Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Effectiveness and Vaught's gap $ \omega $ two-cardinal theorem


Author: James H. Schmerl
Journal: Proc. Amer. Math. Soc. 58 (1976), 237-240
MSC: Primary 02H05; Secondary 02F35
DOI: https://doi.org/10.1090/S0002-9939-1976-0432445-4
MathSciNet review: 0432445
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider functions $ f$ with the property that whenever $ \sigma $ is a sentence in $ {L_{\omega \omega }}$, then $ f(\sigma ) < \omega $, and if $ \sigma $ has a gap $ > f(\sigma )$ model, then $ \sigma $ admits all types. A question of Barwise is answered by showing that no such $ f$ is recursive, and that the least such $ f$ is not co-r.e.


References [Enhancements On Off] (What's this?)

  • [1] K. J. Barwise, Some eastern two cardinal theorems (Fifth Internat. Congress of Logic, Methodology and Philosophy of Science, London, Ontario, 1975).
  • [2] R. B. Jensen, (unpublished).
  • [3] J. H. Schmerl, On $ \kappa $-like models which embed stationary and closed unbounded sets, Ann. Math. Logic (to appear).
  • [4] J. H. Schmerl and S. Shelah, On power-like models for hyperinaccessible cardinals, J. Symbolic Logic 37 (1972), 531-537. MR 47 #6474. MR 0317925 (47:6474)
  • [5] S. Shelah, Generalized quantifiers and compact logic, Trans. Amer. Math. Soc. 204 (1975), 342-364. MR 0376334 (51:12510)
  • [6] -, A two-cardinal theorem and a combinatorial theorem, Proc. Amer. Math. Soc. (to appear). MR 0434800 (55:7764)
  • [7] R. L. Vaught, A Löwenheim-Skolem theorem for cardinals far apart, Theory of Models (Proc. 1963 Internat. Sympos. Berkeley), North-Holland, Amsterdam, 1965, pp. 390-401. MR 35 # 1460. MR 0210573 (35:1460)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 02H05, 02F35

Retrieve articles in all journals with MSC: 02H05, 02F35


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1976-0432445-4
Keywords: Two-cardinal theorems, transfer theorems
Article copyright: © Copyright 1976 American Mathematical Society

American Mathematical Society