A remark on Martin's conjecture (Q2732288)
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scientific article; zbMATH DE number 1623529
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | A remark on Martin's conjecture |
scientific article; zbMATH DE number 1623529 |
Statements
13 January 2003
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atomic Boolean algebras
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elementary theories
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quotient algebras
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Frechet ideals
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countable models
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complete theories
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strong Martin conjecture
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A remark on Martin's conjecture (English)
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It is shown that, for the class of infinite atomic Boolean algebras, the classification of their \((\omega+ \omega)\)-elementary theories can be reduced to the classification of the elementary theories of their quotient algebras modulo the Frechet ideals. Then, by Ershov-Tarski's analysis, the following result is obtained:NEWLINENEWLINENEWLINEThere is a complete consistent theory \(T\) in \(L_{\omega, \omega}\) such that \(T\) has \(2^{\aleph_0}\) countable models, and there are at most countably many models of \(T\) with distinct complete theories in \(L_{\omega+ \omega,\omega}\).NEWLINENEWLINENEWLINEIn particular, the strong Martin conjecture is false.
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