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An example concerning Scott heights

Published online by Cambridge University Press:  12 March 2014

M. Makkai
M. Makkai*
Affiliation:
Mcgill University, Montreal H3A 2K6, Quebec, Canada
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Extract

Unless otherwise stated, every structure in this paper is countable in a countable, actually recursive language and every formula is one of .

The definition of the so-called canonical Scott-sentence of a structure M, CSS(M) (compare Nadel [8]), is based on the ordinal invariant called the Scott-height of M, denoted SH(M) (compare Makkai [5]). To describe SH(M), let “b ≡αα”, for finite tuples of equal lengths b and a of elements of M and any ordinal α, stand for “b and a satisfy (in M) the same formulas of quantifier-rank ≤ α” (for the quantifier rank of a formula, see e.g., Barwise [1]); also, let “b ∼ a” mean that there is an automorphism of M mapping b to a. With a fixed M, and a in M, sh(a) (or shM(a)) denotes the least ordinal α such that for all b in M, b ≡αa implies b ∼ a. SH(M) is the least ordinal α such that for all a and b in M, b ≡αa implies b ∼ a; hence SH(M) = sup{sh(a): a in M}.

Information

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 46 , Issue 2 , June 1981 , pp. 301 - 318
Copyright
Copyright © Association for Symbolic Logic 1981

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References

REFERENCES

[1]Barwise, J., Admissible sets and structures, Springer-Verlag, Berlin and New York, 1975.CrossRefGoogle Scholar
[2]Gregory, J., Uneountable models and infinitary elementary extensions, this Journal, vol. 38 (1973), pp. 460–470.Google Scholar
[3]Harnik, V. and Makkai, M., A tree argument in infinitary model theory, Proceedings of the American Mathematical Society, vol. 67 (1977), pp. 309–314.CrossRefGoogle Scholar
[4]Keisler, H. J., Model theory for infinitary logic, North-Holland, Amsterdam, 1971.Google Scholar
[5]Makkai, M., An “admissible” generalization of a theorem on countable Σ11 sets of reals with applications, Annals of Mathematical Logic, vol. 11 (1977), pp. 1–30.CrossRefGoogle Scholar
[6]Makkai, M., Admissible sets and infinitary logic, Handbook of mathematical logic (Barwise, J., Editor), Chapter A. 7, North-Holland, Amsterdam, 1977.Google Scholar
[7]Morley, M., The number of countable models, this Journal, vol. 35 (1970), pp. 14–18.Google Scholar
[8]Nadel, M., Scott sentences and admissible sets, Annals of Mathematical Logic, vol. 7 (1974), pp. 267–294.CrossRefGoogle Scholar
[9]Ressayre, J. P., Models with compactness properties with respect to logics on admissible sets, Annals of Mathematical Logic, vol. 11 (1977), pp. 31–35.CrossRefGoogle Scholar
[10]Steel, J., On Vaught's conjecture, Cabal Seminar 76–77 (Kechris, A.S. and Moschovakis, Y.N., Editors), Lecture Notes in Mathematics, No. 689, Springer-Verlag, Berlin and New York, 1978, pp. 193–208.CrossRefGoogle Scholar
[11]Shelah, S., Refuting Ehrenfeucht's conjecture on rigid models, Israel Journal of Mathematics, vol. 25 (1976), pp. 273–286.CrossRefGoogle Scholar