Random Reals and Souslin Trees

Richard Laver
1987 Proceedings of the American Mathematical Society
It is consistent that there are no Souslin trees in any random extension of V; thus, the continuum can be singular of cofinality uj\ with Souslin's hypothesis holding. In  , Roitman proved that adding a Cohen real causes MA^ to fail: If g is a Cohen real over V, then in V[g] there is a ccc partial ordering P such that P x P does not have the ccc (see also Galvin , where the existence of such a P is derived from CH). Shelah  improved Roitman's result to: If g is a Cohen real over V,
more » ... en real over V, then in V[g] there is a Souslin tree. Kunen (see  ) noted that Roitman's theorem also holds for random extensions: If r is random over V, then a ccc P, with P x P not ccc, exists in V [r]. Left open was the random real analogue of Shelah's theorem, that is, whether it is a theorem of ZFC that adding a random real adds a Souslin tree. The question whether adding a Sacks real adds a Souslin tree is independent of ZFC + SH. Namely, suppose s is a Sacks real over V. Carlson showed (unpublished) that if V satisfies a fragment of axiom A + 2N° > Ni (which is consistent relative to the consistency of ZFC), then V[s] satisfies MA^, and the author (unpublished) showed that if V satisfies CH, then V[s] satisfies 0Wl. The result of this paper is that if MA^ holds, then the adjunction of any number of random reals does not add a Souslin tree. Recall that if MAK holds, then any tree T of size < k with no paths of length wi is special, that is, T is the union of countably many antichains (Baumgartner, Malitz, and Reinhardt ). THEOREM. If MAK holds and 8 is a measure algebra, then in Vs every tree of size < k with no paths of length wi is special. This theorem fills in the last part to the result that the continuum can be arbitrary with Souslin's hypothesis holding. That Souslin's hypothesis can hold with the continuum being an arbitrary regular cardinal, or an arbitrary singular cardinal of cofinality greater than wi, is the work of Solovay and Tennenbaum  and Jensen (see  ). To get that 2^° can be any singular cardinal of cofinality wi with the Souslin hypothesis holding, begin with a model of MA^, and add A random reals (A any cardinal satisfying cf A = wi and A1*0 = A). Also, by starting with a model of MA^ and adding at least measurably many random reals, the consistency that there is a real valued measurable cardinal < 2N° (see Solovay  ) and Souslin's hypothesis holds, is obtained, relative to the consistency of a measurable cardinal.