then. The
other half, concentrating on model theory, will subsequently appear