550 research outputs found
Linear orders: When embeddability and epimorphism agree
When a linear order has an order preserving surjection onto each of its suborders, we say that it is strongly surjective. We prove that the set of countable strongly surjective linear orders is a D\u2c72(\u3a011)-complete set. Using hypotheses beyond ZFC, we prove the existence of uncountable strongly surjective orders
Set Theory
This workshop included selected talks on pure set theory and its applications, simultaneously showing diversity and coherence of the subject
Club guessing and the universal models
We survey the use of club guessing and other pcf constructs in the context of
showing that a given partially ordered class of objects does not have a
largest, or a universal element
Category forcings, , and generic absoluteness for the theory of strong forcing axioms
We introduce a category whose objects are stationary set preserving complete
boolean algebras and whose arrows are complete homomorphisms with a stationary
set preserving quotient. We show that the cut of this category at a rank
initial segment of the universe of height a super compact which is a limit of
super compact cardinals is a stationary set preserving partial order which
forces and collapses its size to become the second uncountable
cardinal. Next we argue that any of the known methods to produce a model of
collapsing a superhuge cardinal to become the second uncountable
cardinal produces a model in which the cutoff of the category of stationary set
preserving forcings at any rank initial segment of the universe of large enough
height is forcing equivalent to a presaturated tower of normal filters. We let
denote this statement and we prove that the theory of
with parameters in is generically invariant
for stationary set preserving forcings that preserve . Finally we
argue that the work of Larson and Asper\'o shows that this is a next to optimal
generalization to the Chang model of Woodin's generic
absoluteness results for the Chang model . It remains open
whether and are equivalent axioms modulo large cardinals
and whether suffices to prove the same generic absoluteness results
for the Chang model .Comment: - to appear on the Journal of the American Mathemtical Societ
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