THE DECISION PROBLEM FOR RESTRICTED UNIVERSAL QUANTIFICATION IN SET THEORY AND THE AXIOM OF FOUNDATION

1992 ◽  
Vol 38 (1) ◽  
pp. 143-156 ◽  
Author(s):  
Franco Parlamento ◽  
Alberto Policriti
Keyword(s):  
Set Theory ◽  
Decision Problem ◽  
Universal Quantification

scholarly journals Formative Processes with Applications to the Decision Problem in Set Theory

2002 ◽  
Vol 172 (2) ◽  
pp. 165-201 ◽  
Author(s):  
Domenico Cantone ◽  
Pietro Ursino ◽  
Eugenio G. Omodeo
Keyword(s):  
Set Theory ◽  
Decision Problem

Effectively retractable theories and degrees of undecidability

1970 ◽  
Vol 34 (4) ◽  
pp. 597-604
Author(s):  
J. P. Jones
Keyword(s):  
Set Theory ◽  
Decision Problem

In this paper a new property of theories, called effective retractability is introduced and used to obtain a characterization for the degrees of subtheories of arithmetic and set theory. By theory we understand theory in standard formalization as defined by Tarski [10]. The word degree refers to the Kleene-Post notion of degree of recursive unsolvability [2]. By the degree of a theory we mean, of course, the degree associated with its decision problem via Gödel numbering.

A basic logic

1942 ◽  
Vol 7 (3) ◽  
pp. 105-114 ◽  
Author(s):  
Frederic B. Fitch
Keyword(s):  
Finite Number ◽  
Strong Evidence ◽  
Decision Problem ◽  
Simple System ◽  
Basic Logic ◽  
Recursive Methods ◽  
Universal Quantification

This paper is concerned with finding a fairly simple system of logic which is “basic” in the sense that every system of logic is definable in it. If a “system”is regarded as being a class of propositions, rather than a class of sentences, then every class of propositions which is a system should be definable in the basic logic. The system of logic proposed in this paper will not be proved to be a basic logic, but strong evidence that it is basic will be given at the end of the paper. Evidence will also be given that the class of propositions which are not provable in the system is not definable in any system of logic. It will be established that the decision problem is unsolvable for the system. Notable characteristics of the system are its lack of negation and universal quantification, and its similarity to systems proposed by Church, Curry, and Rosser.By a “system” will be meant a class of propositions the membership of which can be specified by ordinary recursive methods, so that if the membership of a given proposition in the class can be established at all, such membership can be established in a finite number of steps. (This is not the same as demanding that a criterion must exist for determining, in a finite number of steps, whether or not some given proposition is a member of the class. Such a demand would require a solution of the decision problem for every system.)

A Course on Set Theory

2009 ◽  
Author(s):  
Ernest Schimmerling
Keyword(s):  
Set Theory
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