151 research outputs found
Subcompact cardinals, squares, and stationary reflection
We generalise Jensen's result on the incompatibility of subcompactness with
square. We show that alpha^+-subcompactness of some cardinal less than or equal
to alpha precludes square_alpha, but also that square may be forced to hold
everywhere where this obstruction is not present. The forcing also preserves
other strong large cardinals. Similar results are also given for stationary
reflection, with a corresponding strengthening of the large cardinal assumption
involved. Finally, we refine the analysis by considering Schimmerling's
hierarchy of weak squares, showing which cases are precluded by
alpha^+-subcompactness, and again we demonstrate the optimality of our results
by forcing the strongest possible squares under these restrictions to hold.Comment: 18 pages. Corrections and improvements from referee's report mad
Tameness in generalized metric structures
We broaden the framework of metric abstract elementary classes (mAECs) in
several essential ways, chiefly by allowing the metric to take values in a
well-behaved quantale. As a proof of concept we show that the result of Boney
and Zambrano on (metric) tameness under a large cardinal assumption holds in
this more general context. We briefly consider a further generalization to
partial metric spaces, and hint at connections to classes of fuzzy structures,
and structures on sheaves
Dense ideals and cardinal arithmetic
From large cardinals we show the consistency of normal, fine,
-complete -dense ideals on for
successor . We explore the interplay between dense ideals, cardinal
arithmetic, and squares, answering some open questions of Foreman
Intrinsic Justification for Large Cardinals and Structural Reflection
We deal with the complex issue of whether large cardinals are intrinsically
justified principles of set theory (we call this the Intrinsicness Issue). In
order to do this, we review, in a systematic fashion, (1.) the abstract
principles that have been formulated to motivate them, as well as (2.) their
mathematical expressions, and assess the justifiability of both on the grounds
of the (iterative) concept of set. A parallel, but closely linked, issue is
whether there exist mathematical principles able to yield all known large
cardinals (we call this the Universality Issue), and we also test principles
for their responses to this issue. Finally, we discuss the first author's
Structural Reflection Principles (SRPs), and their response to Intrinsicness
and Universality. We conclude the paper with some considerations on the global
justifiability of SRPs, and on alternative construals of the concept of set
also potentially able to intrinsically justify large cardinals
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