1994/07/17 by Brendle, Jörg
#FOS: Mathematics #Logic (math.LO)
paper · doi:10.48550/arxiv.math/9407204
With each of the usual tree forcings I (e.g., I = Sacks forcing S, Laver forcing L, Miller forcing M, Mathias forcing R, etc.) we associate a sigma--ideal i0 on the reals as follows: A ∈ i0 iff for all T ∈ I there is S ≤ T (i.e. S is stronger than T or, equivalently, S is a subtree of T) such that A ∩ [S] = ∅, where [S] denotes the set of branches through S. So, s0 is the ideal of Marczewski null sets, r0 is the ideal of Ramsey null sets (nowhere Ramsey sets) etc. We show (in ZFC) that whenever i0, j0 are two such ideals, then i0 \sem j0 ≠ ∅. E.g., for I=S and J=R this gives a Marczewski null set which is not Ramsey, extending earlier partial results by Aniszczyk, Frankiewicz, Plewik, Brown and Corazza and answering a question of the latter. In case I=M and J=L this gives a Miller null set which is not Laver null; this answers a question addressed by Spinas. We also investigate the question which pairs of the ideals considered are orthogonal and which are not. Furthermore we include Mycielski's ideal P2 in our discussion.