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Model constructions in stationary logic. Part I. Forcing

Published online by Cambridge University Press:  12 March 2014

Kim B. Bruce
Kim B. Bruce*
Affiliation:
Princeton University, Princeton, New Jersey 08540 Williams College, Williamstown, Massachusetts 01267
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Extract

In this series of two papers we wish to discuss two new methods of constructing models for an arbitrary theory T of stationary logic, L(aa). The original work in constructing models for this logic appears in Barwise-Kaufmann-Makkai [BKM] (which we shall refer to as BKM for brevity). In that paper the authors prove completeness, compactness, and omitting types theorems for L(aa) by a technique similar to that used by Keisler [1970] in constructing models of L(Q1), logic with the generalized quantifier “there exist uncountably many”. In this paper we give an alternative method of constructing models using a method of model-theoretic forcing similar to that of Bruce [1978]. In a succeeding paper we will give yet another method of constructing models using definable ultrapowers.

As applications of this new forcing construction we give new proofs of the completeness and omitting types theorems for LB(aa), where LB(aa) is any countable fragment of Lω1ω(aa). We also give new proofs of versions of the Barwise completeness and compactness theorems for LA(aa) where A is a countable admissible set. In proofs of these theorems the model constructions via forcing are used in a way very similar to that of consistency properties in Lω1ω.

Aside from the ease in handling the model theory of Lω1ω(aa) we see two other advantages in this sort of model construction. The first is in the ability to prove versions of theorems relativized to formulas of low complexity (e.g. relativized omitting types theorems). The second is that we have a high degree of control over the construction of standard models and hence have a better knowledge of the structure of the model (see the remarks following the proof of Lemma 3.3), something that should prove useful in constructing more complicated models.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 45 , Issue 3 , September 1980 , pp. 439 - 454
Copyright
Copyright © Association for Symbolic Logic 1980

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References

REFERENCES

Barwise, J., Kaufmann, M. and Makkai, M. [BKM], Stationary logic, Annals of Mathematical Logic, vol. 13 (1978), pp. 171–224.CrossRefGoogle Scholar
Barwise, J. [1974], Mostowski's collapsing function and the closed unbounded filter, Fundamenta Mathematicae, vol. 82, pp. 95–103.CrossRefGoogle Scholar
Barwise, J. [1975], Admissible sets and structures, Springer-Verlag, Berlin and New York.CrossRefGoogle Scholar
Bruce, K. [1978], Model-theoretic forcing in logic with a generalized quantifier, Annals of Mathematical Logic, vol. 13, pp. 225–265.CrossRefGoogle Scholar
Bruce, K. and Keisler, H. J. [1979], LA(Ⅎ), this Journal, vol. 44, pp. 15–28.Google Scholar
Chang, C. C. and Keisler, H. J. [1973], Model theory, North-Holland, Amsterdam.Google Scholar
Jech, T. [1973], Some combinatorial problems concerning uncountable cardinals, Annals of Mathematical Logic, vol. 5, pp. 165–198.CrossRefGoogle Scholar
Kaufmann, M. [1978], Some results in stationary logic, Ph.D. Thesis, University of Wisconsin.Google Scholar
Keisler, H. J. [1970], Logic with the quantifier ‘there exist uncountably many’ Annals of Mathematical Logic, vol. 1, pp. 1–93.CrossRefGoogle Scholar
Keisler, H. J. [1971], Model theory for infinitary logic, North-Holland, Amsterdam.Google Scholar
Keisler, H. J. [1973], Forcing and the omitting types theorem, Studies in Model Theory, (Morley, M., Editor), American Mathematical Society, Providence, R.I., pp. 96–133.Google Scholar
Kueker, D. [1977], Countable approximations and Löwenheim-Skolem Theorems, Annals of Mathematical Logic, vol. 11, pp. 57–104.CrossRefGoogle Scholar
Magidor, M. and Malitz, J. [1977], Compact extensions of L(Q). (Part 1a), Annals of Mathematical Logic, vol. 11 (1977), pp. 217–261.CrossRefGoogle Scholar