Strengthening a result of Amir Leshem [7], we prove that the consistency strength of holding GCH together with definable tree property for all successors of regular cardinals is precisely equal to the consistency strength of existence of proper class many Π 1 reflecting cardinals. Moreover it is proved that if κ is a supercompact cardinal and λ > κ is measurable, then there is a generic extensi...

We prove that the PFA lottery preparation of a strongly unfoldable cardinal κ under ¬0 forces PFA(א2-preserving), PFA(א3-preserving) and PFAא2 , with 2 ω = κ = א2. The method adapts to semi-proper forcing, giving SPFA(א2-preserving), SPFA(א3-preserving) and SPFAא2 from the same hypothesis. It follows by a result of Miyamoto that the existence of a strongly unfoldable cardinal is equiconsistent ...

In this paper I will show that it is relatively consistent with the usual axioms of mathematics (ZFC) together with a strong form of the axiom of infinity (the existence of a supercompact cardinal) that the class of uncountable linear orders has a five element basis. In fact such a basis follows from the Proper Forcing Axiom, a strong form of the Baire Category Theorem. The elements are X , ω1,...

We show the relative consistency of the existence of two strongly compact cardinals κ1 and κ2 which exhibit indestructibility properties for their strong compactness, together with level by level equivalence between strong compactness and supercompactness holding at all measurable cardinals except for κ1. In the model constructed, κ1’s strong compactness is indestructible under arbitrary κ1-dir...

The Main Theorem of this article asserts in part that if an extension V ⊆ V satisfies the δ approximation and covering properties, then every embedding j : V → N definable in V with critical point above δ is the lift of an embedding j ↾ V : V → N definable in the ground model V . It follows that in such extensions there can be no new weakly compact cardinals, totally indescribable cardinals, st...

We prove various theorems about the preservation and destruction of the tree property at ω2. Working in a model of Mitchell [9] where the tree property holds at ω2, we prove that ω2 still has the tree property after ccc forcing of size א1 or adding an arbitrary number of Cohen reals. We show that there is a relatively mild forcing in this same model which destroys the tree property. Finally we ...

Three central combinatorial properties in set theory are the tree property, the approachability property and stationary reflection. We prove the mutual independence of these properties by showing that any of their eight Boolean combinations can be forced to hold at κ, assuming that κ = κ and there is a weakly compact cardinal above κ. If in addition κ is supercompact then we can force κ to be א...

We give two results on guessing unbounded subsets of λ+. The first is a positive result and applies to the situation of λ regular and at least equal to א3, while the second is a negative consistency result which applies to the situation of λ a singular strong limit with 2 > λ+. The first result shows that in ZFC there is a guessing of unbounded subsets of S + λ . The second result is a consiste...

Building upon work of Abraham (cf. [1]), Cummings and Foreman have shown in [2], starting from ω many supercompact cardinals, that consistently the following holds. (⋆) For every n < ω, 2n = אn+2 and אn+1 has the tree property. Recall that a cardinal κ is said to have the tree property if there is no Aronszajn κ-tree, i.e. if every tree of height κ all of whose levels have size < κ admits a cof...