399 research outputs found
Dependent choice, properness, and generic absoluteness
We show that Dependent Choice is a sufficient choice principle for developing the basic theory of proper forcing, and for deriving generic absoluteness for the Chang model in the presence of large cardinals, even with respect to -preserving symmetric submodels of forcing extensions. Hence, not only provides the right framework for developing classical analysis, but is also the right base theory over which to safeguard truth in analysis from the independence phenomenon in the presence of large cardinals. We also investigate some basic consequences of the Proper Forcing Axiom in, and formulate a natural question about the generic absoluteness of the Proper Forcing Axiom in and. Our results confirm as a natural foundation for a significant portion of classical mathematics and provide support to the idea of this theory being also a natural foundation for a large part of set theory
Few new reals
We introduce a new method for building models of CH, together with
statements over , by forcing over a model of CH. Unlike similar
constructions in the literature, our construction adds new reals, but only
-many of them. Using this approach, we prove that a very strong form
of the negation of Club Guessing at known as Measuring is consistent
together with CH, thereby answering a well-known question of Moore. The
construction works over any model of ZFC + CH and can be described as a finite
support forcing construction with finite systems of countable models with
markers as side conditions and with strong symmetry constraints on both side
conditions and working parts
The consistency of a club-guessing failure at the successor of a regular cardinal
I answer a question of Shelah by showing that if \k is a regular cardinal such that 2^{{<}\k}=\k, then there is a {<}\k--closed partial order preserving cofinalities and forcing that for every club--sequence \la C_\d\mid \d\in \k^+\cap\cf(\k)\ra with \ot(C_\d)=\k for all \d there is a club D\sub\k^+ such that \{\a<\k\mid \{C_\d(\a+1), C_\d(\a+2)\}\sub D\} is bounded for every \d. This forcing is built as an iteration with {<}\k--supports and with symmetric systems of submodels as side conditions
A Generalization of Martin's Axiom
We define the chain condition. The corresponding forcing axiom
is a generalization of Martin's Axiom and implies certain uniform failures of
club--guessing on that don't seem to have been considered in the
literature before.Comment: 36 page
Separating club-guessing principles in the presence of fat forcing axioms
We separate various weak forms of Club Guessing at in the presence of large, Martin's Axiom, and related forcing axioms. We also answer a question of Abraham and Cummings concerning the consistency of the failure of a certain polychromatic Ramsey statement together with the continuum large. All these models are generic extensions via finite support iterations with symmetric systems of structures as side conditions, possibly enhanced with -sequences of predicates, and in which the iterands are taken from a relatively small class of forcing notions. We also prove that the natural forcing for adding a large symmetric system of structures (the first member in all our iterations) adds -many reals but preserves CH
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
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