90,801 research outputs found
The Strength of Abstraction with Predicative Comprehension
Frege's theorem says that second-order Peano arithmetic is interpretable in
Hume's Principle and full impredicative comprehension. Hume's Principle is one
example of an abstraction principle, while another paradigmatic example is
Basic Law V from Frege's Grundgesetze. In this paper we study the strength of
abstraction principles in the presence of predicative restrictions on the
comprehension schema, and in particular we study a predicative Fregean theory
which contains all the abstraction principles whose underlying equivalence
relations can be proven to be equivalence relations in a weak background
second-order logic. We show that this predicative Fregean theory interprets
second-order Peano arithmetic.Comment: Forthcoming in Bulletin of Symbolic Logic. Slight change in title
from previous version, at request of referee
Kurt Gödel and Computability Theory
Although Kurt Gödel does not figure prominently in the history of computabilty theory, he exerted a significant influence on some of the founders of the field, both through his published work and through personal interaction. In particular, Gödel’s 1931 paper on incompleteness and the methods developed therein were important for the early development of recursive function theory and the lambda calculus at the hands of Church, Kleene, and Rosser. Church and his students studied Gödel 1931, and Gödel taught a seminar at Princeton in 1934. Seen in the historical context, Gödel was an important catalyst for the emergence of computability theory in the mid 1930s
Tarski's influence on computer science
The influence of Alfred Tarski on computer science was indirect but
significant in a number of directions and was in certain respects fundamental.
Here surveyed is the work of Tarski on the decision procedure for algebra and
geometry, the method of elimination of quantifiers, the semantics of formal
languages, modeltheoretic preservation theorems, and algebraic logic; various
connections of each with computer science are taken up
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