TY - JOUR

T1 - Indiscernible sequences for extenders, and the singular cardinal hypothesis

AU - Gitik, Moti

AU - Mitchell, William J.

N1 - Funding Information:
* Corresponding author. E-mail: mitcheil@math.ufl.edu. 1S ome of the results were obtained while Gitik was visiting Los Angeles in Fail 1991. He would like to thank A. Kechris, D. Martin and J. Steel for their hospitality. 2 Mitchell was partially supported by grant number DMS-9240~6 from the National Science Foundation.

PY - 1996/12/15

Y1 - 1996/12/15

N2 - 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 K is a singular strong limit cardinal and 2K ≥ λ where λ is not the successor of a cardinal of cofinality at most K. If cf(K) > ω then it follows that ο(K) ≥ λ, and if cf(K) = ω then either ο(K) ≥ λ or {α : Κ l= ο(α)≥α+n} is confinal in K for each n ∈ ω. We also prove several results which extend or are related to this result, notably Theorem. If 2ω < Nω and 2Nω > Nω1 then there is a sharp for a model with a strong cardinal. In order to prove these theorems we give a detailed analysis of the sequences of indiscernibles which come from applying the covering lemma to nonoverlapping sequences of extenders.

AB - 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 K is a singular strong limit cardinal and 2K ≥ λ where λ is not the successor of a cardinal of cofinality at most K. If cf(K) > ω then it follows that ο(K) ≥ λ, and if cf(K) = ω then either ο(K) ≥ λ or {α : Κ l= ο(α)≥α+n} is confinal in K for each n ∈ ω. We also prove several results which extend or are related to this result, notably Theorem. If 2ω < Nω and 2Nω > Nω1 then there is a sharp for a model with a strong cardinal. In order to prove these theorems we give a detailed analysis of the sequences of indiscernibles which come from applying the covering lemma to nonoverlapping sequences of extenders.

UR - http://www.scopus.com/inward/record.url?scp=0030589734&partnerID=8YFLogxK

U2 - 10.1016/S0168-0072(96)00007-3

DO - 10.1016/S0168-0072(96)00007-3

M3 - מאמר

AN - SCOPUS:0030589734

VL - 82

SP - 273

EP - 316

JO - Annals of Pure and Applied Logic

JF - Annals of Pure and Applied Logic

SN - 0168-0072

IS - 3

ER -