We give a short proof of a lemma which generalizes both the main lemma from the original construction in the author’s thesis of a model with no ω2-Aronszajn trees, and also the “Key Lemma” in Hamkins’s gap forcing theorems. The new lemma directly yields Hamkins’s newer lemma stating that certain forcing notions have the approximation property. According to Hamkins [2], a partial ordering P sati...

For a cardinal κ and a model M of cardinality κ let No(M) denote the number of non-isomorphic models of cardinality κ which are L∞,κequivalent to M . We prove that for κ a weakly compact cardinal, the question of the possible values of No(M) for models M of cardinality κ is equivalent to the question of the possible numbers of equivalence classes of equivalence relations which are Σ1-definable ...

We show that, assuming GCH, if κ is a Ramsey or a strongly Ramsey cardinal and F is a class function on the regular cardinals having a closure point at κ and obeying the constraints of Easton’s theorem, namely, F (α) ≤ F (β) for α ≤ β and α < cf(F (α)), then there is a cofinality preserving forcing extension in which κ remains Ramsey or strongly Ramsey respectively and 2δ = F (δ) for every regu...

We prove that o(κ) = κ is sufficient to construct a model V [C] in which κ is measurable and C is a closed and unbounded subset of κ containing only inaccessible cardinals of V . Gitik proved that o(κ) = κ is necessary. We also calculate the consistency strength of the existence of such a set C together with the assumption that κ is Mahlo, weakly compact, or Ramsey. In addition we consider the ...

The tree property at κ states that there are no Aronszajn trees on κ, or, equivalently, that every κ tree has a cofinal branch. For singular strong limit cardinals κ, there is tension between the tree property at κ and failure of the singular cardinal hypothesis at κ; the former is typically the result of the presence of strongly compact cardinals in the background, and the latter is impossible...

We present a technique for destroying stationary subsets of Pκκ using partial square sequences. We combine this method with Gitik’s poset for changing the cofinality of a cardinal without adding bounded sets to prove a variety of consistency results concerning saturated ideals and the set S(κ, κ+). In this paper we continue our study of consistency results concerning the set S(κ, κ) = {a ∈ Pκκ ...

In their memoir [1, page 54] Alexandroff and Urysohn asked “existe-il un espace compact (bicompact) ne contenant aucun point (κ)?” and went on to remark “La resolution affirmative de ce problème nous donnerait une exemple des espaces compacts (bicompacts) d’une nature toute differente de celle des espaces connus jusqu’à présent”. The ‘compact’ of that memoir is our countably compact, ‘bicompact...

Abstract In a recent paper, Brendle showed the uniqueness of Bryant soliton among 3-dimensional κ-solutions. this we present an alternative proof for fact and show that compact κ-solutions are rotationally symmetric. Our arose from independent work relating to our Strong Stability Theorem singular Ricci flows.

We construct a variety of inner models exhibiting features usually obtained by forcing over universes with large cardinals. For example, if there is a supercompact cardinal, then there is an inner model with a Laver indestructible supercompact cardinal. If there is a supercompact cardinal, then there is an inner model with a supercompact cardinal κ for which 2κ = κ+, another for which 2κ = κ++ ...

For α, β, n ordinals α → (β, n) means that any graph on the ordinal α either contains an independent set of order-type β or a complete subgraph of order-type n. We only consider cases where n is a natural number, hence the naming. The partition relation α → (β, n) for α, β finite multiples of κ where κ = ω or κ is weakly compact is characterized by a finitary problem as is the same for α, β fin...