Saccharinity

A. We present a method to iterate finitely splitting lim-sup tree forcings along non-wellfounded linear orders. We apply this method to construct a forcing (without using an inaccessible or amalgamation) that makes all definable sets of reals measurable with respect to a certain (non-ccc) ideal. 1. I In the seminal paper [10] Solovay proved that in the Levy model (after collapsing an inaccessible) every definable set is measurable and has the Baire property. In [8] Shelah showed that the inaccessible is necessary for measurability, but the Baire property of every definable set can be obtained by a forcing P without the use of an inaccessible (i.e. in ZFC). This forcing P is constructed by amalgamation of universally meager forcings Q. So every Q adds a co-meager set of generics and has many automorphisms, and the forcing P has similar properties to the Levy collapse. The property of Q that implies that Q can be amalgamated is called " sweetness " (a strong version of ccc). One can ask about other ideals than Lebesgue-null and meager (or their defining forc-ings, random and Cohen), and classify such ideals (or forcings). This was done e.g. in Sweet & Sour [7] — sweet forcings are close to Cohen, sour ones close to random. However, there is a completely different construction that brings measurability of all definable sets: Instead iterating basic forcings Q that have many automorphisms and add a measure 1 set of generics, we use a Q that adds only a null set of generics (a single one in our case, and this real remains the only generic over V i even in the final limit). So Q has to be very non-homogeneous. Instead of having many automorphisms in Q, we assume that the skeleton of the iteration has many automorphisms (so in particular a non-wellfounded iterations has to be used). We use the word Saccharinity for this concept: a construction that achieves the same effect as an (amalgamation of) sweet forcings, but using entirely different means. As a first instance of this concept we construct an ideal I on the reals and a forcing P making every projective (or: definable) set measurable modulo I (without using amalgamation or an inaccessible). I will be defined from a tree forcing in the same way that the Marczewski ideal is defined from Sacks forcing. I is not a ccc ideal. Annotated contents. Section 2, p. …