Related papers: Some mutually inconsistent generic large cardinals
We deal with some of problems posed by Monk and related to cardinal invariant of ultraproducts of Boolean algebras. We also introduce and investigate some new cardinal invariants.
We introduce the notion of $\mathcal{C}$-system of filters, generalizing the standard definitions of both extenders and towers of normal ideals. This provides a framework to develop the theory of extenders and towers in a more general and…
This paper is the concise addition to the foregoing work "Inconsistency of Inaccessibility", containing the presentation of main theorem proof (in ZF) about inaccessible cardinals nonexistence. Here some refinement of this presentation is…
The present paper is concerned with the relation between recurrence axioms and Laver-generic large cardinal axioms in light of principles of generic absoluteness and the Ground Axiom. M. Viale proved that Martin's Maximum$^{++}$ together…
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…
The article uses two examples to explore the statement that, contrary to the common wisdom, the properties of singular cardinals are actually more intuitive than those of the regular ones.
We present in an informal way some recent results concerning a possible overlapping between classical unpredictability and quantum indeterminism.
Starting from large cardinals we construct a model of $ZFC$ in which the $GCH$ fails everywhere, but such that $GCH$ holds in its $HOD$. The result answers a question of Sy Friedman. Also, relative to the existence of large cardinals, we…
Using GCH, we force the following: There are continuum many simple cardinal characteristics with pairwise different values.
C-cross topologies are introduced. Modifcations of the Kuratowski-Ulam Theorem are considered. Cardinal invariants add, cof, cov and non with respect to meager or nowhere dense subsets are compared. Remarks on invariants cof(nwdY) are…
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 and study a new type of compactness principle for strong logics that, roughly speaking, infers the consistency of a theory from the consistency of its small fragments in certain outer models of the set-theoretic universe. We…
If we assume the axiom of choice, then every two cardinal numbers are comparable. In the absence of the axiom of choice, this is no longer so. For a few cardinalities related to an arbitrary infinite set, we will give all the possible…
Exacting and ultraexacting cardinals are large cardinal numbers compatible with the Zermelo-Fraenkel axioms of set theory, including the Axiom of Choice. In contrast with standard large cardinal notions, their existence implies that the…
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…
We show that various tameness assertions about abstract elementary classes imply the existence of large cardinals under mild cardinal arithmetic assumptions.
We define Magidor cardinals as J\'onsson cardinals upon replacing colorings of finite subsets by colorings of $\aleph_0$-bounded subsets. Unlike J\'onsson cardinals which appear at some low level of large cardinals, we prove the consistency…
Assuming that there is no inner model with a Woodin cardinal, we obtain a characterization of $\lambda$-tall cardinals in extender models that are iterable. In particular we prove that in such extender models, a cardinal $\kappa$ is a tall…
We introduce the split principles and show that they bear tight connections to large cardinal properties such as inaccessibility, weak compactness, subtlety, almost ineffability and ineffability, as well as classical combinatorial objects…
We show that many large cardinal notions can be characterized in terms of the existence of certain elementary embeddings between transitive set-sized structures, that map their critical point to the large cardinal in question. In…