The tree property and the continuum function below ℵω

We say that a regular cardinal kappa, kappa > aleph(0), has the tree property if there are no kappa-Aronszajn trees; we say that has the weak tree property if there are no special kappa-Aronszajn trees. Starting with infinitely many weakly compact cardinals, we show that the tree property at every even cardinal aleph(2n,) 0 < n <omega, is consistent with an arbitrary continuum function below aleph(omega) which satisfies 2(aleph 2n) > aleph(2n+1), n < omega. Next, starting with infinitely many Mahlo cardinals, we show that the weak tree property at every cardinal aleph(n), 1 < n < omega, is consistent with an arbitrary continuum function below aleph(omega) which satisfies 2(aleph n) > aleph(n+1), n < omega. Thus the tree property has no provable effect on the continuum function below aleph(omega) except for the trivial requirement that the tree property at kappa(++) implies 2(kappa)> kappa(+) for every infinite kappa. (C) 2018 WILEY- VCHVerlag GmbH & Co. KGaA, Weinheim.