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

Paul Iverson

Colloquium Mathematicae (1991)

  • Volume: 62, Issue: 2, page 181-187
  • ISSN: 0010-1354

Abstract

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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.

How to cite

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Iverson, Paul. "The number of countable isomorphism types of complete extensions of the theory of Boolean algebras." Colloquium Mathematicae 62.2 (1991): 181-187. <http://eudml.org/doc/210106>.

@article{Iverson1991,
abstract = {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.},
author = {Iverson, Paul},
journal = {Colloquium Mathematicae},
keywords = {number of nonisomorphic models; denumerable models; complete extensions of the theory of Boolean algebras},
language = {eng},
number = {2},
pages = {181-187},
title = {The number of countable isomorphism types of complete extensions of the theory of Boolean algebras},
url = {http://eudml.org/doc/210106},
volume = {62},
year = {1991},
}

TY - JOUR
AU - Iverson, Paul
TI - The number of countable isomorphism types of complete extensions of the theory of Boolean algebras
JO - Colloquium Mathematicae
PY - 1991
VL - 62
IS - 2
SP - 181
EP - 187
AB - 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.
LA - eng
KW - number of nonisomorphic models; denumerable models; complete extensions of the theory of Boolean algebras
UR - http://eudml.org/doc/210106
ER -

References

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  1. [1] S. Burris, Boolean powers, Algebra Universalis 5 (1975), 341-360. 
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  11. [11] A. Mostowski and A. Tarski, Boolesche Ringe mit geordnete Basis, Fund. Math. 32 (1939), 69-86. Zbl0021.10903
  12. [12] A. Mostowski and A. Tarski, Arithmetical classes and types of well ordered systems, Preliminary report, Bull. Amer. Math. Soc. 55 (1949), 65. 
  13. [13] G. C. Nelson, Boolean powers, recursive models, and the Horn theory of a structure, Pacific J. Math. 114 (1984), 207-220. Zbl0505.03014
  14. [14] G. C. Nelson, Ultrafilters in Boolean algebras, to appear. 
  15. [15] J. G. Rosenstein, Linear Orderings, Academic Press, New York 1982. 
  16. [16] A. Tarski, Arithmetical classes and types of Boolean algebras, Preliminary report, Bull. Amer. Math. Soc. 55 (1949), 64. 
  17. [17] R. Vaught, Denumerable models of complete theories, in: Infinitistic Methods, Pergamon Press, London 1961, 303-321. 
  18. [18] J. Waszkiewicz, ∀ n -theories of Boolean algebras, Colloq. Math. 30 (1974), 171-175. Zbl0301.02046

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