Translator Disclaimer
September 2004 Continuum-many Boolean algebras of the form 𝒫(ω )/ℐ, ℐ Borel
Michael Ray Oliver
J. Symbolic Logic 69(3): 799-816 (September 2004). DOI: 10.2178/jsl/1096901768


We examine the question of how many Boolean algebras, distinct up to isomorphism, that are quotients of the powerset of the naturals by Borel ideals, can be proved to exist in ZFC alone. The maximum possible value is easily seen to be the cardinality of the continuum 20; earlier work by Ilijas Farah had shown that this was the value in models of Martin’s Maximum or some similar forcing axiom, but it was open whether there could be fewer in models of the Continuum Hypothesis. We develop and apply a new technique for constructing many ideals whose quotients must be nonisomorphic in any model of ZFC. The technique depends on isolating a kind of ideal, called shallow, that can be distinguished from the ideal of all finite sets even after any isomorphic embedding, and then piecing together various copies of the ideal of all finite sets using distinct shallow ideals. In this way we are able to demonstrate that there are continuum-many distinct quotients by Borel ideals, indeed by analytic P-ideals, and in fact that there is in an appropriate sense a Borel embedding of the Vitali equivalence relation into the equivalence relation of isomorphism of quotients by analytic P-ideals. We also show that there is an uncountable definable wellordered collection of Borel ideals with distinct quotients.


Download Citation

Michael Ray Oliver. "Continuum-many Boolean algebras of the form 𝒫(ω )/ℐ, ℐ Borel." J. Symbolic Logic 69 (3) 799 - 816, September 2004.


Published: September 2004
First available in Project Euclid: 4 October 2004

zbMATH: 1070.03030
MathSciNet: MR2078923
Digital Object Identifier: 10.2178/jsl/1096901768

Rights: Copyright © 2004 Association for Symbolic Logic


This article is only available to subscribers.
It is not available for individual sale.

Vol.69 • No. 3 • September 2004
Back to Top