Corpus ID: 235436199

The $\omega$-th inner mantle

@inproceedings{Williams2021TheI,
  title={The \$\omega\$-th inner mantle},
  author={Kameryn J. Williams},
  year={2021}
}
This article investigates pathological behavior at the first limit stage in the sequence of inner mantles, obtained by iterating the definition of the mantle to get smaller and smaller inner models. I show: (A) it is possible that the ω-th inner mantle is not a definable class; and (B) it is possible that the ω-th inner mantle is a definable class but does not satisfy AC. This answers a pair of questions of Fuchs, Hamkins, and Reitz [FHR15]. 

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References

SHOWING 1-10 OF 19 REFERENCES
Inner mantles and iterated HOD
TLDR
A class forcing notion is presented, uniformly definable for ordinals $\eta$, which forces the ground model to be the $\eta$-th inner mantle of the extension, in which the sequence of inner mantles has length at least $\eta$. Expand
The downward directed grounds hypothesis and very large cardinals
TLDR
It is shown that if the universe has some very large cardinal, then the mantle must be a ground, and some fundamental theorems on the forcing method and the set-theoretic geology are established. Expand
Inner Models for Set Theory - Part III
TLDR
The present paper is concerned exclusively with a particular kind of model, the ‘super-complete models’ defined in section 2.4 of I, where ψ ( U ) is the propositional function expressing in terms of U the fact that the model determined by U satisfies the relativization of axioms A, B, C. Expand
Set Theory, Arithmetic, and Foundations of Mathematics: The continuum hypothesis, the generic-multiverse of sets, and the Ω conjecture
Is this really evidence (as is often cited) that the Continuum Hypothesis has no answer? Another prominent problem from the early 20th century concerns the projective sets, [8]; these are the subsetsExpand
Cohen forcing and inner models
  • J. Reitz
  • Mathematics, Computer Science
  • Math. Log. Q.
  • 2020
TLDR
The results are generalized to $Add(\kappa,\lambda)$, and to iterations of Cohen forcing where the poset at each stage comes from an arbitrary intermediate inner model. Expand
Set-theoretic geology
TLDR
This theorem is proved while also controlling the HOD of the final model, as well as the generic HOD, which is the intersection of all HODs of all set-forcing extensions of V. Expand
Certain very large cardinals are not created in small forcing extensions
  • R. Laver
  • Mathematics, Computer Science
  • Ann. Pure Appl. Log.
  • 2007
TLDR
The reverse directions of the large cardinal axioms of the title are proved and V is definable using the parameter V δ + 1, where δ = P = + . Expand
Generalized iteration of forcing
Generalized iteration extends the usual notion of iterated forcing from iterating along an ordinal to iterating along any partially ordered set. We consider a class of forcings called perfect treeExpand
Iterating ordinal definability
TLDR
A uniform way of obtaining ordinal definability by forcing descending sequences of the iterations of HOD is presented and their structure is described and the following results are proved. Expand
A minimal model for set theory
In the proof of the consistency of the Continuum Hypothesis and the Axiom of Choice with the other axioms of set theory, Godel [ l ] introduced the notion of a constructible set and showed that theExpand
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