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The tree property and the continuum function
Stejskalová, Šárka ; Honzík, Radek (advisor) ; Cummings, James (referee) ; Brooke-Taylor, Andrew (referee)
The continuum function is a function which maps every infinite cardinal κ to 2κ. We say that a regular uncountable cardinal κ has the tree property if every κ-tree has a cofinal branch, or equivalently if there are no κ-Aronszajn trees. We say that a regular uncountable cardinal κ has the weak tree property if there are no special κ-Aronszajn trees. It is known that the tree property, and the weak tree property, have the following non-trivial effect on the continuum function: (∗) If the (weak) tree property holds at κ++, then 2κ ≥ κ++. In this thesis we show several results which suggest that (∗) is the only restriction which the tree property and the weak tree property put on the continuum function in addition to the usual restrictions provable in ZFC (monotonicity and the fact that the cofinality of 2κ must be greater than κ; let us denote these conditions by (∗∗)). First we show that the tree property at ℵ2n for every 1 ≤ n < ω, and the weak tree property at ℵn for 2 ≤ n < ω, does not restrict the continuum function below ℵω more than is required by (∗), i.e. every behaviour of the continuum function below ℵω which satisfies the conditions (∗) and (∗∗) is realisable in some generic extension. We use infinitely many weakly compact cardinals (for the tree property) and infinitely many Mahlo...
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The tree property and the continuum function
Stejskalová, Šárka ; Honzík, Radek (advisor) ; Cummings, James (referee) ; Brooke-Taylor, Andrew (referee)
The continuum function is a function which maps every infinite cardinal κ to 2κ. We say that a regular uncountable cardinal κ has the tree property if every κ-tree has a cofinal branch, or equivalently if there are no κ-Aronszajn trees. We say that a regular uncountable cardinal κ has the weak tree property if there are no special κ-Aronszajn trees. It is known that the tree property, and the weak tree property, have the following non-trivial effect on the continuum function: (∗) If the (weak) tree property holds at κ++, then 2κ ≥ κ++. In this thesis we show several results which suggest that (∗) is the only restriction which the tree property and the weak tree property put on the continuum function in addition to the usual restrictions provable in ZFC (monotonicity and the fact that the cofinality of 2κ must be greater than κ; let us denote these conditions by (∗∗)). First we show that the tree property at ℵ2n for every 1 ≤ n < ω, and the weak tree property at ℵn for 2 ≤ n < ω, does not restrict the continuum function below ℵω more than is required by (∗), i.e. every behaviour of the continuum function below ℵω which satisfies the conditions (∗) and (∗∗) is realisable in some generic extension. We use infinitely many weakly compact cardinals (for the tree property) and infinitely many Mahlo...
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