A premouse inheriting strong cardinals from V

@article{Schlutzenberg2015API,
  title={A premouse inheriting strong cardinals from V},
  author={Farmer Schlutzenberg},
  journal={Ann. Pure Appl. Log.},
  year={2015},
  volume={171},
  pages={102826}
}

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References

SHOWING 1-10 OF 31 REFERENCES

Reconstructing resurrection.

Let $R$ be an iterable weak coarse premouse and let $N$ be a premouse with Mitchell-Steel indexing, produced by a fully backgrounded $L[\mathbb{E}]$-construction of $R$. We identify and correct a

Fine structure for tame inner models

In this paper, we solve the strong uniqueness problem posed in [St2]. That is, we extend the full fine structure theory of [MiSt] to backgrounded models all of whose levels are tame (defined in [St2]

THE MAXIMALITY OF THE CORE MODEL

Our main results are: 1) every countably certified extender that coheres with the core model K is on the extender sequence of K, 2) K computes successors of weakly compact cardinals correctly, 3)

Local K constructions

As one might suspect, the more useful answer would be “yes”. For theK-construction of [2], this question is open. The problem is that the construction of [2] is not local: because of the

Local Kc constructions

  • J. Steel
  • Mathematics
    Journal of Symbolic Logic
  • 2007
It is suspect that in fact strong cardinals in V may fail to be strong in V, if is the output of the construction of [2], and one might ask for a construction with output such that iterations on V can be lifted to iteration trees on V.

Measures in Mice

This thesis analyses extenders in fine structural mice. Kunen showed that in the inner model for one measurable cardinal, there is a unique measure. This result is generalized, in various ways, to

Inner Models and Large Cardinals

  • R. Jensen
  • Mathematics
    Bulletin of Symbolic Logic
  • 1995
The development of two important themes of modern set theory are sketched, both of which can be regarded as growing out of work of Kurt Godel.

AD and J\'onsson cardinals in L(R)

Assume ZF+AD+V=L(R) and let \kappa<\Theta\ be an uncountable cardinal. We show that \kappa\ is J\'onsson, and that if cof(\kappa)=\omega\ then \kappa\ is Rowbottom. We also establish some other

An Outline of Inner Model Theory

This paper outlines the basic theory of canonical inner models satisfying large cardinal hypotheses. It begins with the definition of the models, and their fine structural analysis modulo iterability

Stacking mice

The main new technical result, which is due to the first author, is a weak covering theorem for the model obtained by stacking mice over Kc∥κ, and it is shown that either of the following hypotheses imply that there is an inner model with a proper class of strong cardinals and a properclass of Woodin cardinals.