The tree property and the continuum function below

Mathematical Logic Quarterly 64 (1-2):89-102 (2018)
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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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Citations of this work

Easton's theorem for the tree property below ℵ.Šárka Stejskalová - 2021 - Annals of Pure and Applied Logic 172 (7):102974.

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References found in this work

Aronszajn trees on ℵ2 and ℵ3.Uri Abraham - 1983 - Annals of Mathematical Logic 24 (3):213-230.
The fine structure of the constructible hierarchy.R. Björn Jensen - 1972 - Annals of Mathematical Logic 4 (3):229.
The tree property up to אω+1.Itay Neeman - 2014 - Journal of Symbolic Logic 79 (2):429-459.
Fragility and indestructibility of the tree property.Spencer Unger - 2012 - Archive for Mathematical Logic 51 (5-6):635-645.

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