2.
Local initial segments of the Turing degrees - Kjos-Hanssen, Bjørn
Recent results on initial segments of the Turing degrees are
presented, and some conjectures about initial segments that have
implications for the existence of nontrivial automorphisms of the
Turing degrees are indicated.
5.
Symmetry and interactivity in programming - Curien, P.-L.
We recall some of the early occurrences of the notions of
interactivity and symmetry in the operational and denotational
semantics of programming languages. We suggest some connections with
ludics.
6.
Two spaces looking for a geometer - Parisi, Giorgio
In this talk I will introduce two spaces: the first space is
the usual n-dimensional vector space with the unusual feature that
n is a non-integer; the second space is composed of the linear
matrices acting on the previous space (physicists are particularly
interested in studying the limit as n goes to zero). These two
spaces are not known to most mathematicians, but they are
widely used by physicists. It is possible that, by extending the
notion of space, they can become well defined mathematical objects.
8.
Foundations and applications: axiomatization and education - Lawvere, F. William
Foundations and Applications depend ultimately for their existence
on each other.
The main links between them are education and the axiomatic method.
Those links can
be strengthened with the help of a categorical method which was concentrated
forty years ago by Cartier, Grothendieck, Isbell, Kan, and Yoneda.
I extended that method
to extract some essential features of the category of categories in 1965,
and I apply it here in section 3 to sketch a similar foundation within the
smooth categories which provide the setting for the mathematics of change.
The possibility that other methods may be needed to clarify a
contradiction introduced by Cantor,
now embedded in mathematical practice, is discussed in...
12.
Computable and continuous partial homomorphisms on metric partial algebras - Stoltenberg-Hansen, Viggo; Tucker, John V.
We analyse the connection between the
computability and continuity of functions
in the case of homomorphisms between
topological algebraic structures. Inspired
by the Pour-El and Richards equivalence theorem
between computability and boundedness for closed
linear operators on Banach spaces, we study
the rather general situation of partial homomorphisms
between metric partial universal algebras. First, we
develop a set of basic notions and results that reveal
some of the delicate algebraic, topological and effective
properties of partial algebras. Our main computability concepts
are based on numerations and include those of effective metric
partial algebras and effective partial homomorphisms. We prove a
general equivalence theorem that includes a version of
the Pour-El and Richards
Theorem, and has other...
13.
Survey of the Steinhaus tiling problem - Jackson, Steve; Mauldin, R. Daniel
We survey some results and problems arising from a classic problem of
Steinhaus: Is there a subset S of ?2 such that each
isometric copy of ?2 (the lattice points in the plane)
meets S in exactly one point.
14.
A universal approach to self-referential paradoxes, incompleteness and fixed points - Yanofsky, Noson S.
Following F. William Lawvere, we show that many
self-referential paradoxes, incompleteness theorems and fixed
point theorems fall out of the same simple scheme. We demonstrate
these similarities by showing how this simple scheme encompasses
the semantic paradoxes, and how they arise as diagonal arguments
and fixed point theorems in logic, computability theory,
complexity theory and formal language theory.
15.
Finite conformal hypergraph covers and Gaifman cliques in finite structures - Hodkinson, Ian; Otto, Martin
We provide a canonical construction of conformal covers for finite
hypergraphs and present two immediate applications to the finite model
theory of relational structures. In the setting of relational
structures, conformal covers serve to construct guarded bisimilar
companion structures that avoid all incidental Gaifman cliquesthus
serving as a partial analogue in finite model theory for the usually
infinite guarded unravellings. In hypergraph theoretic terms, we show
that every finite hypergraph admits a bisimilar cover by a finite
conformal hypergraph. In terms of relational structures, we show that
every finite relational structure admits a guarded bisimilar cover by
a finite structure whose Gaifman cliques are guarded. One of our
applications answers an...
18.
$rec.titulo - Doen, Kosta
Some thirty years ago, two proposals were made concerning criteria
for identity of proofs. Prawitz proposed to analyze identity of proofs in
terms of the equivalence relation based on reduction to normal form in
natural deduction. Lambek worked on a normalization proposal analogous to
Prawitzs, based on reduction to cut-free form in sequent systems, but he
also suggested understanding identity of proofs in terms of an equivalence
relation based on generality, two derivations having the same generality if
after generalizing maximally the rules involved in them they yield the same
premises and conclusions up to a renaming of variables. These two proposals
proved to be extensionally equivalent only for...
1.
