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Strict-Π11 predicates on countable and cofinality ω transitive sets1

Published online by Cambridge University Press:  12 March 2014

Philip W. Grant
Philip W. Grant*
Affiliation:
University College, Swansea, Wales, UK Sunderland Polytechnic, Tyne and Weald, England
*
Current address: University College Swansea, Wales, Uk
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Extract

Throughout the paper A will be a transitive set closed under finite subsets and the formulas in various classes mentioned are allowed to contain parameters from A (or from B in §2).

By use of a refinement of Moschovakis' notion of the game-quantifier [13], [14], [15] we are able to obtain a game-theoretic description of s-Π11 predicates over countable sets which then leads to a classification of positive Σ1 inductive sets.

Similar results are then proved for certain sets of cofinality ω. As a consequence we obtain the compactness results of Green [8], [11], Nyberg [16] and Makkai [12].

The use of games to classify inductive sets was initiated by Moschovakis [13], [14], [15] and has been extended to Q-inductive sets by Aczel [2]. Games were also used in a slightly different setting by Vaught [18] and Makkai [12]. In fact, Vaught's proof of the compactness theorem is very close to our proof in §1 and Makkai's extension to cofinality ω sets uses a result similar to Theorem 3 in §2.

We are indebted to the referee for many helpful suggestions, in particular, for bringing to our attention the related works of Vaught and Makkai cited above.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 42 , Issue 2 , June 1977 , pp. 161 - 173
Copyright
Copyright © Association for Symbolic Logic 1977

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Footnotes

1

Most of these results appear in the author's doctoral dissertation written under the supervision of R. O. Gandy to whom he would like to express his sincere thanks.

References

REFERENCES

[1]Aczel, P., Implicit and inductive definability, this Journal, Abstract, vol. 35 (1970), p. 599.Google Scholar
[2]Aczel, P., Quantifiers, games and inductive definitions, Proceedings of the Third Scandinavian Logic Symposium (Kanger, S., Editor), North-Holland, Amsterdam, 1975, pp. 1–14.Google Scholar
[3]Barwise, J., Infinitary logic and admissible sets, this Journal, vol. 34 (1969), pp. 226–252.Google Scholar
[4]Barwise, J., Applications of strict Π11 predicates to infinitary logic, this Journal, vol. 34 (1969), pp. 409–423.Google Scholar
[5]Chang, C. C. and Moschovakis, Y. N., The Suslin–Kleene Theorem for Vk with cofinality (k) = ω, Pacific Journal of Mathematics, vol. 35 (1970), pp. 565–569.CrossRefGoogle Scholar
[6]Gandy, R. O., Inductive definitions, Generalised recursion theory (Fenstad, J. and Hinman, P., Editors), North-Holland, Amsterdam, 1974.Google Scholar
[7]Grant, P. W., Inductive definitions and recursion theory, Ph.D. Dissertation, Oxford, 1973.Google Scholar
[8]Green, J., Σ1compactness for next admissible sets, this Journal, vol. 39 (1974), pp. 105–116.Google Scholar
[9]Jensen, R. and Karp, C., Primitive recursive set functions, Proceedings of Symposia in Pure Mathematics, vol. XIII, Part 1, pp. 143–176, American Mathematical Society, 1971.Google Scholar
[10]Karp, C., An algebraic proof of the Barwise compactness theorem, The syntax and semantics of infinitary languages (Barwise, J., Editor), Lecture Notes in Mathematics, Springer-Verlag, Berlin, vol. 72 (1968), pp. 80–95.CrossRefGoogle Scholar
[11]Karp, C., From countable to cofinality ω in infinitary model theory, this Journal, Abstract, vol. 37 (1972), pp. 430–431.Google Scholar
[12]Makkai, M., Generalising Vaught sentences from ω to strong cofinality ω, Fundamenta Mathematical vol. 82 (1974), pp. 105–119.Google Scholar
[13]Moschovakis, Y. N., The Suslin–Kleene Theorem for countable structures, Duke Mathematical Journal, vol. 37 (1970), pp. 341–352.Google Scholar
[14]Moschovakis, Y. N., The game quantifier, Proceedings of the American Mathematical Society, vol. 31 (1972), pp. 245–250.Google Scholar
[15]Moschovakis, Y. N., Elementary induction on abstract structures, North-Holland, Amsterdam, 1974.Google Scholar
[16]Nyberg, A. M., Uniform inductive definability and infinitary logic, this Journal, vol. 41 (1976), pp. 109–120.Google Scholar
[17]Nyberg, A. M., Applications of model theory to recursion theory on structures of strong cofinality ω, preprint series, no. 17, Institute of Mathematics, University of Oslo, 1974.Google Scholar
[18]Vaught, R., Descriptive set theory in Lω1ω, Proceedings of the Cambridge School in Mathematical Logic, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 337 (1973), pp. 574–598.Google Scholar