Let be the Cohen forcing which adds one Cohen real.
Then the following holds:
This proves that whenever if a universe is some generic extension of some definable inner model with the Cohen forcing, there must be a Souslin tree.
On the other hand, it is well-known that the set-theoretical principle implies the existence of Souslin trees.
In particular, Souslin trees exists in , the Godel's constructible universe.
This shows that the existence of Souslin trees is compatible with both of combinatorics and the universe having many generic objects.
This is an interesting phenomenon.
I am interested also in the properties which are compatible both with combinatorics and large cardinal combinatorics.
An important case is the Singular Cardinal Hypothesis(SCH).
SCH is the following statement that Generalize the generalized Continuum Hypothesis(GCH):
For every singlar cardinal ,
It is known that SCH follows from the GCH and hence is a consequence of combinatorics.
But Solovay showed that SCH holds for all cardinal "above" a supercompact cardinal, which is a very large cardinal.
I'm seeking the reason why such phenomena are observed and expecting that Souslin trees will play a crucial role in analysis.