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 2â”0; earlier work by Ilijas Farah had shown
that this was the value in models of Martinâs 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.