Skip to main content Accessibility help

A fixed point for the jump operator on structures

  • Antonio Montalbán (a1)


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.



Hide All
[AKMS89]Ash, Chris, Knight, Julia, Manasse, Mark, and Slaman, Theodore, Generic copies of countable structures, Annals of Pure and Applied Logic, vol. 42 (1989), no. 3, pp. 195205.
[AK00]Ash, C.J. and Knight, J., Computable structures and the hyperarithmetical hierarchy, Studies in Logic and the Foundations of Mathematics, vol. 144, Elsevier Science, 2000.
[Bal06]Baleva, V., The jump operation for structure degrees, Archive for Mathematical Logic, vol. 45 (2006), no. 3, pp. 249265.
[Chi90]Chisholm, John, Effective model theory vs. recursive model theory, this Journal, vol. 55 (1990), no. 3, pp. 11681191.
[Dev84]Devlin, Keith J., Constructibility, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1984.
[DJ94]Downey, Rod and Jockusch, Carl G., Every low Boolean algebra is isomorphic to a recursive one. Proceedings of the American Mathematical Society, vol. 122 (1994), no. 3, pp. 871880.
[EP70]Enderton, H. B. and Putnam, Hilary, A note on the hyperarithmetical hierarchy, this Journal, vol. 35 (1970), pp. 429430.
[Har68]Harrison, J., Recursive pseudo-well-orderings, Transactions of the American Mathematical Society, vol. 131 (1968), pp. 526543.
[Kal09]Kalimullin, I. Sh., Relations between algebraic reducibilities of algebraic systems, Izvestiya Vysshikh Uchebnykh ZavedeniǏ. Matematika, vol. 53 (2009), no. 6, pp. 7172.
[Khi04]Khisamiev, A. N., On the Ershov upper semilattice LE, SibirskiǏ MatematicheskiǏ Zhurnal, vol. 45 (2004), no. 1, pp. 211228.
[Mon09]Montalbán, Antonio, Notes on the jump of a structure. Mathematical theory and computational practice, CiE 2009 (Ambos-Spies, Klaus, Löwe, Benedikt, and Merkle, Wolfgang, editors), Lecture Notes in Computer Science, vol. 5635, Springer, 2009, pp. 372378.
[Mon10]Montalbán, Antonio, Coding and definability in computable structures, Notre Dame Journal of Formal Logic, to be published. Notes from a course at Notre Dame University, 2010.
[Mor04]Morozov, A. S., On the relation of Σ-reducibility between admissible sets, SibirskiǏ MatematicheskiǏ Zhurnal, vol. 45 (2004), no. 3, pp. 634652.
[Puz09]Puzarenko, V. G., On a certain reducibility on admissible sets, SibirskiǏ MatematicheskiǏ Zhurnal, vol. 50 (2009), no. 2, pp. 415429.
[Puz11]Puzarenko, Vadim, Fixed points of the jump operator, Algebra and Logic, vol. 50 (2011), no. 5, pp. 418438.
[Ric77]Richter, Linda, Degrees of unsolvability of models, Ph.D. thesis, University of Illinois at Urbana-Champaign, 1977.
[Sos07]Soskova, Alexandra A., A jump inversion theorem for the degree spectra, Computation and logic in the real world, CiE 2007 (Cooper, S. Barry, Löwe, Benedikt, and Sorbi, Andrea, editors), Lecture Notes in Computer Science, vol. 4497, Springer-Verlag, 2007, pp. 716726.
[SS09]Soskova, Alexandra A. and Soskov, Ivan N., A jump inversion theorem for the degree spectra, Journal of Logic and Computation, vol. 19 (2009), no. 1, pp. 199215.
[Stu07]Stukachev, A. I., Degrees of presentability of models. I, Algebra Logika, vol. 46 (2007), no. 6, pp. 763–788 and 793794.
[Stu08]Stukachev, A. I., On degrees of presentability of models. II, Algebra Logika, vol. 47 (2008), no. 1, pp. 108–126 and 131.
[Stu10]Stukachev, A. I., A jump inversion theorem for the semilattices of Sigma-degrees, Siberian Advances in Mathematics, vol. 20 (2010), no. 1, pp. 6874.
[Stu]Stukachev, A. I., Effective model theory via the σ-definability approach, in a Lecture Notes in Logic volume, Association of Symbolic Logic and Cambridge University Press, to appear.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed