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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

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