33,064 research outputs found
A note on the axioms for Zilber's pseudo-exponential fields
We show that Zilber's conjecture that complex exponentiation is isomorphic to
his pseudo-exponentiation follows from the a priori simpler conjecture that
they are elementarily equivalent. An analysis of the first-order types in
pseudo-exponentiation leads to a description of the elementary embeddings, and
the result that pseudo-exponential fields are precisely the models of their
common first-order theory which are atomic over exponential transcendence
bases. We also show that the class of all pseudo-exponential fields is an
example of a non-finitary abstract elementary class, answering a question of
Kes\"al\"a and Baldwin.Comment: 10 pages, v2: substantial alteration
Imaginaries in separably closed valued fields
We show that separably closed valued fields of finite imperfection degree
(either with lambda-functions or commuting Hasse derivations) eliminate
imaginaries in the geometric language. We then use this classification of
interpretable sets to study stably dominated types in those structures. We show
that separably closed valued fields of finite imperfection degree are
metastable and that the space of stably dominated types is strict
pro-definable
A Categorical Construction of Bachmann-Howard Fixed Points
Peter Aczel has given a categorical construction for fixed points of normal
functors, i.e. dilators which preserve initial segments. For a general dilator
we cannot expect to obtain a well-founded fixed point, as the
order type of may always exceed the order type of . In the present
paper we show how to construct a Bachmann-Howard fixed point of , i.e. an
order with an "almost" order preserving collapse
. Building
on previous work, we show that -comprehension is equivalent to the
assertion that is well-founded for any dilator .Comment: This version has been accepted for publication in the Bulletin of the
London Mathematical Societ
Checking Zenon Modulo Proofs in Dedukti
Dedukti has been proposed as a universal proof checker. It is a logical
framework based on the lambda Pi calculus modulo that is used as a backend to
verify proofs coming from theorem provers, especially those implementing some
form of rewriting. We present a shallow embedding into Dedukti of proofs
produced by Zenon Modulo, an extension of the tableau-based first-order theorem
prover Zenon to deduction modulo and typing. Zenon Modulo is applied to the
verification of programs in both academic and industrial projects. The purpose
of our embedding is to increase the confidence in automatically generated
proofs by separating untrusted proof search from trusted proof verification.Comment: In Proceedings PxTP 2015, arXiv:1507.0837
Categoricity in Quasiminimal Pregeometry Classes
Quasiminimal pregeometry classes were introduces by Zilber [2005a] to isolate
the model theoretical core of several interesting examples. He proves that a
quasiminimal pregeometry class satisfying an additional axiom, called
excellence, is categorical in all uncountable cardinalities. Recently Bays et
al. [2014] showed that excellence follows from the rest of axioms. In this
paper we present a direct proof of the categoricity result without using
excellence
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