12,638 research outputs found
First-Order Logical Duality
From a logical point of view, Stone duality for Boolean algebras relates
theories in classical propositional logic and their collections of models. The
theories can be seen as presentations of Boolean algebras, and the collections
of models can be topologized in such a way that the theory can be recovered
from its space of models. The situation can be cast as a formal duality
relating two categories of syntax and semantics, mediated by homming into a
common dualizing object, in this case 2. In the present work, we generalize the
entire arrangement from propositional to first-order logic. Boolean algebras
are replaced by Boolean categories presented by theories in first-order logic,
and spaces of models are replaced by topological groupoids of models and their
isomorphisms. A duality between the resulting categories of syntax and
semantics, expressed first in the form of a contravariant adjunction, is
established by homming into a common dualizing object, now \Sets, regarded
once as a boolean category, and once as a groupoid equipped with an intrinsic
topology. The overall framework of our investigation is provided by topos
theory. Direct proofs of the main results are given, but the specialist will
recognize toposophical ideas in the background. Indeed, the duality between
syntax and semantics is really a manifestation of that between algebra and
geometry in the two directions of the geometric morphisms that lurk behind our
formal theory. Along the way, we construct the classifying topos of a decidable
coherent theory out of its groupoid of models via a simplified covering theorem
for coherent toposes.Comment: Final pre-print version. 62 page
Godel's Incompleteness Phenomenon - Computationally
We argue that Godel's completeness theorem is equivalent to completability of
consistent theories, and Godel's incompleteness theorem is equivalent to the
fact that this completion is not constructive, in the sense that there are some
consistent and recursively enumerable theories which cannot be extended to any
complete and consistent and recursively enumerable theory. Though any
consistent and decidable theory can be extended to a complete and consistent
and decidable theory. Thus deduction and consistency are not decidable in
logic, and an analogue of Rice's Theorem holds for recursively enumerable
theories: all the non-trivial properties of such theories are undecidable
Infinite time Turing machines and an application to the hierarchy of equivalence relations on the reals
We describe the basic theory of infinite time Turing machines and some recent
developments, including the infinite time degree theory, infinite time
complexity theory, and infinite time computable model theory. We focus
particularly on the application of infinite time Turing machines to the
analysis of the hierarchy of equivalence relations on the reals, in analogy
with the theory arising from Borel reducibility. We define a notion of infinite
time reducibility, which lifts much of the Borel theory into the class
in a satisfying way.Comment: Submitted to the Effective Mathematics of the Uncountable Conference,
200
Effective lambda-models vs recursively enumerable lambda-theories
A longstanding open problem is whether there exists a non syntactical model
of the untyped lambda-calculus whose theory is exactly the least lambda-theory
(l-beta). In this paper we investigate the more general question of whether the
equational/order theory of a model of the (untyped) lambda-calculus can be
recursively enumerable (r.e. for brevity). We introduce a notion of effective
model of lambda-calculus calculus, which covers in particular all the models
individually introduced in the literature. We prove that the order theory of an
effective model is never r.e.; from this it follows that its equational theory
cannot be l-beta or l-beta-eta. We then show that no effective model living in
the stable or strongly stable semantics has an r.e. equational theory.
Concerning Scott's semantics, we investigate the class of graph models and
prove that no order theory of a graph model can be r.e., and that there exists
an effective graph model whose equational/order theory is minimum among all
theories of graph models. Finally, we show that the class of graph models
enjoys a kind of downwards Lowenheim-Skolem theorem.Comment: 34
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