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### Indiscernible Sequences for Extenders, and the Singular Cardinal Hypothesis [article]

Moti Gitik, William Mitchell
1995 arXiv   pre-print
We prove several results giving lower bounds for the large cardinal strength of a failure of the singular cardinal hypothesis.  ...  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.  ...  For other applications, such as the singular cardinal hypothesis, this is not possible.  ...

### Indiscernible sequences for extenders, and the singular cardinal hypothesis

Moti Gitik, William J. Mitchell
1996 Annals of Pure and Applied Logic
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.  ...  The applications to the singular cardinal hypothesis are given in Section 3, and some open problems are stated in Section 4.  ...  For other applications, such as the singular cardinal hypothesis, this is not possible.  ...

### 0# and inner models

SY D. Friedman
2002 Journal of Symbolic Logic (JSL)
The principal tools are the Covering Theorem for L and the technique of reverse Easton iteration. Let I denote the class of Silver indiscernibles for L and 〈i α ∣ α ϵ ORD〉 its increasing enumeration.  ...  Also fix an inner model M of GCH not containing 0# and let ω α denote the ω α of the model M[0#], the least inner model containing M as a submodel and 0# as an element.  ...  By hypothesis, the M-cardinality of κ is greater than ω M 1 .  ...

### On the singular cardinal hypothesis

W. J. Mitchell
1992 Transactions of the American Mathematical Society
We use core model theory to obtain the following lower bounds to the consistency strength for the failure of the Singular Cardinal Hypothesis: Suppose that k is a singular strong limit cardinal such that  ...  Then there is an inner model K such that o(k) = «c++ in K if k has uncountable cofinality, and Va < k3u < k o(k) > u in K otherwise.  ...  Introduction The Singular Cardinal Hypothesis (SCH) asserts that if k is any singular strong limit cardinal then 2K = k+ .  ...

### On the Singular Cardinal Hypothesis

W. J. Mitchell
1992 Transactions of the American Mathematical Society
We use core model theory to obtain the following lower bounds to the consistency strength for the failure of the Singular Cardinal Hypothesis: Suppose that k is a singular strong limit cardinal such that  ...  Then there is an inner model K such that o(k) = «c++ in K if k has uncountable cofinality, and Va < k3u < k o(k) > u in K otherwise.  ...  Introduction The Singular Cardinal Hypothesis (SCH) asserts that if k is any singular strong limit cardinal then 2K = k+ .  ...

### On gaps under GCH type assumptions

Moti Gitik
2003 Annals of Pure and Applied Logic
We prove equiconsistency results concerning gaps between a singular strong limit cardinal κ of cofinality ℵ 0 and its power under assumptions that 2 κ = κ +δ+1 for δ < κ and some weak form of the Singular  ...  Cardinal Hypothesis below κ.  ...  If 2 κ ≥ κ +δ and the Singular Cardinal Hypothesis holds below κ at least for cardinals of cofinality cf δ, then in the core model either (i) o(κ) ≥ κ +δ+1 + 1 or (ii) {α < κ | o(α) ≥ α +δ+1 + 1} is unbounded  ...

### On the Singular Cardinal Hypothesis [article]

William J. Mitchell
1992 arXiv   pre-print
We use the core model for sequences of measures to prove a new lower bound for the consistency strength of the failure of the SCH: THEOREM (i) If there is a singular strong limit cardinal κ such that 2  ...  (ii) If there is a singular strong limit cardinal κ of uncountable cofinality such that 2^κ > κ^+ then there is an inner model with o(κ) = κ^++.  ...  -Introduction The Singular Cardinal Hypothesis (SCH) asserts that if κ is any singular strong limit cardinal then 2 κ = κ + .  ...

### On Gaps under GCH Type Assumptions [article]

M. Gitik
2000 arXiv   pre-print
This basically completes the study of consistency strength of various gaps between a strong limit singular cardinal of cofinality omega and its power under GCH type assumptions below.  ...  The results of the previous version are impoved.  ...  We are grateful to Saharon Shelah for many helpful conversations and for explanations that he gave on the pcf-theory.  ...

### Definable Singularity

William J. Mitchell
1991 Transactions of the American Mathematical Society
In the course of the proof we develop a simplified statement of the covering lemma for sequences of measures which avoids the use of mice.  ...  The main result of this paper is a characterization of singular cardinals in terms of the core model, assuming that there is no model of 3/c o(k) = k++ .  ...  Is there, under any large cardinal hypothesis, a class model M of ZFC which contains a singular cardinal k such that there is no witness in M to the singularity of k which is, in some reasonable sense,  ...

### Definable singularity

William J. Mitchell
1991 Transactions of the American Mathematical Society
In the course of the proof we develop a simplified statement of the covering lemma for sequences of measures which avoids the use of mice.  ...  The main result of this paper is a characterization of singular cardinals in terms of the core model, assuming that there is no model of 3/c o(k) = k++ .  ...  Is there, under any large cardinal hypothesis, a class model M of ZFC which contains a singular cardinal k such that there is no witness in M to the singularity of k which is, in some reasonable sense,  ...

### On possible non-homeomorphic substructures of the real line

P. D. Welch
2002 Proceedings of the American Mathematical Society
This requires large cardinals, and we obtain an exact consistency strength: Theorem 1.  ...  We consider the problem, raised by Kunen and Tall, of whether the real continuum can have non-homeomorphic versions in different submodels of the universe of all sets.  ...  To make c singular and Jónsson requires the consistency of larger cardinals. Theorem 2.6.  ...

### Exact saturation in pseudo-elementary classes for simple and stable theories [article]

Itay Kaplan, Nicholas Ramsey, Saharon Shelah
2022 arXiv   pre-print
PC-exact saturation at singular cardinals, satisfying mild set-theoretic hypotheses, which had previously been open even for the random graph.  ...  We also consider the local analogue of PC-exact saturation, showing that local PC-exact saturation for singular cardinals of countable cofinality characterizes supershort theories.  ...  For example if I is an indiscernible set, then the type omitted is that of a new element in the sequence. Question 6.2. Which NIP theories have local exact saturation at singular cardinals as above?  ...

### Exact saturation in simple and NIP theories [article]

Itay Kaplan, Saharon Shelah, Pierre Simon
2015 arXiv   pre-print
A theory T is said to have exact saturation at a singular cardinal κ if it has a κ-saturated model which is not κ^+-saturated.  ...  Also, an NIP theory has exact saturation if and only if it is not distal. This gives a new characterization of distality.  ...  Let d = (I 1 + J, I 2 ), which we identify with the corresponding cut in the extended sequence I ′ = I ∪ J.  ...

### Perfect trees and elementary embeddings

Sy-David Friedman, Katherine Thompson
2008 Journal of Symbolic Logic (JSL)
This technique is crucial both in the study of large cardinal preservation and of internal consistency.  ...  In easy cases, such as when forcing to make the GCH hold while preserving a measurable cardinal (via a reverse Easton iteration of α-Cohen forcing for successor cardinals α), the generic G* is simply generated  ...  By adding a Prikry sequence s through κ over V [G][g], we obtain a failure of the singular cardinal hypothesis, as in V [G][g][s], κ is a singular strong limit cardinal of cofinality ω and 2 κ = κ ++ .  ...

### The maximality of the core model [article]

Ernest Schimmerling, John R. Steel
1997 arXiv   pre-print
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.  ...  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 Determinacy holds. o If square-kappa-finite  ...  The research of the second author was partially supported by the National Science Foundation.  ...
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