63 research outputs found
Tameness from two successive good frames
We show, assuming a mild set-theoretic hypothesis, that if an abstract
elementary class (AEC) has a superstable-like forking notion for models of
cardinality and a superstable-like forking notion for models of
cardinality , then orbital types over models of cardinality
are determined by their restrictions to submodels of cardinality
. By a superstable-like forking notion, we mean here a good frame, a
central concept of Shelah's book on AECs.
It is known that locality of orbital types together with the existence of a
superstable-like notion for models of cardinality implies the
existence of a superstable-like notion for models of cardinality ,
but here we prove the converse. An immediate consequence is that forking in
can be described in terms of forking in .Comment: 27 page
Forking and superstability in tame AECs
We prove that any tame abstract elementary class categorical in a suitable
cardinal has an eventually global good frame: a forking-like notion defined on
all types of single elements. This gives the first known general construction
of a good frame in ZFC. We show that we already obtain a well-behaved
independence relation assuming only a superstability-like hypothesis instead of
categoricity. These methods are applied to obtain an upward stability transfer
theorem from categoricity and tameness, as well as new conditions for
uniqueness of limit models.Comment: 33 page
Downward categoricity from a successor inside a good frame
We use orthogonality calculus to prove a downward transfer from categoricity
in a successor in abstract elementary classes (AECs) that have a good frame (a
forking-like notion for types of singletons) on an interval of cardinals:
Let be an AEC and let be cardinals.
If has a type-full good -frame and is categorical in
both and , then is categorical in all .
We deduce improvements on the threshold of several categoricity transfers
that do not mention frames. For example, the threshold in Shelah's transfer can
be improved from to
assuming that the AEC is -tame. The successor hypothesis can also be removed from Shelah's result by
assuming in addition either that the AEC has primes over sets of the form or (using an unpublished claim of Shelah) that the weak generalized
continuum hypothesis holds.Comment: 63 pages. Was previously named "A downward categoricity transfer for
tame abstract elementary classes
Shelah's eventual categoricity conjecture in universal classes: part I
We prove:
Let be a universal class. If is categorical in cardinals of
arbitrarily high cofinality, then is categorical on a tail of cardinals.
The proof stems from ideas of Adi Jarden and Will Boney, and also relies on a
deep result of Shelah. As opposed to previous works, the argument is in ZFC and
does not use the assumption of categoricity in a successor cardinal. The
argument generalizes to abstract elementary classes (AECs) that satisfy a
locality property and where certain prime models exist. Moreover assuming
amalgamation we can give an explicit bound on the Hanf number and get rid of
the cofinality restrictions:
Let be an AEC with amalgamation. Assume that is fully
-tame and short and has primes over sets of the form . Write . If is categorical in a , then
is categorical in all .Comment: 51 page
On categoricity in successive cardinals
We investigate, in ZFC, the behavior of abstract elementary classes (AECs)
categorical in many successive small cardinals. We prove for example that a
universal sentence categorical on an end
segment of cardinals below must be categorical also everywhere
above . This is done without any additional model-theoretic
hypotheses (such as amalgamation or arbitrarily large models) and generalizes
to the much broader framework of tame AECs with weak amalgamation and coherent
sequences.Comment: 19 page
Shelah's eventual categoricity conjecture in tame AECs with primes
A new case of Shelah's eventual categoricity conjecture is established:
Let be an AEC with amalgamation. Write . Assume
that is -tame and has primes over sets of the form . If is categorical in some , then is
categorical in all .
The result had previously been established when the stronger locality
assumptions of full tameness and shortness are also required.
An application of the method of proof of the theorem is that Shelah's
categoricity conjecture holds in the context of homogeneous model theory (this
was known, but our proof gives new cases):
Let be a homogeneous diagram in a first-order theory . If is
categorical in a , then is categorical in all .Comment: 16 pages. Generalizes arXiv:1506.0702
Saturation and solvability in abstract elementary classes with amalgamation
Let be an abstract elementary class (AEC) with amalgamation and no
maximal models. Let . If is categorical in
, then the model of cardinality is Galois-saturated.
This answers a question asked independently by Baldwin and Shelah. We deduce
several corollaries: has a unique limit model in each cardinal below
, (when is big-enough) is weakly tame below ,
and the thresholds of several existing categoricity transfers can be improved.
We also prove a downward transfer of solvability (a version of superstability
introduced by Shelah):
Let be an AEC with amalgamation and no maximal models. Let . If is solvable in , then is solvable in
.Comment: 19 page
Indiscernible extraction and Morley sequences
We present a new proof of the existence of Morley sequences in simple
theories. We avoid using the Erd\H{o}s-Rado theorem and instead use only
Ramsey's theorem and compactness. The proof shows that the basic theory of
forking in simple theories can be developed using only principles from
"ordinary mathematics", answering a question of Grossberg, Iovino and Lessmann,
as well as a question of Baldwin.Comment: Shortened to 6 page
Infinitary stability theory
We introduce a new device in the study of abstract elementary classes (AECs):
Galois Morleyization, which consists in expanding the models of the class with
a relation for every Galois type of length less than a fixed cardinal .
We show:
(The semantic-syntactic correspondence)
An AEC is fully -tame and type short if and only if Galois
types are syntactic in the Galois Morleyization.
This exhibits a correspondence between AECs and the syntactic framework of
stability theory inside a model. We use the correspondence to make progress on
the stability theory of tame and type short AECs. The main theorems are:
Let be a -tame AEC with amalgamation. The following are
equivalent:
* is Galois stable in some .
* does not have the order property (defined in terms of Galois types).
* There exist cardinals and with such that is Galois stable in any with .
Let be a fully -tame and type short AEC with amalgamation,
. If is Galois stable, then the
class of -Galois saturated models of admits an independence notion
(-coheir) which, except perhaps for extension, has the properties of
forking in a first-order stable theory.Comment: 34 pages (v1 was split into this paper and arXiv:1503.01366
The lazy model theoretician's guide to Shelah's eventual categoricity conjecture in universal classes
We give a short overview of the proof of Shelah's eventual categoricity
conjecture in universal classes with amalgamation in arXiv:1506.07024 .Comment: 6 page
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