Indiscernible Sequences for Extenders, and the Singular Cardinal Hypothesis

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 $\kappa$ is a singular strong limit cardinal and $2^\kappa >= \lambda$ where $\lambda$ is not the successor of a cardinal of cofinality at most $\kappa$. (i) If \cofinality(\kappa)>\gw then $o(\kappa)\ge\lambda$. (ii) If \cofinality(\kappa)=\gw then either $o(\kappa)\ge\lambda$ or \set{\ga:K\sat o(\ga)\ge\ga^{+n}} is cofinal in $\kappa$ 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

Logical Dreams

We discuss the past and future of set theory, axiom systems and independence results. We deal in particular with cardinal arithmetic

A general tool for consistency results related to I1

In this paper we provide a general tool to prove the consistency of $I1(\lambda)$ with various combinatorial properties at $\lambda$ typical at settings with $2^\lambda>\lambda^+$, that does not need a profound knowledge of the forcing notions involved. Examples of such properties are the first failure of GCH, a very good scale and the negation of the approachability property, or the tree property at $\lambda^+$ and $\lambda^{++}$

Stationary reflection principles and two cardinal tree properties

We study consequences of stationary and semi-stationary set reflection. We show that the semi stationary reflection principle implies the Singular Cardinal Hypothesis, the failure of weak square principle, etc. We also consider two cardinal tree properties introduced recently by Weiss and prove that they follow from stationary and semi stationary set reflection augmented with a weak form of Martin's Axiom. We also show that there are some differences between the two reflection principles which suggest that stationary set reflection is analogous to supercompactness whereas semi-stationary set reflection is analogous to strong compactness.Comment: 19 page