Related papers: Strong Compactness and the Ultrapower Axiom
Assuming an abstract comparison principle called the Ultrapower Axiom, which is motivated by the comparison process of inner model theory and generalizes the statement that the Mitchell order is linear on normal ultrafilters, we…
The relationship between the large cardinal notions of strong compactness and supercompactness cannot be determined under the standard ZFC axioms of set theory. Under a hypothesis called the Ultrapower Axiom, we prove that the notions are…
The inner model problem for supercompact cardinals, one of the central open problems in modern set theory, asks whether there is a canonical model of set theory with a supercompact cardinal. The problem is closely related to the more…
We prove that the Generalized Continuum Hypothesis holds above a supercompact cardinal assuming the Ultrapower Axiom, an abstract comparison principle motivated by inner model theory at the level of supercompact cardinals.
This paper establishes a number of constraints on the structure of large cardinals under strong compactness assumptions. These constraints coincide with those imposed by the Ultrapower Axiom, a principle that is expected to hold in Woodin's…
A diagonal version of the strong reflection principle is introduced, along with fragments of this principle associated to arbitrary forcing classes. The relationships between the resulting principles and related principles, such as the…
We characterize sums of normal ultrafilters after the Magidor iteration (product) of Prikry forcings over a discrete set of measurable cardinals. We apply this to show that the weak Ultrapower Axiom is not equivalent to the Ultrapower…
We introduce a general notion of covering property, of which many classical definitions are particular instances. Notions of closure under various sorts of convergence, or, more generally, under taking kinds of accumulation points, are…
We begin the study of the consequences of the existence of certain infinite matrices. Our present application is to compactness of products of topological spaces.
We introduce combinatorial principles that characterize strong compactness and supercompactness for inaccessible cardinals but also make sense for successor cardinals. Their consistency is established from what is supposedly optimal.…
We study relationships between various set theoretic compactness principles, focusing on the interplay between the three families of combinatorial objects or principles mentioned in the title. Specifically, we show the following. (1) Strong…
We study some topics about \L o\'s's theorem without assuming the Axiom of Choice. We prove that \L o\'s's fundamental theorem of ultraproducts is equivalent to a weak form that every ultrapower is elementary equivalent to its source…
In the absence of the Axiom of Choice, the "small" cardinal $\omega_1$ can exhibit properties more usually associated with large cardinals, such as strong compactness and supercompactness. For a local version of strong compactness, we say…
We discuss some well-known compactness principles for uncountable structures of small regular sizes ($\omega_n$ for $2 \le n<\omega$, $\aleph_{\omega+1}$, $\aleph_{\omega^2+1}$, etc.), consistent from weakly compact (the size-restricted…
We study consequences of stationary and semi-stationary set reflection. We show that the semi stationary reflection principle implies the Singular Cardinal Hypothesis, the failure of weak square principle, etc. We also consider two cardinal…
Recently the second author introduced combinatorial principles that characterize supercompactness for inaccessible cardinals but can also hold true for small cardinals. We prove that the proper forcing axiom PFA implies these principles…
We show that supercompactness and strong compactness can be equivalent even as properties of pairs of regular cardinals. Specifically, we show that if V models ZFC + GCH is a given model (which in interesting cases contains instances of…
We prove a compactness theorem for pseudopower operations of the form $pp_{\Gamma(\mu,\sigma)}(\mu)$ where $\aleph_0<\sigma=cf(\sigma)\leq cf(\mu)$. Our main tool is a result that has Shelah's cov vs. pp Theorem as a consequence. We also…
One of the consequences of the Compactness Principle in structural Ramsey theory is that the small Ramsey degrees cannot exceed the corresponding big Ramsey degrees, thereby justifying the choice of adjectives. However, it is unclear what…
In the present paper we prove the compactness theorem with respect to partial structures and quasi-truth, using the technique of ultraproducts. Partial structures and quasi-truth are two notions developed within the partial structures…