317 research outputs found
The Complexity of Orbits of Computably Enumerable Sets
The goal of this paper is to announce there is a single orbit of the c.e.
sets with inclusion, \E, such that the question of membership in this orbit
is -complete. This result and proof have a number of nice
corollaries: the Scott rank of \E is \wock +1; not all orbits are
elementarily definable; there is no arithmetic description of all orbits of
\E; for all finite , there is a properly
orbit (from the proof).
A few small corrections made in this versionComment: To appear in the Bulletion of Symbolic Logi
First Order Theories of Some Lattices of Open Sets
We show that the first order theory of the lattice of open sets in some
natural topological spaces is -equivalent to second order arithmetic. We
also show that for many natural computable metric spaces and computable domains
the first order theory of the lattice of effectively open sets is undecidable.
Moreover, for several important spaces (e.g., , , and the
domain ) this theory is -equivalent to first order arithmetic
The Small-Is-Very-Small Principle
The central result of this paper is the small-is-very-small principle for
restricted sequential theories. The principle says roughly that whenever the
given theory shows that a property has a small witness, i.e. a witness in every
definable cut, then it shows that the property has a very small witness: i.e. a
witness below a given standard number.
We draw various consequences from the central result. For example (in rough
formulations): (i) Every restricted, recursively enumerable sequential theory
has a finitely axiomatized extension that is conservative w.r.t. formulas of
complexity . (ii) Every sequential model has, for any , an extension
that is elementary for formulas of complexity , in which the
intersection of all definable cuts is the natural numbers. (iii) We have
reflection for -sentences with sufficiently small witness in any
consistent restricted theory . (iv) Suppose is recursively enumerable
and sequential. Suppose further that every recursively enumerable and
sequential that locally inteprets , globally interprets . Then,
is mutually globally interpretable with a finitely axiomatized sequential
theory.
The paper contains some careful groundwork developing partial satisfaction
predicates in sequential theories for the complexity measure depth of
quantifier alternations
Computability Theory
Computability is one of the fundamental notions of mathematics, trying to capture the effective content of mathematics. Starting from Gödel’s Incompleteness Theorem, it has now blossomed into a rich area with strong connections with other areas of mathematical logic as well as algebra and theoretical computer science
Changing a semantics: opportunism or courage?
The generalized models for higher-order logics introduced by Leon Henkin, and
their multiple offspring over the years, have become a standard tool in many
areas of logic. Even so, discussion has persisted about their technical status,
and perhaps even their conceptual legitimacy. This paper gives a systematic
view of generalized model techniques, discusses what they mean in mathematical
and philosophical terms, and presents a few technical themes and results about
their role in algebraic representation, calibrating provability, lowering
complexity, understanding fixed-point logics, and achieving set-theoretic
absoluteness. We also show how thinking about Henkin's approach to semantics of
logical systems in this generality can yield new results, dispelling the
impression of adhocness. This paper is dedicated to Leon Henkin, a deep
logician who has changed the way we all work, while also being an always open,
modest, and encouraging colleague and friend.Comment: 27 pages. To appear in: The life and work of Leon Henkin: Essays on
his contributions (Studies in Universal Logic) eds: Manzano, M., Sain, I. and
Alonso, E., 201
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