The tree property and the continuum function below |
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| 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. |
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