Effectiveness and Vaught's gap 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

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Abstract | References | Similar Articles | Additional Information

Abstract: We consider functions with the property that whenever is a sentence in , then , and if has a gap model, then admits all types. A question of Barwise is answered by showing that no such is recursive, and that the least such is not co-r.e.

**[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 Congress of Logic, Methodology and Philosophy of Science, London, Ontario, 1975).**[2]**R. B. Jensen, (unpublished).**[3]**J. H. Schmerl,*On -like models which embed stationary and closed unbounded sets*, Ann.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)**

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DOI:
https://doi.org/10.1090/S0002-9939-1976-0432445-4

Keywords:
Two-cardinal theorems,
transfer theorems

Article copyright:
© Copyright 1976
American Mathematical Society