We prove several results giving lower bounds for the large cardinal strength of a failure of the singular cardinal hypothesis. The main result is the following theorem:
Theorem: Suppose $κ$ is a singular strong limit cardinal and $2^κ>= λ$ where $λ$ is not the successor of a cardinal of cofinality at most $κ$.
(i) If $\cofinality(κ)>\gw$ then $o(κ)\geλ$.
(ii) If $\cofinality(κ)=\gw$ then either $o(κ)\geλ$ or $\set{\ga:K\sat o(\ga)\ge\ga^{+n}}$ is cofinal in $κ$ for each $n\in\gw$.
In order to prove this theorem we give a detailed analysis of the sequences of indiscernibles which come from applying the covering lemma to nonoverlapping sequences of extenders.
A model with a sequence of indiscernibles depending on a particular precovering set is constructed.The initial assumption is as follows: for every n<omega the set {alpha | o(alpha)=alpha^+n } is unbounded in kappa.
Assuming that there is no inner model with a Woodin cardinal, we obtain a characterization of $λ$-tall cardinals in extender models that are iterable. In particular we prove that in such extender models, a cardinal $κ$ is a tall cardinal if and only if it is either a strong cardinal or a measurable limit of strong cardinals.
We investigate the circumstances under which there exist a singular cardinal $μ$ and a short $(κ, μ)$-extender $E$ witnessing "$κ$ is $μ$-strong", such that $μ$ is singular in $\Ult(V, E)$.
We define extender sequences, generelizing measure sequences from Radin Forcing. Using the extender sequences we combine Gitik-Magidor forcing for adding many Prikry sequences with Radin forcing. This forcing satisfies Prikry like condition, destroys no cardinals, and has a kind of properness. Depending on the large cardinal we start with, this forcing can blow the power of a cardinal together with changing its' cofinality to a prescribe value. If this prescribed value is the cardinal itself then it remains regular and even can remain measurable.
Supercompact extender based forcings are used to construct models with HOD cardinal structure different from those of V. In particular, a model with all regular uncountable cardinals measurable in HOD is constructed.
We build a supercompact version of the forcing defined in \cite{gitik2019}. For each singular cardinal in the ground model with any fixed cofinality, which is a limit of supercompact cardinals, it is possible to force so that the size of the powerset of the singular cardinal is arbitrarily large, while preserving the singular cardinal. An important feature of this forcing is that it is possible to define the forcing so that the successor of the singular cardinal is collapsed, but all the cardinals above it are preserved.
We present a modification to the Prikry on Extenders forcing notion allowing the blow up of the power set of a large cardinal, change its cofinality to omega without adding bounded subsets, working directly from arbitrary extender (e.g., n-huge extender). Using this forcing, starting from a superstrong cardinal, we construct a model in which the added Prikry sequences are a scale in the normal Prikry sequence.
We introduce the notion of $\mathcal{C}$-system of filters, generalizing the standard definitions of both extenders and towers of normal ideals. This provides a framework to develop the theory of extenders and towers in a more general and concise way. In this framework we investigate the topic of definability of generic large cardinals properties.
We study the notion of tightly stationary sets which was introduced by Foreman and Magidor in \cite{ForMag-MS}. We obtain two consistency results which show that it is possible for a sequence of regular cardinals $( κ_n )_{n < ω}$ to have the property that for every sequence $\vec{S}$, of some fixed-cofinality stationary sets $S_n \subseteq κ_n$, $\vec{S}$ is tightly stationary in a generic extension. The results are obtained using variations of the short-extenders forcing method.
An optimal extension of the Jensen covering lemma, within the limits imposed by Prikry forcing, is proved. If L[E] is an "iterable" weasel with no measurable cardinals, then either L[E] has "indiscernibles", or every uncountable set of ordinals is contained in a set in L[E] of the same cardinality. (The terms "iterable" and "indiscernibles" are made precise in the paper.) Most importantly, there is no hypothesis explicitly limiting the large cardinals which are consistent in L[E].
Foreman proved the Duality Theorem, which gives an algebraic characterization of certain ideal quotients in generic extensions. As an application he proved that generic supercompactness of $ω_1$ is preserved by any proper forcing. We generalize portions of Foreman's Duality Theorem to the context of generic extender embeddings and ideal extenders (as introduced by Claverie in his PhD Thesis, Universitat Munster, 2010). As an application we prove that if $ω_1$ is generically strong, then it remains so after adding any number of Cohen subsets of $ω_1$; however many other $ω_1$-closed posets---su…
Assume $AD+V=L(\mathbb{R})$. Let $κ=\utildeδ^2_1$, the supremum of all $\utildeΔ^2_1$ prewellorderings. We prove that extenders on the sequence of $\H$ that have critical point $κ$ are generated by countably complete measures. This provides a partial reversal of Woodin's result that the $<Θ$-strongness of $κ$ in $\H$ is witnessed by $κ$-complete ultrafilters on $\k$. The aforementioned characterization of extenders works in a more general setting for all cutpoint measurable cardinals of $\H$ in all models of determinacy where the fine structural analysis of $\H$ has been carried out. For examp…
If T is an iteration tree on K and F is a countably certified extender that coheres with the final model of T, then F is on the extender sequence of the final model of T. Several applications of maximality are proved, including: o K computes successors of weakly compact cardinals correctly. o K^c is an iterate of K. o (with Mitchell) If alpha is a cardinal > aleph_1, then K-restriction-alpha is universal for mice of height alpha. Other results in this paper, when combined with work of Woodin, imply: o If square-kappa-finite fails and kappa is a singular, strong limit cardinal, then Inductive D…
Assume ZF (without the Axiom of Choice). Let $j:V_\varepsilon\to V_δ$ be a non-trivial $\in$-cofinal $Σ_1$-elementary embedding, where $\varepsilon,δ$ are limit ordinals. We prove some restrictions on the constructibility of $j$ from $V_δ$, mostly focusing on the case $\varepsilon=δ$. In particular, if $\varepsilon=δ$ and $j\in L(V_δ)$ then $δ$ has cofinality $ω$. However, assuming ZFC+I$_3$, with the appropriate $\varepsilon=δ$, one can force to get such $j\in L(V^{V[G]}_δ)$. Assuming Dependent Choice and that $δ$ has cofinality $ω$ (but not assuming $V=L(V_δ)$), and $j:V_δ\to V_δ$ is $Σ_1$-e…
We give an application of our extender based Radin forcing to cardinal arithmetic. Using a preparation forcing and interleaving of Cohen and Levy forcings in the normal Radin sequence we get a model with a power function having a fixed finite gap on all cardinals.
We extend the concept of "almost indiscernible theory" introduced by Pillay and Sklinos in [Bull. Symb. Log., 2015] (which was itself a modernization and expansion of Baldwin and Shelah [Algebra Universalis, 1983]), to uncountable languages and uncountable parameter sequences. Roughly speaking a theory $T$ is almost indiscernible if some saturated model is in the algebraic closure of an indiscernible set of sequences. We show that such a theory $T$ is nonmultidimensional, superstable, and stable in all cardinals $\ge |T |$ . We prove a structure theorem for sufficiently large $a$-models $M$: T…
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 mice below a superstrong cardinal. The analysis is then used to show that certain tame mice satisfy $V=\mathsf{HOD}$. In particular, the approach proides a new proof of this result for the inner model $M_n$ for $n$ Woodin cardinals. It is also shown that in $M_n$, all homogeneously Suslin sets of reals are $\mathbfΔ^1_{n+1}$.
We study the consistency and consistency strength of various configurations concerning the cardinal characteristics $\mathfrak{s}_θ,\mathfrak{p}_θ,\mathfrak{g}_θ,\mathfrak{r}_θ,\mathfrak{t}_θ$ at uncountable regular cardinals $θ$. Motivated by a theorem of Raghavan-Shelah who proved that $\mathfrak{s}_θ\leq\mathfrak{b}_θ$, we explore in the first part of the paper the consistency of inequalities comparing $\mathfrak{s}_θ$ with $\mathfrak{p}_θ$ and $\mathfrak{g}_θ$. In the second part of the paper we study variations of the extender-based Radin forcing to establish several consistency results c…
