Journal ArticleDOI
On Recursively Enumerable and Arithmetic Models of Set Theory
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A very simple proof is obtained, not involving any formal constructions within the system of the notions of truth and satisfiability, of an extension of the Kreisel-Mostowski theorems, that set theory with the single non-logical constant ϵ does not possess any recursively enumerable model.Abstract:
In this note we shall prove a certain relative recursiveness lemma concerning countable models of set theory (Lemma 5). From this lemma will follow two results about special types of such models. Kreisel [5] and Mostowski [6] have shown that certain (finitely axiomatized) systems of set theory, formulated by means of the ϵ relation and certain additional non-logical constants, do not possess recursive models. Their purpose in doing this was to construct consistent sentences without recursive models. As a first corollary of Lemma 5, we obtain a very simple proof, not involving any formal constructions within the system of the notions of truth and satisfiability, of an extension of the Kreisel-Mostowski theorems. Namely, set theory with the single non-logical constant ϵ does not possess any recursively enumerable model. Thus we get, as a side product, an easy example of a consistent sentence containing a single binary relation which does not possess any recursively enumerable model; this sentence being the conjunction of the (finitely many) axioms of set theory.read more
Citations
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Journal ArticleDOI
Hiearchies of Boolean Algebras
TL;DR: A denumerable structure is said to be recursive iff its universe is a recursive subset of the natural numbers and its relations and operations are recursive.
Proceedings ArticleDOI
Clock construction in fully asynchronous parallel systems and PRAM simulation
Yonatan Aumann,Michael O. Rabin +1 more
TL;DR: The authors show how to harness this clock to drive a PRAM simulation on an asynchronous system, which is more efficient than existing ones, while actually relaxing the assumptions on the underlying asynchronous system.
Book ChapterDOI
Decision Problems in Predicate Logic
TL;DR: The main concern is to reveal the deep structural and combinatorial simularities between computations and logical deductions, which bring out explicitely the fundamental and uniform reason for many undecidability and complexity results forCombinatorial and for logical decision problems.
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Thirty Years of Foundational Studies Lectures on the Development of Mathematical Logic and the Study of the Foundations of Mathematics in 1930–1964
TL;DR: In this article, the authors present sixteen lectures on the development of mathematical logic and the study of the foundations of mathematics in the years 1930-1964, delivered by the author in the Summer School in Vaasa, Finland in the summer of 1964.
References
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Journal ArticleDOI
The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis
TL;DR: Godel as discussed by the authors proved the incompleteness theorem of the Continuum Hypothesis, which states that there is no set of numbers between the integers and real numbers, which was later included as the first of mathematician David Hilbert's twenty-three unsolved math problems.
Book
The Consistency of the axiom of choice and of the generalized continuum-hypothesis with the axioms of set theory
TL;DR: Kurt Godel, mathematician and logician, was one of the most influential thinkers of the twentieth century and ranked higher than fellow scientists Edwin Hubble, Enrico Fermi, John Maynard Keynes, James Watson, Francis Crick, and Jonas Salk.
Book ChapterDOI
On a System of Axioms Which Has no Recursively Enumerable Arithmetic Model
TL;DR: In this article, it was shown that every consistent axiomatic system S based on the functional calculus of the first order has an interpretation in the set of positive integers, where A is the conjunction of the axioms of S 2 and R 1, R 2,,, R p are the predicates that occur in A.