The tree property and the continuum function below ?_ω

摘要

We say that a regular cardinal κ,κ > ?_0, has the tree property if there are no κ-Aronszajn trees; we say that κ has the weak tree property if there are no special κ-Aronszajn trees. Starting with infinitely many weakly compact cardinals, we show that the tree property at every even cardinal ?_(2n), 0 < n < ω, is consistent with an arbitrary continuum function below ?_ω which satisfies 2~(?_(2n)) > ?_(2n+1), n < ω. Next, starting with infinitely many Mahlo cardinals, we show that the weak tree property at every cardinal ?_n, 1 < n < ω, is consistent with an arbitrary continuum function below ?ω which satisfies 2~(?_n) > ?_(n+1), n < ω. Thus the tree property has no provable effect on the continuum function below ?_ω except for the trivial requirement that the tree property at κ~(++) implies 2κ > κ~+ for every infinite κ.