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Constructive definition of certain analytic sets of numbers

Published online by Cambridge University Press:  12 March 2014

P. Lorenzen  and
J. Myhill
P. Lorenzen
Affiliation:
The Institute for Advanced Study
J. Myhill
Affiliation:
The Institute for Advanced Study
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Extract

According to Kleene [8] and Post [13] the definition of recursive relations (or sets) can be given most simply via the definition of recursively enumerable (r.e.) relations. The latter can be defined by finite systems of finite rules for deriving strings of symbols, each rule being of the form

each formula or being obtained by juxtaposition from atomic symbols and variables ranging over strings of atomic symbols.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 24 , Issue 1 , March 1959 , pp. 37 - 49
Copyright
Copyright © Association for Symbolic Logic 1959

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References

[1]Addison, J. W., Separation principles in the hierarchies of classical and effective descriptive set theory, Bulletin of the American Mathematical Society, vol. 63 (1957), pp. 397–8.Google Scholar
[2]Addison, J. W., Separation principles in the hierarchies of classical and effective descriptive set theory, to appear in Fundamenta mathematicae.Google Scholar
[3]Addison, J. W. and Kleene, S. C., A note on function quantification, Proceedings of the American Mathematical Society, vol. 8 (1957), pp. 1002–1006.CrossRefGoogle Scholar
[4]Brouwer, L. E. J., Beweis, dass jede volle Funktion gleichmässig stetig ist, Koninklijke Nederlandse Akademie van Wetenschappen, vol. 27 (1924), pp. 189–193.Google Scholar
[5]Fitch, F. B., An extension of basic logic, this Journal, vol. 13 (1948), pp. 95–106.Google Scholar
[6]Fitch, F. B., A simplification of basic logic, this Journal, vol. 18 (1953), pp. 317–325.Google Scholar
[7]Kleene, S. C., On notations for ordinal numbers, this Journal, vol. 3 (1938), pp. 150–155.Google Scholar
[8]Kleene, S. C., Recursive predicates and quantifiers, Transactions of the American Mathematical Society, vol. 53 (1943), pp. 41–73.CrossRefGoogle Scholar
[9]Kleene, S. C., Hierarchies of number-theoretic predicates, Bulletin of the American Mathematical Society, vol. 61 (1955), pp. 193–213.CrossRefGoogle Scholar
[10]Kleene, S. C., Arithmetical predicates and function quantifiers, Transactions of the American Mathematical Society, vol. 79 (1955), pp. 312–340.CrossRefGoogle Scholar
[11]Kleene, S. C., On the forms of the predicates in the theory of constructive ordinals (second paper), American journal of mathematics, vol. 77 (1955), pp. 405–428.CrossRefGoogle Scholar
[12]Lorenzen, P., Einführung in die operative Logik und Mathematik, Berlin 1955.CrossRefGoogle Scholar
[13]Post, E. L., Recursively enumerable sets of positive integers and their decision problems, Bulletin of the American Mathematical Society, vol. 50 (1944), pp. 284–316.CrossRefGoogle Scholar
[14]Sierpinski, W., Les ensembles projectifs et analytiques, Mémorial des sciences mathématiques, vol. 112 (1950), pp. 1–80.Google Scholar
[15]Spector, C., Recursive well-orderings, this Journal, vol. 20 (1955), pp. 151–163.Google Scholar
[16]Spector, C., Recursive ordinals and predicative set-theory. Summaries of talks at the Summer Institute of Symbolic Logic in 1957 at Cornell University, vol. 3, pp. 377–382.Google Scholar