37 research outputs found
Exercices de style: A homotopy theory for set theory II
This is the second part of a work initiated in \cite{GaHa}, where we
constructed a model category, \Qt, for set theory. In the present paper we
use this model category to introduce homotopy-theoretic intuitions to set
theory. Our main observation is that the homotopy invariant version of
cardinality is the covering number of Shelah's PCF theory, and that other
combinatorial objects, such as Shelah's revised power function - the cardinal
function featuring in Shelah's revised GCH theorem - can be obtained using
similar tools. We include a small "dictionary" for set theory in \QtNaamen,
hoping it will help in finding more meaningful homotopy-theoretic intuitions in
set theory
Strongly Dependent Ordered Abelian Groups and Henselian Fields
Strongly dependent ordered abelian groups have finite dp-rank. They are
precisely those groups with finite spines and . We apply this to show that if is a
strongly dependent field, then is strongly dependent for any henselian
valuation
Reducts of Hrushovski's constructions of a higher geometrical arity
Let denote the structure obtained from Hrushovski's (non
collapsed) construction with an n-ary relation and its
associated pre-geometry. It was shown by Evans and Ferreira that
. We show that has a
reduct, such that . To achieve this we show that is a
slightly generalised Fra\"iss\'e-Hrushovski limit incorporating into the
construction non-eliminable imaginary sorts in
Unstable structures definable in o-minimal theories
Let M be an o-minimal structure with elimination of imaginaries, N an
unstable structure definable in M. Then there exists X, interpretable in N,
such that X with all the structure induced from N is o-minimal. In particular X
is linearly ordered.
As part of the proof we show: Theorem 1: If the M-dimenson of N is 1 then any
1-N-type is either strongly stable or finite by o-minimal. Theorem 2: If N is
N-minimal then it is 1-M-dimensional
Eliminating Field Quantifiers in Strongly Dependent Henselian Fields
We prove elimination of field quantifiers for strongly dependent henselian
fields in the Denef-Pas language. This is achieved by proving the result for a
class of fields generalizing algebraically maximal Kaplansky fields. We deduce
that if is strongly dependent then so is its henselization.Comment: The original version was for algebraically maximal kaplansky fields.
This new version is a generalizatio
Strongly minimal groups in o-minimal structures
We prove Zilber's Trichotomy Conjecture for strongly minimal expansions of
two-dimensional groups, definable in o-minimal structures:
Theorem. Let M be an o-minimal expansion of a real closed field, (G;+) a
2-dimensional group definable in M, and D = (G;+,...) a strongly minimal
structure, all of whose atomic relations are definable in M. If D is not
locally modular, then an algebraically closed field K is interpretable in D,
and the group G, with all its induced D-structure, is definably isomorphic in D
to an algebraic K-group with all its induced K-structure
Definable Valuations induced by multiplicative subgroups and NIP Fields
We study the algebraic implications of the non-independence property (NIP)
and variants thereof (dp-minimality) on infinite fields, motivated by the
conjecture that all such fields which are neither real closed nor separably
closed admit a definable henselian valuation. Our results mainly focus on Hahn
fields and build up on Will Johnson's preprint "dp-minimal fields", arXiv:
1507.02745v1, July 2015.Comment: There was a mistake in Theorem 3.2 (4) of the first arxiv version
posted April 10, we corrected it and modified the introduction and section 3
accordingly in the second version, a few other issues are fixed in the third
version. The fourth version is a revision according to the referee's
comments. To appear in AFM
On definable Skolem functions in weakly o-minimal non-valuational structures
We prove that all known examples of weakly o-minimal non-valuational
structures have no definable Skolem functions. We show, however, that such
structures eliminate imaginaries up to (definable families of) definable cuts.
Along the way we give some new examples of weakly o-minimal non-valuational
structures
A theory of pairs for non-valuational structures
Given a weakly o-minimal structure and its o-minimal completion
, we first associate to a canonical
language and then prove that determines . We then investigate the theory of the pair in the spirit of the theory of dense pairs of o-minimal structures, and
prove, among other results, that it is near model complete, and every -definable open subset of is already definable in . We give an example of a weakly o-minimal structure which
interprets and show that it is not elementarily equivalent
to any reduct of an o-minimal trace
Minimal types in super-dependent theories
We give necessary and sufficient geometric conditions for a theory definable
in an o-minimal structure to interpret a real closed field. The proof goes
through an analysis of thorn-minimal types in super-rosy dependent theories of
finite rank. We prove that such theories are coordinatised by thorn-minimal
types and that such a type is unstable if an only if every non-algebraic
extension thereof is. We conclude that a type is stable if and only if it
admits a coordinatisation in thorn-minimal stable types. We also show that
non-trivial thorn-minimal stable types extend stable sets