Abstract

We say that a regular cardinal κ,, 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,, is consistent with an arbitrary continuum function below which satisfies,. Next, starting with infinitely many Mahlo cardinals, we show that the weak tree property at every cardinal,, is consistent with an arbitrary continuum function below which satisfies,. Thus the tree property has no provable effect on the continuum function below except for the trivial requirement that the tree property at implies for every infinite κ.