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

Colloquium Mathematicae (1991)

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

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topIverson, 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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- [8] R. D. Mayer and R. S. Pierce, Boolean algebras with ordered bases, Pacific J. Math. 10 (1960), 925-942. Zbl0097.02003
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- [11] A. Mostowski and A. Tarski, Boolesche Ringe mit geordnete Basis, Fund. Math. 32 (1939), 69-86. Zbl0021.10903
- [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. Zbl0505.03014
- [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, ${\forall}_{n}$-theories of Boolean algebras, Colloq. Math. 30 (1974), 171-175. Zbl0301.02046

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