305 research outputs found
The Proper Forcing Axiom, Prikry forcing, and the Singular Cardinals Hypothesis
The purpose of this paper is to present some results which suggest that the
Singular Cardinals Hypothesis follows from the Proper Forcing Axiom. What will
be proved is that a form of simultaneous reflection follows from the Set
Mapping Reflection Principle, a consequence of PFA. While the results fall
short of showing that MRP implies SCH, it will be shown that MRP implies that
if SCH fails first at kappa then every stationary subset of S_{kappa^+}^omega =
{a < kappa^+ : cf(a) = omega} reflects. It will also be demonstrated that MRP
always fails in a generic extension by Prikry forcing.Comment: 7 page
Club-guessing, stationary reflection, and coloring theorems
We obtain strong coloring theorems at successors of singular cardinals from
failures of certain instances of simultaneous reflection of stationary sets.
Along the way, we establish new results in club-guessing and in the general
theory of ideals.Comment: Initial public versio
The Proper Forcing Axiom and the Singular Cardinal Hypothesis
We show that the Proper Forcing Axiom implies the Singular Cardinal
Hypothesis. The proof is by interpolation and uses the Mapping Reflection
Principle.Comment: 10 page
Saturated filters at successors of singulars, weak reflection and yet another weak club principle
Suppose that lambda is the successor of a singular cardinal mu whose
cofinality is an uncountable cardinal kappa. We give a sufficient condition
that the club filter of lambda concentrating on the points of cofinality kappa
is not lambda^+-saturated. The condition is phrased in terms of a notion that
we call weak reflection. We discuss various properties of weak reflectio
Stationary reflection principles and two cardinal tree properties
We study consequences of stationary and semi-stationary set reflection. We
show that the semi stationary reflection principle implies the Singular
Cardinal Hypothesis, the failure of weak square principle, etc. We also
consider two cardinal tree properties introduced recently by Weiss and prove
that they follow from stationary and semi stationary set reflection augmented
with a weak form of Martin's Axiom. We also show that there are some
differences between the two reflection principles which suggest that stationary
set reflection is analogous to supercompactness whereas semi-stationary set
reflection is analogous to strong compactness.Comment: 19 page
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