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Note on the Kondo-Addison theorem

Published online by Cambridge University Press:  12 March 2014

David Guaspari*
Affiliation:
Churchill College, Cambridge, England

Extract

The Kondo-Addison theorem says that every set A of reals contains a real which is implicit (for short, Imp). Any real which is Imp is constructible, and it is the main purpose of this note to show that a close examination of the usual Kondo-Addison argument allows us to compute (in terms of the ordinals associated with A by the classical representation theorem) where in the constructible hierarchy an element of A ∩ Imp must occur. This result is related to a basis theorem of Barwise and Fisher [1] and is accompanied by a strong counterexample which shows that certain tepid improvements of their theorem (and, a fortiori, of the main theorem in this note) are impossible. As a final flourish, the ‘classifying ordinals’ used to compute the locations of the singletons are themselves characterized using an idea from [1].

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1974

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References

REFERENCES

[1]Barwise, J. and Fisher, E., The Shoenfield absoluteness lemma, Israel Journal of Mathematics, vol. 8 (1967), pp. 329339.CrossRefGoogle Scholar
[2]Shoenfield, J. R., Mathematical logic, Addison-Wesley, Reading, Mass., 1967.Google Scholar
[3]Rogers, H., Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967.Google Scholar