Journal of Symbolic Logic

Continuum-many Boolean algebras of the form 𝒫(ω )/ℐ, ℐ Borel

Michael Ray Oliver

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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.

Article information

J. Symbolic Logic, Volume 69, Issue 3 (2004), 799-816.

First available in Project Euclid: 4 October 2004

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Oliver, Michael Ray. Continuum-many Boolean algebras of the form 𝒫(ω )/ℐ, ℐ Borel. J. Symbolic Logic 69 (2004), no. 3, 799--816. doi:10.2178/jsl/1096901768.

Export citation


  • C. C. Chang and H. Jerome Keisler Model theory, third ed., North-Holland,1990.
  • Ilijas Farah Analytic quotients: Theory of liftings for quotients over analytic ideals on the integers, Memoirs of the AMS, vol. 148 (2000), no. 702.
  • Winfried Just and Adam Krawczyk On certain Boolean algebras $\mathscrP\omega/I$, Transactions of the American Mathematical Society, vol. 285 (1984), no. 1, pp. 411--429.
  • Winfried Just and Žarko Mijajlović Separation properties of ideals over $\omega$, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 33 (1987), no. 3, pp. 267--276.
  • Alexander S. Kechris Classical descriptive set theory, Springer--Verlag,1995.
  • Alain Louveau and Boban Veličkovič A note on Borel equivalence relations, Proceedings of the American Mathematical Society, vol. 120 (1994), no. 1, pp. 255--259.
  • Yiannis Moschovakis Descriptive set theory, North--Holland,1980.
  • Michael Ray Oliver An inquiry into the number of isomorphism classes of Boolean algebras and the Borel cardinality of certain Borel equivalence relations, Ph.D. thesis, UCLA, April 2003.
  • Sławomir Solecki Analytic ideals and their applications, Annals of Pure and Applied Logic, vol. 99 (1999), no. 1--3, pp. 51--72.
  • Juris Steprāns Many quotient algebras of the integers modulo co-analytic ideals, preprint, York University, 2003.
  • Jacques Stern Évaluation du rang de Borel de certains ensembles, Comptes Rendus Hebdomaires des Séances de l'Académie des Sciences, vol. 286 (1978), no. 20, pp. A855--857, Série A--B.
  • Samy Zafrany Borel ideals vs. Borel sets of countable relations and trees, Annals of Pure and Applied Logic, vol. 43 (1989), no. 2, pp. 161--195.