Corpus ID: 228743803

The Virtual Large Cardinal Hierarchy

@inproceedings{Dimopoulos2020TheVL,
  title={The Virtual Large Cardinal Hierarchy},
  author={Stamatis Dimopoulos and Victoria Gitman and Dan Saattrup Nielsen},
  year={2020}
}
We continue the study of the virtual large cardinal hierarchy, initiated in [5], by analysing virtual versions of superstrong, Woodin, Vopěnka, and Berkeley cardinals. Gitman and Schindler showed that virtualizations of strong and supercompact cardinals yield the same large cardinal notion [5]. We show the same result for a (weak) virtualization of Woodin and a virtualization of Vopěnka cardinals. We also show that there is a virtually Berkeley cardinal if and only if the virtual Vopěnka… Expand

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References

SHOWING 1-10 OF 18 REFERENCES
Virtual large cardinals
TLDR
The concept of virtual large cardinals is introduced and a hierarchy of new large cardinal notions between ineffable cardinals and 0 # is obtained. Expand
Ramsey-like cardinals II
TLDR
It is shown that the α-iterable cardinals form a strict hierarchy for α ≤ ω1, that they are downward absolute to L for , and that the consistency strength of Schindler's remarkable cardinals is strictly between 1-iterables and 2-iterably cardinals. Expand
A hierarchy of Ramsey-like cardinals
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 onExpand
Berkeley Cardinals and the search for V
This thesis is concerned with Berkeley Cardinals, very large cardinal axioms inconsistent with the Axiom of Choice. These notions have been recently introduced by J. Bagaria, P. Koellner and W. H.Expand
Proper Forcing and Remarkable Cardinals
  • R. Schindler
  • Mathematics, Computer Science
  • Bulletin of Symbolic Logic
  • 2000
The present paper investigates the power of proper forcings to change the shape of the universe, in a certain well-defined respect. It turns out that the ranking among large cardinals can be used asExpand
The higher infinite : large cardinals in set theory from their beginnings
The theory of large cardinals is currently a broad mainstream of modern set theory, the main area of investigation for the analysis of the relative consistency of mathematical propositions andExpand
Semi-proper forcing, remarkable cardinals, and Bounded Martin's Maximum
TLDR
It is shown that Bounded Martin's Maximum (BMM) is much stronger than BSPFA in that if BMM holds, then for every X ∈ V, X# exists. Expand
WEAKLY REMARKABLE CARDINALS, ERDŐS CARDINALS, AND THE GENERIC VOPĚNKA PRINCIPLE
TLDR
It is shown that the ${Sigma_2}-reflecting weakly remarkableCardinals are exactly the remarkable cardinals, and that the existence of a non-${{\rm{\Sigma }}_ 2}$- ReflectingWeakly remarkable cardinal has higher consistency strength: it is equiconsistent with theexistence of an ω-Erdős cardinal. Expand
A model of the generic Vopěnka principle in which the ordinals are not Mahlo
TLDR
The generic Vopěnka principle is proved to be relatively consistent with the ordinals being non-Mahlo, and the generic Vopska scheme is relatively inconsistent with the existence of a Delta-Δ2-definable class containing no regular cardinals. Expand
The consistency strength of the perfect set property for universally Baire sets of reals.
We show that the statement "every universally Baire set of reals has the perfect set property" is equiconsistent modulo ZFC with the existence of a cardinal that we call a virtually Shelah cardinal.Expand
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