6 research outputs found
Arithmetical conservation results
In this paper we present a proof of Goodman's Theorem, a classical result in
the metamathematics of constructivism, which states that the addition of the
axiom of choice to Heyting arithmetic in finite types does not increase the
collection of provable arithmetical sentences. Our proof relies on several
ideas from earlier proofs by other authors, but adds some new ones as well. In
particular, we show how a recent paper by Jaap van Oosten can be used to
simplify a key step in the proof. We have also included an interesting
corollary for classical systems pointed out to us by Ulrich Kohlenbach
More on Geometric Morphisms between Realizability Toposes
Geometric morphisms between realizability toposes are studied in terms of
morphisms between partial combinatory algebras (pcas). The morphisms inducing
geometric morphisms (the {\em computationally dense\/} ones) are seen to be the
ones whose `lifts' to a kind of completion have right adjoints. We characterize
topos inclusions corresponding to a general form of relative computability. We
characterize pcas whose realizability topos admits a geometric morphism to the
effective topos.Comment: 20 page
Computability in partial combinatory algebras
We prove a number of elementary facts about computability in partial
combinatory algebras (pca's). We disprove a suggestion made by Kreisel about
using Friedberg numberings to construct extensional pca's. We then discuss
separability and elements without total extensions. We relate this to Ershov's
notion of precompleteness, and we show that precomplete numberings are not 1-1
in general
Rethinking the notion of oracle: A link between synthetic descriptive set theory and effective topos theory
We present three different perspectives of oracle. First, an oracle is a
blackbox; second, an oracle is an endofunctor on the category of represented
spaces; and third, an oracle is an operation on the object of truth-values.
These three perspectives create a link between the three fields, computability
theory, synthetic descriptive set theory, and effective topos theory
A general form of relative recursion
The purpose of this note is to observe a generalization of the concept "computable in..." to arbitrary partial combinatory algebras. For every partial combinatory algebra (pca) A and every partial endofunction on A, a pca A[f] is constructed such that in A[f], the function f is representable by an element; a universal property of the construction is formulated in terms of Longley's 2-category of pcas and decidable applicative morphisms. It is proved that there is always a geometric inclusion from the realizability topos on A[f] into the one on A and that there is a meaningful preorder on the partial endofunctions on A which generalizes Turing reducibility
A general form of relative recursion
The purpose of this note is to observe a generalization of the concept “computable in... ” to arbitrary partial combinatory algebras. For every partial combinatory algebra (pca) A and every partial endofunction on A, a pca A[f] is constructed such that in A[f], the function f is representable by an element; a universal property of the construction is formulated in terms of Longley’s 2-category of pcas and decidable applicative morphisms. It is proved that there is always a geometric inclusion from the realizability topos on A[f] into the one on A, and that there is a meaningful preorder on the partial endofunctions on A which generalizes Turing reducibility. AMS Subject Classification (2000): 03B40,68N1