64 research outputs found
On absoluteness of categoricity in abstract elementary classes
"Vegeu el resum a l'inici del document del fitxer adjunt"
Amalgamation, absoluteness, and categoricity
"Vegeu el resum a l'inici del document del fitxer adjunt"
Beginning of stability theory for Polish Spaces
We consider stability theory for Polish spaces and more generally for
definable structures. We succeed to prove existence of indiscernibles under
reasonable conditions
Indeterminateness and `The' Universe of Sets: Multiversism, Potentialism, and Pluralism
In this article, I survey some philosophical attitudes to talk concerning `the' universe of sets. I separate out four different strands of the debate, namely: (i) Universism, (ii) Multiversism, (iii) Potentialism, and (iv) Pluralism. I discuss standard arguments and counterarguments concerning the positions and some of the natural mathematical programmes that are suggested by the various views
Non-Absoluteness of Model Existence at
In [FHK13], the authors considered the question whether model-existence of
-sentences is absolute for transitive models of ZFC, in
the sense that if are transitive models of ZFC with the same
ordinals, and , then if and only if .
From [FHK13] we know that the answer is positive for and under
the negation of CH, the answer is negative for all . Under GCH, and
assuming the consistency of a supercompact cardinal, the answer remains
negative for each , except the case when which is an
open question in [FHK13].
We answer the open question by providing a negative answer under GCH even for
. Our examples are incomplete sentences. In fact, the same
sentences can be used to prove a negative answer under GCH for all
assuming the consistency of a Mahlo cardinal. Thus, the large cardinal
assumption is relaxed from a supercompact in [FHK13] to a Mahlo cardinal.
Finally, we consider the absoluteness question for the
-amalgamation property of -sentences (under
substructure). We prove that assuming GCH, -amalgamation is
non-absolute for . This answers a question from [SS]. The
cases and infinite remain open. As a corollary we get that
it is non-absolute that the amalgamation spectrum of an
-sentence is empty
The non-absoluteness of model existence in uncountable cardinals for Lw1,w
"Vegeu el resum a l'inici del document del fitxer adjunt"
A presentation theorem for continuous logic and Metric Abstract Elementary Classes
We give a presentation theorem for continuous first-order logic and Metric
Abstract Elementary classes in terms of and Abstract
Elementary Classes, respectively. This presentation is accomplished by
analyzing dense subsets that are closed under functions. We extend this
correspondence to types and saturation
The set-theoretic multiverse
The multiverse view in set theory, introduced and argued for in this article,
is the view that there are many distinct concepts of set, each instantiated in
a corresponding set-theoretic universe. The universe view, in contrast, asserts
that there is an absolute background set concept, with a corresponding absolute
set-theoretic universe in which every set-theoretic question has a definite
answer. The multiverse position, I argue, explains our experience with the
enormous diversity of set-theoretic possibilities, a phenomenon that challenges
the universe view. In particular, I argue that the continuum hypothesis is
settled on the multiverse view by our extensive knowledge about how it behaves
in the multiverse, and as a result it can no longer be settled in the manner
formerly hoped for.Comment: 35 page
Strong subgroup chains and the Baer-Specker group
Examples are given of non-elementary properties that are preserved under
C-filtrations for various classes C of Abelian groups. The Baer-Specker group
is never the union of a chain of proper subgroups with cotorsionfree quotients.
Cotorsion-free groups form an abstract elementary class (AEC). The Kaplansky
invariants of the Baer-Specker group are used to determine the AECs defined by
the perps of the Baer-Specker quotient groups that are obtained by factoring
the Baer-Specker group B of a ZFC extension by the Baer-Specker group A of the
ground model, under various hypotheses, yielding information about its
stability spectrum.Comment: 12 page
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