A premouse inheriting strong cardinals from V

  title={A premouse inheriting strong cardinals from V},
  author={Farmer Schlutzenberg},
  journal={Ann. Pure Appl. Log.},

Fine structure from normal iterability.

We show that (i) the standard fine structural properties for premice follow from normal iterability (whereas the classical proof relies on iterability for stacks of normal trees), and (ii) every

Ordinal definability in $L[\mathbb{E}]$

Let $M$ be a tame mouse modelling ZFC. We show that $M$ satisfies ``there is a real $x$ such that every set is OD in $x$'', and that the restriction of the extender sequence of $M$ to the interval


It is proved that under the same hypothesis, the hereditarily ordinal definable sets $\operatorname {HOD}^{M_n(x)[g]} $ satisfies $(GCH)”, which is a partial iteration strategy for $\mathcal {M}_{\infty }$ .

Full normalization for transfinite stacks

We describe the extension of normal iteration strategies with appropriate condensation properties to strategies for stacks of normal trees, with full normalization. Given a regular uncountable

Local mantles of $L[x]$

Assume ZFC. Let κ be a cardinal. Recall that a < κ-ground is a transitive proper class W modelling ZFC such that V is a generic extension of W via a forcing P ∈ W of cardinality < κ, and the κ-mantle

The definability of the extender sequence E from E ↾ א 1 in L

  • 2019
Let M be an iterable fine structural mouse. We prove that if E ∈ M and M |=“E is a countably complete short extender whose support is a cardinal θ and Hθ ⊆ Ult(V,E)”, then E is in the extender

J an 2 02 1 Background construction for λ-indexed mice

Let M be a λ-indexed (that is, Jensen indexed) premouse. We prove that M is iterable with respect to standard λ-iteration rules iff M is iterable with respect to a natural version of Mitchell-Steel

Iterability for (transfinite) stacks

A natural condensation property for iteration strategies, \emph{inflation condensation}, is defined and it is shown that if $\Sigma$ has inflation condensation then $M$ is $(m,\Omega, \Omega+1)^*$-iterable.

The definability of the extender sequence $\mathbb{E}$ from $\mathbb{E}{\upharpoonright}\aleph_1$ in $L[\mathbb{E}]$.

Let $M$ be an iterable fine structural mouse. We prove that if $E\in M$ and $M\models$``$E$ is a countably complete short extender whose support is a cardinal $\theta$ and



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]


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.