Related papers: Reinhardt cardinals in inner models
Let "ex" be the cardinality of the smallest independent family of subsets of omega (independent means that all nontrivial Boolean combinations are infinite) which cannot be extended to a homogeneous independent family. "Homogeneous" means…
We show relative to strong hypotheses that patterns of compact cardinals in the universe, where a compact cardinal is one which is either strongly compact or supercompact, can be virtually arbitrary. Specifically, we prove if V is a model…
Under some cardinal arithmetic assumptions, we prove that every stationary subset of lambda of a right cofinality has the weak diamond. This is a strong negation of uniformization. We then deal with a weaker version of the weak diamond-…
We investigate large set axioms defined in terms of elementary embeddings over constructive set theories, focusing on $\mathsf{IKP}$ and $\mathsf{CZF}$. Most previously studied large set axioms, notably the constructive analogues of large…
Justin Moore's weak club-guessing principle $\mho$ admits various possible generalizations to the second uncountable cardinal. One of them was shown to hold in ZFC by Shelah. A stronger one was shown to follow from several consequences of…
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 show that if the weak compactness of a cardinal is made indestructible by means of any preparatory forcing of a certain general type, including any forcing naively resembling the Laver preparation, then the cardinal was originally…
We prove that, consistently, there exists a weakly but not strongly inaccessible cardinal $\lambda$ for which the sequence $\langle 2^\theta:\theta<\lambda\rangle$ is not eventually constant and the weak diamond fails at $\lambda$. We also…
We show that the traditional criterion for a simplex to belong to the Delaunay triangulation of a point set is equivalent to a criterion which is a priori weaker. The argument is quite general; as well as the classical Euclidean case, it…
We summarize the known methods of producing a non-supercompact strongly compact cardinal and describe some new variants. Our Main Theorem shows how to apply these methods to many cardinals simultaneously and exactly control which cardinals…
It is well-known that if $Y$ is a Banach space the $weak^{*}$-fragmentability of its dual space by some metric $\rho$ implies that $Y^{*}$ belongs to the Stegall class -- the former for shortly $\mathcal{W^*}F$, being the latter…
When one studies the structure (e.g. graded ideals, graded subspaces, radicals, ...) or graded polynomial identities of graded algebras, the grading group itself does not play an important role, but can be replaced by any other group that…
We prove the consistency of the failure of the weak diamond $\Phi_\lambda$ at strongly inaccessible cardinals. On the other hand, we show that the very weak diamond $\Psi_\lambda$ is equivalent to the statement $2^{<\lambda}<2^\lambda$ and…
We continue the work from [8] and make a small -- but significant -- improvement to the definition of $j$-decomposable system. This provides us with a better lifting of elementary embeddings to symmetric extensions. In particular, this…
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.…
This paper explores several topics related to Woodin's HOD conjecture. We improve the large cardinal hypothesis of Woodin's HOD dichotomy theorem from an extendible cardinal to a strongly compact cardinal. We show that assuming there is a…
Suppose that there is a measurable cardinal. If \aleph_\omega is a strong limit cardinal, but the power of \aleph_\omega is bigger than \aleph_{\omega_1}, then there is an inner model with a Woodin cardinal. Modulo the need of the…
A proof will be presented that the existence of a non-trivial $\Sigma_1$-elementary embedding $j: V_{\lambda+3} \prec V_{\lambda+3}$ is inconsistent with $\textsf{ZF}$. Sections 1 and 2 shall review various important contributions from the…
We combine several folklore observations to provide a working framework for iterating constructions which contradict the axiom of choice. We use this to define a model in which any kind of structural failure must fail with a proper class of…
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…