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Applications of Strict Π11 predicates to infinitary logic1

Published online by Cambridge University Press:  12 March 2014

Jon Barwise
Jon Barwise*
Affiliation:
Yale University
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Extract

Consider the predicate of natural numbers defined by:

where R is recursive. If, as usual, the variable ƒ ranges over ωω (the set of functions from natural numbers to natural numbers) then this is just the usual normal form for Π11 sets. If, however, ƒ ranges over 2ω (the set of functions from ω into {0, 1}) then every such predicate is recursively enumerable.3 Thus the second type of formula is generally ignored. However, the reduction just mentioned requires proof, and the proof uses some form of the Brower-König Infinity Lemma.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 34 , Issue 3 , 17 November 1969 , pp. 409 - 423
Copyright
Copyright © Association for Symbolic Logic 1969

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References

[1]Barwise, J., Infinitary logic and admissible sets, this Journal, vol. 34 (1969), pp. 226–252.Google Scholar
[2]Barwise, J., Implicit definability and compactness in infinitary languages, The syntax and semantics of infinitary languages, Lecture notes in mathematics, no. 72, Springer-Verlag, 1968, pp. 1–35.CrossRefGoogle Scholar
[3]Barwise, J., Gandy, R. and Moschovakis, Y., The next admissible set (in preparation).Google Scholar
[4]Jensen, J. and Karp, C., Primitive recursive functions (to appear).Google Scholar
[5]Karp, C., Nonaxiomatizability results for infinitary systems, this Journal, vol. 32 (1967), pp. 367–384.Google Scholar
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[8]Kunen, K., Implicit definability and infinitary languages, this Journal, vol. 33 (1968), pp. 446–451.Google Scholar
[9]Scott, D., Logic with denumerably long formulas and finite strings of quantifiers, Theory of models, North Holland, Amsterdam, 1965, pp. 329–341.Google Scholar
[10]Shoenfild, J., Mathematical logic, Addison-Wesley, Reading, Mass., 1967.Google Scholar