arXiv subjects

Marwan Salam Mohammd

Publications and source records attributed to Marwan Salam Mohammd.

3 recordsLinked to original sources

arXiv2025

Berkeley Cardinals and Vopěnka's Principle

We introduce "$n$-choiceless" supercompact and extendible cardinals in Zermelo-Fraenkel set theory without the Axiom of Choice. We prove relations between these cardinals and Vopěnka's Principle similar to those of Bagaria's work in his papers "$C^{(n)}$-Cardinals" and "More on the Preservation of Large Cardinals Under Class Forcing." We use these relations to characterize Berkeley cardinals in terms of a restricted form of Vopěnka's Principle. Finally, we establish the equiconsistency of the "$n$-choiceless" extendible cardinals with their original counterparts, and study the consistency strength of other relevant theories.

math.LO↗

arXiv2025

Strong Rigidity and Elementary Embeddings

We present a method for producing elementary embeddings from homomorphisms. This method is utilized in the study of the "strongly rigid relation principle" as defined by Hamkins and Palumbo in their paper "The Rigid Relation Principle, a New Weak Choice Principle." We establish that the strongly rigid relation principle is also a weak choice principle that is independent of ZF. Finally, we characterize proto Berkeley cardinals in terms of a strong failure of the strongly rigid relation principle.

math.LO↗

arXiv2025

Reinhardt Cardinals and Eventually Dominating Functions

We prove a result concerning elementary embeddings of the set-theoretic universe into itself (Reinhardt embeddings) and functions on ordinals that "eventually dominate" such embeddings. We apply that result to show the existence of elementary embeddings satisfying some strict conditions and that are also reminiscent of extendibility in a more local setting. Building further on these concepts, we make precise the nature of some large cardinals whose existence under Reinhardt embeddings was proven by Gabriel Goldberg in his paper "Measurable Cardinals and Choiceless Axioms." Finally, these ideas are used to present another proof of the Kunen inconsistency.

math.LO↗