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A fixed point for the jump operator on structures

Published online by Cambridge University Press:  12 March 2014

Antonio Montalbán
Antonio Montalbán*
Affiliation:
Department of Mathematics, University of Chicago, 5734 S. University Avenue Chicago, IL 60637, USA, E-mail:antonio@math.uchicago.edu, URL: http://www.math.uchicago.edu/~antonio/index.html
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Abstract

Assuming that 0# exists, we prove that there is a structure that can effectively interpret its own jump. In particular, we get a structure such that

where is the set of Turing degrees which compute a copy of

More interesting than the result itself is its unexpected complexity. We prove that higher-order arithmetic, which is the union of full “nth-order arithmetic for all n, cannot prove the existence of such a structure.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 78 , Issue 2 , June 2013 , pp. 425 - 438
Copyright
Copyright © Association for Symbolic Logic 2013

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References

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