5 research outputs found
Ladder system uniformization on trees I & II
Given a tree of height , we say that a ladder system colouring
has a -uniformization if there is a
function defined on a subtree of so that for any of limit height and almost all ,
. In sharp contrast to the
classical theory of uniformizations on , J. Moore proved that CH is
consistent with the statement that any ladder system colouring has a
-uniformization (for any Aronszajn tree ). Our goal is to present a fine
analysis of ladder system uniformization on trees pointing out the analogies
and differences between the classical and this new theory. We show that if
is a Suslin tree then (i) CH implies that there is a ladder system colouring
without -uniformization; (ii) the restricted forcing axiom implies
that any ladder system colouring has an -uniformization. For an
arbitrary Aronszajn tree , we show how diamond-type assumptions affect the
existence of ladder system colourings without a -uniformization.
Furthermore, it is consistent that for any Aronszajn tree and ladder system
there is a colouring of without a -uniformization;
however, and quite surprisingly, implies that for any ladder
system there is an Aronszajn tree so that any monochromatic
colouring of has a -uniformization. We also prove positive
uniformization results in ZFC for some well-studied trees of size continuum,
and finish with a list of open problems.Comment: Revised version with improved presentation and updated problem list;
30 pages and 2 figures; submitted for publicatio
Uncountable strongly surjective linear orders
A linear order is strongly surjective if can be mapped onto any of
its suborders in an order preserving way. We prove various results on the
existence and non-existence of uncountable strongly surjective linear orders
answering questions of Camerlo, Carroy and Marcone. In particular,
implies the existence of a lexicographically ordered
Suslin-tree which is strongly surjective and minimal; every strongly surjective
linear order must be an Aronszajn type under or in
the Cohen and other canonical models (where );
finally, we prove that it is consistent with CH that there are no uncountable
strongly surjective linear orders at all. We end the paper with a healthy list
of open problems.Comment: 21 pages, revised version; to appear in Order; comments are very
welcom
Laver and set theory
In this commemorative article, the work of Richard Laver is surveyed in its full range and extent.Accepted manuscrip
Hausdorff gaps and towers in P(\omega)/Fin
We define and study two classes of uncountable -chains:
Hausdorff towers and Suslin towers. We discuss their existence in various
models of set theory. Then, some of the results and methods are used to provide
examples of indestructible gaps not equivalent to a Hausdorff gap. Also, we
indicate possible ways of developing a structure theory for towers.Comment: 34 page
Rethinking the notion of oracle: A link between synthetic descriptive set theory and effective topos theory
We present three different perspectives of oracle. First, an oracle is a
blackbox; second, an oracle is an endofunctor on the category of represented
spaces; and third, an oracle is an operation on the object of truth-values.
These three perspectives create a link between the three fields, computability
theory, synthetic descriptive set theory, and effective topos theory