49,658 research outputs found
Computability and analysis: the legacy of Alan Turing
We discuss the legacy of Alan Turing and his impact on computability and
analysis.Comment: 49 page
Unifying Functional Interpretations: Past and Future
This article surveys work done in the last six years on the unification of
various functional interpretations including G\"odel's dialectica
interpretation, its Diller-Nahm variant, Kreisel modified realizability,
Stein's family of functional interpretations, functional interpretations "with
truth", and bounded functional interpretations. Our goal in the present paper
is twofold: (1) to look back and single out the main lessons learnt so far, and
(2) to look forward and list several open questions and possible directions for
further research.Comment: 18 page
Perspectives for proof unwinding by programming languages techniques
In this chapter, we propose some future directions of work, potentially
beneficial to Mathematics and its foundations, based on the recent import of
methodology from the theory of programming languages into proof theory. This
scientific essay, written for the audience of proof theorists as well as the
working mathematician, is not a survey of the field, but rather a personal view
of the author who hopes that it may inspire future and fellow researchers
On what I do not understand (and have something to say): Part I
This is a non-standard paper, containing some problems in set theory I have
in various degrees been interested in. Sometimes with a discussion on what I
have to say; sometimes, of what makes them interesting to me, sometimes the
problems are presented with a discussion of how I have tried to solve them, and
sometimes with failed tries, anecdote and opinion. So the discussion is quite
personal, in other words, egocentric and somewhat accidental. As we discuss
many problems, history and side references are erratic, usually kept at a
minimum (``see ... '' means: see the references there and possibly the paper
itself).
The base were lectures in Rutgers Fall'97 and reflect my knowledge then. The
other half, concentrating on model theory, will subsequently appear
W-types in setoids
W-types and their categorical analogue, initial algebras for polynomial
endofunctors, are an important tool in predicative systems to replace
transfinite recursion on well-orderings. Current arguments to obtain W-types in
quotient completions rely on assumptions, like Uniqueness of Identity Proofs,
or on constructions that involve recursion into a universe, that limit their
applicability to a specific setting. We present an argument, verified in Coq,
that instead uses dependent W-types in the underlying type theory to construct
W-types in the setoid model. The immediate advantage is to have a proof more
type-theoretic in flavour, which directly uses recursion on the underlying
W-type to prove initiality. Furthermore, taking place in intensional type
theory and not requiring any recursion into a universe, it may be generalised
to various categorical quotient completions, with the aim of finding a uniform
construction of extensional W-types.Comment: 17 pages, formalised in Coq; v2: added reference to formalisatio
Generalized probabilities in statistical theories
In this review article we present different formal frameworks for the
description of generalized probabilities in statistical theories. We discuss
the particular cases of probabilities appearing in classical and quantum
mechanics, possible generalizations of the approaches of A. N. Kolmogorov and
R. T. Cox to non-commutative models, and the approach to generalized
probabilities based on convex sets
Canonical extensions and ultraproducts of polarities
J{\'o}nsson and Tarski's notion of the perfect extension of a Boolean algebra
with operators has evolved into an extensive theory of canonical extensions of
lattice-based algebras. After reviewing this evolution we make two
contributions. First it is shown that the failure of a variety of algebras to
be closed under canonical extensions is witnessed by a particular one of its
free algebras. The size of the set of generators of this algebra can be made a
function of a collection of varieties and is a kind of Hanf number for
canonical closure. Secondly we study the complete lattice of stable subsets of
a polarity structure, and show that if a class of polarities is closed under
ultraproducts, then its stable set lattices generate a variety that is closed
under canonical extensions. This generalises an earlier result of the author
about generation of canonically closed varieties of Boolean algebras with
operators, which was in turn an abstraction of the result that a first-order
definable class of Kripke frames determines a modal logic that is valid in its
so-called canonical frames
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