216 research outputs found
Toward categoricity for classes with no maximal models
We provide here the first steps toward Classification Theory of Abstract
Elementary Classes with no maximal models, plus some mild set theoretical
assumptions, when the class is categorical in some lambda greater than its
Lowenheim-Skolem number. We study the degree to which amalgamation may be
recovered, the behaviour of non mu-splitting types. Most importantly, the
existence of saturated models in a strong enough sense is proved, as a first
step toward a complete solution to the Los Conjecture for these classes
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
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
On the structure of categorical abstract elementary classes with amalgamation
For an abstract elementary class with amalgamation and no maximal models,
we show that categoricity in a high-enough cardinal implies structural
properties such as the uniqueness of limit models and the existence of good
frames. This improves several classical results of Shelah.
Let . If is categorical in a , then:
1) Whenever are such that and are limit
over , we have .
2) If , the model of size is -saturated.
3) If and , then there exists a type-full good
-frame with underlying class the saturated models in .
Our main tool is the symmetry property of splitting (previously isolated by
the first author). The key lemma deduces symmetry from failure of the order
property.Comment: 19 pages. This has since been merged with arXiv:1508.0325
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
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
Equivalent definitions of superstability in tame abstract elementary classes
In the context of abstract elementary classes (AECs) with a monster model,
several possible definitions of superstability have appeared in the literature.
Among them are no long splitting chains, uniqueness of limit models, and
solvability. Under the assumption that the class is tame and stable, we show
that (asymptotically) no long splitting chains implies solvability and
uniqueness of limit models implies no long splitting chains. Using known
implications, we can then conclude that all the previously-mentioned
definitions (and more) are equivalent:
Let be a tame AEC with a monster model. Assume that is stable in a
proper class of cardinals. The following are equivalent:
1) For all high-enough , has no long splitting chains.
2) For all high-enough , there exists a good -frame on a
skeleton of .
3) For all high-enough , has a unique limit model of cardinality
.
4) For all high-enough , has a superlimit model of cardinality
.
5) For all high-enough , the union of any increasing chain of
-saturated models is -saturated.
6) There exists such that for all high-enough , is
-solvable.
This gives evidence that there is a clear notion of superstability in the
framework of tame AECs with a monster model.Comment: 24 page
Categoricity may fail late
We build an example that generalizes math.LO/9201240 to uncountable cases. In
particular, our example yields a sentence psi in L_{(2^lambda)^+, omega} that
is categorical in lambda, lambda^+, ..., lambda^{+k} but not in
beth_{k+1}(lambda)^+. This is connected with the Los Conjecture and with
Shelah's own conjecture and construction of excellent classes for the psi in
L_{omega_1,omega} case
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
The categoricity spectrum of large abstract elementary classes
The categoricity spectrum of a class of structures is the collection of
cardinals in which the class has a single model up to isomorphism. Assuming
that cardinal exponentiation is injective (a weakening of the generalized
continuum hypothesis, GCH), we give a complete list of the possible
categoricity spectrums of an abstract elementary class with amalgamation and
arbitrarily large models. Specifically, the categoricity spectrum is either
empty, an end segment starting below the Hanf number, or a closed interval
consisting of finite successors of the L\"owenheim-Skolem-Tarski number (there
are examples of each type). We also prove (assuming a strengthening of the GCH)
that the categoricity spectrum of an abstract elementary class with no maximal
models is either bounded or contains an end segment. This answers several
longstanding questions around Shelah's categoricity conjecture.Comment: 50 page
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