[Paper Review] Indiscernible Sequences for Extenders, and the Singular Cardinal Hypothesis

Moti Gitik, William Mitchell|arXiv (Cornell University)|Jul 27, 1995

Advanced Topology and Set Theory11 references4 citations

TL;DR

This paper establishes lower bounds on the large cardinal strength required for failures of the Singular Cardinal Hypothesis (SCH) by analyzing indiscernible sequences derived from extenders. Using a detailed covering lemma analysis of nonoverlapping extenders, it proves that if a singular strong limit cardinal κ satisfies 2^κ ≥ λ (with λ not a successor of a cardinal of cofinality ≤ κ), then either o(κ) ≥ λ or certain sets of cardinals with high order of reflection are cofinal in κ, depending on cofinality(κ).

ABSTRACT

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.

Motivation & Objective

• To determine the large cardinal strength required for a failure of the Singular Cardinal Hypothesis (SCH) at a singular strong limit cardinal κ.
• To investigate the consistency strength of 2^κ ≥ λ when λ is not a successor of a cardinal of cofinality ≤ κ.
• To analyze the structure of indiscernible sequences arising from nonoverlapping sequences of extenders via the covering lemma.
• To establish conditions under which the Mitchell order o(κ) must be at least λ or certain reflection properties hold in κ.
• To clarify the role of cofinality(κ) in determining the required large cardinal assumptions for SCH failure.

Proposed method

• Applies the covering lemma to nonoverlapping sequences of extenders to extract indiscernible sequences.
• Uses the structure of these indiscernible sequences to analyze the Mitchell order o(κ) at singular strong limit cardinals.
• Considers two cases based on cofinality(κ): uncountable cofinality and countable cofinality (ω).
• For cofinality(κ) > ω, proves that o(κ) ≥ λ is necessary if 2^κ ≥ λ.
• For cofinality(κ) = ω, shows that either o(κ) ≥ λ or the set {γ < κ : K ⊨ o(γ) ≥ γ^{+n}} is cofinal in κ for every n ∈ ω.
• Employs model-theoretic and inner model-theoretic techniques to derive reflection and indiscernibility properties from extender sequences.

Experimental results

Research questions

• RQ1 What is the minimum large cardinal strength required for the failure of the Singular Cardinal Hypothesis at a singular strong limit cardinal κ?
• RQ2 How does the cofinality of κ affect the necessary Mitchell order o(κ) when 2^κ ≥ λ?
• RQ3 Under what conditions does the existence of a cardinal λ with 2^κ ≥ λ and λ not a successor of a small cofinality force o(κ) ≥ λ?
• RQ4 When cofinality(κ) = ω, what alternative reflection properties must hold if o(κ) < λ?
• RQ5 How do indiscernible sequences derived from nonoverlapping extenders constrain the consistency strength of SCH failures?

Key findings

• If κ is a singular strong limit cardinal with cofinality(κ) > ω and 2^κ ≥ λ where λ is not a successor of a cardinal of cofinality ≤ κ, then o(κ) ≥ λ.
• If cofinality(κ) = ω and 2^κ ≥ λ under the same condition on λ, then either o(κ) ≥ λ or the set {γ < κ : K ⊨ o(γ) ≥ γ^{+n}} is cofinal in κ for every n ∈ ω.
• The analysis of indiscernible sequences from extenders provides a precise mechanism to derive lower bounds on the consistency strength of SCH failures.
• The results show that SCH failure at singular strong limit cardinals of uncountable cofinality requires at least λ-many Mitchell order at κ.
• For singular cardinals of cofinality ω, SCH failure implies a strong reflection property unless the Mitchell order is at least λ.
• The covering lemma is instrumental in connecting extender sequences to indiscernible structures and deriving large cardinal lower bounds.