Under Martin’s axiom, collapsing of the continuum by Sacks forcing is characterized by the additivity of Marczewski’s ideal (see [4]). We show that the same characterization holds true if proving that under this hypothesis there are no small uncountable maximal antichains in . We also construct a partition of into perfect sets which is a maximal antichain in and show that -sets are exactly (subsets of) selectors of maximal antichains of perfect sets.
I isolate a simple condition that is equivalent to preservation of P-points in definable proper forcing.
We show that in the presence of large cardinals proper forcings do not change the theory of with real and ordinal parameters and do not code any set of ordinals into the reals unless that set has already been so coded in the ground model.
We shall investigate some properties of forcing which are preserved by finite support iterations and which ensure that unbounded families in given partially ordered sets remain unbounded.
Using the theory of rudimentary recursion and provident sets expounded in [MB], we give a treatment of set forcing appropriate for working over models of a theory PROVI which may plausibly claim to be the weakest set theory supporting a smooth theory of set forcing, and of which the minimal model is Jensen’s . Much of the development is rudimentary or at worst given by rudimentary recursions with parameter the notion of forcing under consideration. Our development eschews the power set axiom. We...