The structure of the ideals of meager and null sets
by
Jörg Brendle
Dartmouth College

Given sets X and Y as well as ideals I subset P(X) and J subset P(Y), we say I and J are isomorphic if there is a bijection f: X --> Y such that A in I iff f[A] in J for all A subset X. It is well-known that under the continuum hypothesis CH, the null ideal N on the reals R is isomorphic to the ideal on \omega₁ ×\omega₁ generated by sets of the form A ×\omega₁ where A subset \omega₁ is countable. The same is true for the ideal of meager sets M on R. In fact, this forms part of the Sierpinski-Erdos Duality Theorem. A similar statement holds if CH is replaced by the equality \AddN = \CofN, but not in ZFC alone. Here, for a given ideal I, the additivity of I, \AddI, is the size of the smallest F subset or equal I with \cup F not in I, while the cofinality of I, \CofI, is the cardinality of the smallest F subset or equal I such that each member of I is included in a member of F. This led Cichon to suggest

* that one investigates whether N can be consistently isomorphic to some given "nice" ideal I which satisfies \AddI < \CofI.

We show that if the real line is the disjoint union of \kappa meager sets such that every meager set is contained in a countable union of them, then \kappa = \omega₁. This result entails for example that M cannot be isomorphic to the ideal on \omega₂ ×\omega₂ generated by rectangles of the form A ×\omega₂ where A subset or equal \omega₂ is countable. We also prove two theorems saying roughly that any attempt to produce the isomorphism type of the meager ideal in the Cohen real and the random real extensions must fail. In my talk, I will give a sketch of the proof of these results which hold for meager replaced by null, as well. Unfortunately, they all are on the non-structural side, and the question whether * can have a positive answer remains open. Specifically we suggest

** that one studies the structure of M in some particular forcing extension satisfying \AddM < \CofM, for example in the Laver real or in the Miller real model.

Another open problem is whether M and N can be isomorphic in a non-trivial way.

Date received: March 21, 1996

Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caac-16.