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Colloquium Mathematicum

1991 | 62 | 2 | 181-187
Tytuł artykułu

The number of countable isomorphism types of complete extensions of the theory of Boolean algebras

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
There is a conjecture of Vaught [17] which states: Without The Generalized Continuum Hypothesis one can prove the existence of a complete theory with exactly $ω_1$ nonisomorphic, denumerable models. In this paper we show that there is no such theory in the class of complete extensions of the theory of Boolean algebras. More precisely, any complete extension of the theory of Boolean algebras has either 1 or $2^ω$ nonisomorphic, countable models. Thus we answer this conjecture in the negative for any complete extension of the theory of Boolean algebras. In Rosenstein [15] there is a similar conjecture that any countable complete theory which has uncountably many denumerable models must have $2^ω$ nonisomorphic denumerable models, and this is true without using the Continuum Hypothesis. This paper is an excerpt of the author's thesis, which was written under the guidance of Professor G. C. Nelson. A more detailed exposition of the material may be found there.
Słowa kluczowe
Czasopismo
Rocznik
Tom
Numer
Strony
181-187
Opis fizyczny
Daty
wydano
1991
otrzymano
1989-05-18
poprawiono
1990-06-04
Twórcy
autor
• 3 S 464 Warren Ave., Warrenville, Illinois 60555, U.S.A
Bibliografia
• [1] S. Burris, Boolean powers, Algebra Universalis 5 (1975), 341-360.
• [2] S. Burris and H. P. Sankappanavar, A Course in Universal Algebra, Springer, New York 1981.
• [3] C. C. Chang and H. J. Keisler, Model Theory, North-Holland, Amsterdam 1973.
• [4] H. B. Enderton, A Mathematical Introduction to Logic, Academic Press, New York 1972.
• [5] Yu. L. Ershov, Decidability of the elementary theory of relatively complemented distributive lattices and of the theory of filters, Algebra i Logika Sem. 3 (3) (1964), 17-38 (in Russian).
• [6] L. Feiner, Orderings and Boolean algebras not isomorphic to recursive ones, Ph.D. Thesis, M.I.T., Cambridge, Mass., 1967.
• [7] P. Iverson, Derivatives of linear orderings with applications to the first order theories of Boolean algebras, Ph.D. Thesis, University of Iowa, Iowa City 1988.
• [8] R. D. Mayer and R. S. Pierce, Boolean algebras with ordered bases, Pacific J. Math. 10 (1960), 925-942.
• [9] J. Mead, Prime models and model companions for the theories of Boolean algebras, Ph.D. Thesis, University of Iowa, Iowa City 1975.
• [10] J. Mead, Recursive prime models for Boolean algebras, Colloq. Math. 41 (1979), 25-33.
• [11] A. Mostowski and A. Tarski, Boolesche Ringe mit geordnete Basis, Fund. Math. 32 (1939), 69-86.
• [12] A. Mostowski and A. Tarski, Arithmetical classes and types of well ordered systems, Preliminary report, Bull. Amer. Math. Soc. 55 (1949), 65.
• [13] G. C. Nelson, Boolean powers, recursive models, and the Horn theory of a structure, Pacific J. Math. 114 (1984), 207-220.
• [14] G. C. Nelson, Ultrafilters in Boolean algebras, to appear.
• [15] J. G. Rosenstein, Linear Orderings, Academic Press, New York 1982.
• [16] A. Tarski, Arithmetical classes and types of Boolean algebras, Preliminary report, Bull. Amer. Math. Soc. 55 (1949), 64.
• [17] R. Vaught, Denumerable models of complete theories, in: Infinitistic Methods, Pergamon Press, London 1961, 303-321.
• [18] J. Waszkiewicz, $∀_n$-theories of Boolean algebras, Colloq. Math. 30 (1974), 171-175.
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Bibliografia
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