17,107 research outputs found
The Schroder-Bernstein property for a-saturated models
A first-order theory T has the Schr\"oder-Bernstein (SB) property if any pair
of elementarily bi-embeddable models are isomorphic. We prove that T has an
expansion by constants that has the SB property if and only if T is superstable
and non-multidimensional. We also prove that among superstable theories T, the
class of a-saturated models of T has the SB property if and only if T has no
nomadic types.Comment: 13 page
Pseudosaturation and the Interpretability Orders
We streamline treatments of the interpretability orders
of Shelah, the key new notion being that of
pseudosaturation. Extending work of Malliaris and Shelah, we classify the
interpretability orders on the stable theories. As a further application, we
prove that for all countable theories , if is unsupersimple,
then if and only if . We thus deduce that simplicity is a dividing
line in , and that consistently,
characterizes maximality in ; previously these
results were only known for .Comment: 42 page
Henkin constructions of models with size continuum
We survey the technique of constructing customized models of size continuum
in omega steps and illustrate the method by giving new proofs of mostly old
results within this rubric. One new theorem, which is joint with Saharon
Shelah, is that a pseudominimal theory has an atomic model of size continuum
Independence relations in randomizations
The randomization of a complete first order theory is the complete
continuous theory with two sorts, a sort for random elements of models of
, and a sort for events in an underlying probability space. We study various
notions of independence in models of .Comment: 45 page
The uncountable spectra of countable theories
Let T be a complete, first-order theory in a finite or countable language
having infinite models. Let I(T,kappa) be the number of isomorphism types of
models of T of cardinality \kappa. We denote by \mu (respectively \hat\mu) the
number of cardinals (respectively infinite cardinals) less than or equal to
\kappa. We prove that I(T,\kappa), as a function of \kappa > \aleph_0, is the
minimum of 2^{\kappa} and one of the following functions:
1. 2^{\kappa};
2. the constant function 1;
3. |\hat\mu^n/{\sim_G}|-|(\hat\mu - 1)^n/{\sim_G}| if \hat\mu<\omega for some
1= \omega some group G <= Sym(n);
4. the constant function \beth_2;
5. \beth_{d+1}(\mu) for some infinite, countable ordinal d;
6. \sum_{i=1}^d \Gamma(i) where d is an integer greater than 0 (the depth of
T) and \Gamma(i) is either \beth_{d-i-1}(\mu^{\hat\mu}) or
\beth_{d-i}(\mu^{\sigma(i)} + \alpha(i)), where \sigma(i) is either 1, \aleph_0
or \beth_1, and \alpha(i) is 0 or \beth_2; the first possibility for \Gamma(i)
can occur only when d-i > 0.Comment: 51 pages, published version, abstract added in migratio
Semi-isolation and the strict order property
We study semi-isolation as a binary relation on the locus of a complete type
and prove that under some additional assumptions it induces the strict order
property.Comment: Submitted to Notre Dame Journal of Formal Logi
A multiverse perspective on the axiom of constructiblity
I shall argue that the commonly held V not equal L via maximize position,
which rejects the axiom of constructibility V = L on the basis that it is
restrictive, implicitly takes a stand in the pluralist debate in the philosophy
of set theory by presuming an absolute background concept of ordinal. The
argument appears to lose its force, in contrast, on an upwardly extensible
concept of set, in light of the various facts showing that models of set theory
generally have extensions to models of V = L inside larger set-theoretic
universes.Comment: 21 pages. This article expands on an argument that I made during my
talk at the Asian Initiative for Infinity: Workshop on Infinity and Truth,
held July 25--29, 2011 at the Institute for Mathematical Sciences, National
University of Singapore. Commentary concerning this paper can be made at
http://jdh.hamkins.org/multiverse-perspective-on-constructibilit
On uncountable hypersimple unidimensional theories
We extend a dichotomy between 1-basedness and supersimplicity proved in a
previous paper. The generalization we get is to arbitrary language, with no
restrictions on the topology (we do not demand type-definabilty of the open set
in the definition of essential 1-basedness). We conclude that every (possibly
uncountable) hypersimple unidimensional theory that is not s-essentially
1-based by means of the forking topology is supersimple. We also obtain a
strong version of the above dichotomy in the case where the language is
countable
Properly ergodic structures
We consider ergodic -invariant probability measures
on the space of -structures with domain (for a countable
relational language), and call such a measure a properly ergodic structure when
no isomorphism class of structures is assigned measure . We characterize
those theories in countable fragments of for
which there is a properly ergodic structure concentrated on the models of the
theory. We show that for a countable fragment of the almost-sure -theory of a properly ergodic structure has
continuum-many models (an analogue of Vaught's Conjecture in this context), but
its full almost-sure -theory has no models. We
also show that, for an -theory , if there is some properly ergodic
structure that concentrates on the class of models of , then there are
continuum-many such properly ergodic structures.Comment: 41 page
Generalized amalgamation and homogeneity
In this paper we shall prove that any -transitive finitely homogeneous
structure with a supersimple theory satisfying a generalized amalgamation
property is a random structure. In particular, this adapts a result of Koponen
for binary homogeneous structures to arbitrary ones without binary relations.
Furthermore, we point out a relation between generalized amalgamation,
triviality and quantifier elimination in simple theories
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