17 research outputs found
Combinatorial Properties and Dependent choice in symmetric extensions based on L\'{e}vy Collapse
We work with symmetric extensions based on L\'{e}vy Collapse and extend a few
results of Arthur Apter. We prove a conjecture of Ioanna Dimitriou from her
P.h.d. thesis. We also observe that if is a model of ZFC, then
can be preserved in the symmetric extension of in terms of
symmetric system , if
is -distributive and is -complete.
Further we observe that if is a model of ZF + , then
can be preserved in the symmetric extension of in terms of
symmetric system , if
is -strategically closed and is
-complete.Comment: Revised versio
Capturing sets of ordinals by normal ultrapowers
We investigate the extent to which ultrapowers by normal measures on
can be correct about powersets for . We
consider two versions of this questions, the capturing property
and the local capturing property
. holds if there is
an ultrapower by a normal measure on which correctly computes
. is a weakening of
which holds if every subset of is
contained in some ultrapower by a normal measure on . After examining
the basic properties of these two notions, we identify the exact consistency
strength of . Building on results of Cummings,
who determined the exact consistency strength of
, and using a forcing due to Apter and Shelah, we
show that can hold at the least measurable
cardinal.Comment: 20 page
Joint Laver diamonds and grounded forcing axioms
I explore two separate topics: the concept of jointness for set-theoretic
guessing principles, and the notion of grounded forcing axioms. A family of
guessing sequences is said to be joint if all of its members can guess any
given family of targets independently and simultaneously. I primarily
investigate jointness in the case of various kinds of Laver diamonds. In the
case of measurable cardinals I show that, while the assertions that there are
joint families of Laver diamonds of a given length get strictly stronger with
increasing length, they are all equiconsistent. This is contrasted with the
case of partially strong cardinals, where we can derive additional consistency
strength, and ordinary diamond sequences, where large joint families exist
whenever even one diamond sequence does. Grounded forcing axioms modify the
usual forcing axioms by restricting the posets considered to a suitable ground
model. I focus on the grounded Martin's axiom which states that Martin's axioms
holds for posets coming from some ccc ground model. I examine the new axiom's
effects on the cardinal characteristics of the continuum and show that it is
quite a bit more robust under mild forcing than Martin's axiom itself.Comment: This is my PhD dissertatio