235 research outputs found
The strong Prikry property
I isolate a combinatorial property of a poset that I call the
strong Prikry property, which implies the existence of an ultrafilter on the
complete Boolean algebra of such that one inclusion
of the Boolean ultrapower version of the so-called \Bukovsky-Dehornoy
phenomenon holds with respect to and . I show that in all cases
that were previously studied, and for which it was shown that they come with a
canonical iterated ultrapower construction whose limit can be described as a
single Boolean ultrapower, the posets in question satisfy this property: Prikry
forcing, Magidor forcing and generalized Prikry forcing
Subcomplete forcing principles and definable well-orders
It is shown that the boldface maximality principle for subcomplete forcing,
together with the assumption that the universe has only set-many grounds,
implies the existence of a (parameter-free) definable well-ordering of
. The same conclusion follows from the boldface
maximality for subcomplete forcing, assuming there is no inner model with an
inaccessible limit of measurable cardinals. Similarly, the bounded subcomplete
forcing axiom, together with the assumption that does not exist, for
some , implies the existence of a well-order of
which is -definable without parameters, and
-definable using a subset of as a parameter.
This well-order is in . Enhanced version of bounded
forcing axioms are introduced that are strong enough to have the implications
of the maximality principles mentioned above.Comment: 23 pages, sections on "more reflection" and enhanced bounded forcing
axioms adde
Subcomplete forcing, trees and generic absoluteness
We investigate properties of trees of height and their
preservation under subcomplete forcing. We show that subcomplete forcing cannot
add a new branch to an -tree. We introduce fragments of
subcompleteness which are preserved by subcomplete forcing, and use these in
order to show that certain strong forms of rigidity of Suslin trees are
preserved by subcomplete forcings. Finally, we explore under what circumstances
subcomplete forcing preserves Aronszajn trees of height and width .
We show that this is the case if CH fails, and if CH holds, then this is the
case iff the bounded subcomplete forcing axiom holds. Finally, we explore the
relationships between bounded forcing axioms, preservation of Aronszajn trees
of height and width and generic absoluteness of
-statements over first order structures of size , also
for other canonical classes of forcing.Comment: Some results were added and some arguments streamline
Weak square and stationary reflection
It is well-known that the square principle entails the
existence of a non-reflecting stationary subset of , whereas the
weak square principle does not. Here we show that if
for all , then
entails the existence of a non-reflecting stationary subset
of in the forcing extension for adding a
single Cohen subset of . It follows that indestructible forms of
simultaneous stationary reflection entail the failure of weak square. We
demonstrate this by settling a question concerning the subcomplete forcing
axiom (SCFA), proving that SCFA entails the failure of for
every singular cardinal of countable cofinality.Comment: 11 page
Boolean ultrapowers, the Bukovsky-Dehornoy phenomenon, and iterated ultrapowers
We show that while the length iterated ultrapower by a normal
ultrafilter is a Boolean ultrapower by the Boolean algebra of Prikry forcing,
it is consistent that no iteration of length greater than (of the same
ultrafilter and its images) is a Boolean ultrapower. For longer iterations,
where different ultrafilters are used, this is possible, though, and we give
Magidor forcing and a generalization of Prikry forcing as examples. We refer to
the discovery that the intersection of the finite iterates of the universe by a
normal measure is the same as the generic extension of the direct limit model
by the critical sequence as the Bukovsky-Dehornoy phenomenon, and we develop a
sufficient criterion (the existence of a simple skeleton) for when a version of
this phenomenon holds in the context of Boolean ultrapowers. Assuming that the
canonical generic filter over the Boolean ultrapower model has what we call a
continuous representation, we show that the Boolean model consists precisely of
those members of the intersection model that have continuously and eventually
uniformly represented codes
Degrees of rigidity for Souslin trees
We investigate various strong notions of rigidity for Souslin trees,
separating them under Diamond into a hierarchy. Applying our methods to the
automorphism tower problem in group theory, we show under Diamond that there is
a group whose automorphism tower is highly malleable by forcing.Comment: 33 page
Changing the Heights of Automorphism Towers by Forcing with Souslin Trees over L
We prove that there are groups in the constructible universe whose
automorphism towers are highly malleable by forcing. This is a consequence of
the fact that, under a suitable diamond hypothesis, there are sufficiently many
highly rigid non-isomorphic Souslin trees whose isomorphism relation can be
precisely controlled by forcing.Comment: 23 page
Aronszajn tree preservation and bounded forcing axioms
I investigate the relationships between three hierarchies of reflection
principles for a forcing class : the hierarchy of bounded forcing
axioms, of -absoluteness and of Aronszajn tree preservation
principles. The latter principle at level says that whenever is a
tree of height and width that does not have a branch of
order type , and whenever is a forcing notion in , then
it is not the case that forces that has such a branch.
-absoluteness serves as an intermediary between these principles
and the bounded forcing axioms. A special case of the main result is that for
forcing classes that don't add reals, the three principles at level
are equivalent. Special attention is paid to certain subclasses of subcomplete
forcing, since these are natural forcing classes that don't add reals
Incomparable -like models of set theory
We show that the analogues of the Hamkins embedding theorems, proved for the
countable models of set theory, do not hold when extended to the uncountable
realm of -like models of set theory. Specifically, under the
hypothesis and suitable consistency assumptions, we show that
there is a family of many -like models of ZFC, all
with the same ordinals, that are pairwise incomparable under embeddability;
there can be a transitive -like model of ZFC that does not embed into
its own constructible universe; and there can be an -like model of PA
whose structure of hereditarily finite sets is not universal for the
-like models of set theory.Comment: 15 pages. Commentary concerning this article can be made at
http://jdh.hamkins.org/incomparable-omega-one-like-models-of-set-theor
Ehrenfeucht's lemma in set theory
Ehrenfeucht's lemma (1973) asserts that whenever one element of a model of
Peano arithmetic is definable from another, then they satisfy different types.
We consider here the analogue of Ehrenfeucht's lemma for models of set theory.
The original argument applies directly to the ordinal-definable elements of any
model of set theory, and in particular, Ehrenfeucht's lemma holds fully for
models of set theory satisfying . We show that the lemma can fail,
however, in models of set theory with , and it necessarily fails in
the forcing extension to add a generic Cohen real. We go on to formulate a
scheme of natural parametric generalizations of Ehrenfeucht's lemma, namely,
the principles of the form , which asserts that whenever an object
is definable from some using parameters in , with ,
then the types of and over are different. We also consider various
analogues of Ehrenfeucht's lemma obtained by using algebraicity in place of
definability, where a set is algebraic in if it is a member of a finite
set definable from (as in Hamkins, Leahy arXiv:1305.5953). Ehrenfeucht's
lemma holds for the ordinal-algebraic sets, we prove, if and only if the
ordinal-algebraic and ordinal-definable sets coincide. Using similar analysis,
we answer two open questions posed by Hamkins and Leahy, by showing that (i)
algebraicity and definability need not coincide in models of set theory and
(ii) the internal and external notions of being ordinal algebraic need not
coincide.Comment: 13 pages. Commentary concerning this paper can be made at
http://jdh.hamkins.org/ehrenfeuchts-lemma-in-set-theor
- β¦