Skip to main content Accessibility help
×
Home

Continuum-many Boolean algebras of the form Borel

  • Michael Ray Oliver (a1)

Abstract.

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 ; 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.

Copyright

References

Hide All
[CK90]Chang, C. C. and Keisler, H. Jerome, Model theory, third ed., North-Holland, 1990.
[Far00a]Farah, Ilijas, Analytic quotients: Theory of liftings for quotients over analytic ideals on the integers, Memoirs of the AMS, vol. 148 (2000), no. 702.
[Far00b]Farah, Ilijas, Rigidity conjectures, Proceedings of Logic Colloquium 2000, 2000.
[Far02]Farah, Ilijas, How many Boolean algebras P(ℕ)/I are there?, Illinois Journal of Mathematics, vol. 46 (2002), no. 4, pp. 9991033.
[JK84]Just, Winfried and Krawczyk, Adam, On certain Boolean algebras Pω/I, Transactions of the American Mathematical Society, vol. 285 (1984), no. 1, pp. 411429.
[JM87]Just, Winfried and Mijajlović, Žarko, Separation properties of ideals over ω, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 33 (1987), no. 3, pp. 267276.
[Kec95]Kechris, Alexander S., Classical descriptive set theory, Springer-Verlag, 1995.
[LV94]Louveau, Alain and Veličkovič, Boban, A note on Borel equivalence relations, Proceedings of the American Mathematical Society, vol. 120 (1994), no. 1, pp. 255259.
[Mos80]Moschovakis, Yiannis, Descriptive set theory, North-Holland, 1980.
[Oli03]Oliver, Michael Ray, An inquiry into the number of isomorphism classes of Boolean algebras and the Borel cardinality of certain Borel equivalence relations, Ph.D. thesis, UCLA, 04 2003.
[Sol99]Solecki, Slawomir, Analytic ideals and their applications, Annals of Pure and Applied Logic, vol. 99 (1999), no. 1–3, pp. 5172.
[Step03]Steprāns, Juris, Many quotient algebras of the integers modulo co-analytic ideals, preprint, York University, 2003.
[Ster78]Stern, Jacques, Évaluation du rang de Borel de certains ensembles, Comptes Rendus Hebdomaires des Séances de l'Académie des Sciences, vol. 286 (1978), no. 20, pp. A855857, Série A–B.
[Zaf89]Zafrany, Samy, Borel ideals vs. Borel sets of countable relations and trees, Annals of Pure and Applied Logic, vol. 43 (1989), no. 2, pp. 161195.

Related content

Powered by UNSILO

Continuum-many Boolean algebras of the form Borel

  • Michael Ray Oliver (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.