Related papers: The iterability hierarchy above I3
This paper continues the study of the Ramsey-like large cardinals. Ramsey-like cardinals are defined by generalizing the characterization of Ramsey cardinals via the existence of elementary embeddings. Ultrafilters derived from such…
We introduce a hierarchy of large cardinals between weakly compact and measurable cardinals, that is closely related to the Ramsey-like cardinals introduced by Victoria Gitman, and is based on certain infinite filter games, however also has…
We prove that if there is an elementary embedding from the universe to itself, then there is a proper class of measurable successor cardinals.
We unveil new patterns of Structural Reflection in the large-cardinal hierarchy below the first measurable cardinal. Namely, we give two different characterizations of strongly unfoldable and subtle cardinals in terms of a weak form of the…
We introduce a hierarchy of fast-growing complexity classes and show its suitability for completeness statements of many non elementary problems. This hierarchy allows the classification of many decision problems with a non-elementary…
We present a novel, perspicuous framework for building iterated ultrapowers. Furthermore, our framework naturally lends itself to the construction of a certain type of order indiscernibles, here dubbed tight indiscernibles, which are shown…
This paper contributes to the theory of large cardinals beyond the Kunen inconsistency, or choiceless large cardinal axioms, in the context where the Axiom of Choice is not assumed. The first part of the paper investigates a periodicity…
We isolate a new large cardinal concept, "remarkability." Consistencywise, remarkable cardinals are between ineffable and omega-Erdos cardinals. They are characterized by the existence of "0^sharp-like" embeddings; however, they relativize…
The purpose of this paper is to provide an introductory overview of the large cardinal hierarchy in set theory. By a large cardinal, we mean any cardinal $\kappa$ whose existence is strong enough of an assumption to prove the consistency of…
We try to control many cardinal characteristics by working with a notion of orthogonality between two families of forcings. We show that b^+<g is consistent
We use a reverse Easton forcing iteration to obtain a universe with a definable well-ordering, while preserving the GCH and proper classes of a variety of very large cardinals. This is achieved by coding using the principle diamond star at…
In this paper we consider successive iterations of the first-order differential operations in space ${\bf R}^3.$
We show that many large cardinal notions up to measurability can be characterized through the existence of certain filters for small models of set theory. This correspondence will allow us to obtain a canonical way in which to assign ideals…
We derive, in order of magnitude, the observed astrophysical and cosmological scales in the Universe, from neutron stars to superclusters of galaxies, up to, asymptotically, the observed radius of the Universe. This result is obtained by…
Large cardinals arising from the existence of arbitrarily long end elementary extension chains over models of set theory are studied here. In particular, we show that the large cardinals obtained that way (`Unfoldable cardinals') behave as…
I analyze the hierarchy of large cardinals between a supercompact cardinal and an almost-huge cardinal. Many of these cardinals are defined by modifying the definition of a high-jump cardinal. A high-jump cardinal is defined as the critical…
Despite being an established notion in the large cardinal hierarchy, results about Woodin cardinals are sparse in the literature. Here we gather known results about the preservation of Woodin cardinals under certain forcing extensions, as…
We deal with (< kappa)-supported iterated forcing notions which are (E_0,E_1)-complete, have in mind problems on Whitehead groups, uniformizations and the general problem. We deal mainly with the successor of a singular case. This continues…
We construct models, by three-dimensional arrays of ccc posets, where many classical cardinal characteristics of the continuum are pairwise different.
Matrices over the dual numbers are considered. We propose an approach to classify these matrices up to similarity. Some preliminary results on the realization of this approach are obtained. In particular, we produce explicitly canonical…