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A finite family weak square principle

Published online by Cambridge University Press:  12 March 2014

Ernest Schimmerling
Ernest Schimmerling*
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA E-mail: [email protected]
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Definition 1.1. Suppose that λ ≤ κ are cardinals and Γ is a subset of (κ, κ+). By , we mean the principle asserting that there is a sequence 〈Fν | ν ∈ lim(Γ)〉 such that for every ν ∈ lim(Γ), the following hold.

(1) 1 ≤ card(Fν) < λ.

(2) The following hold for every C ∈ Fν.

(a) C ⊆ ν ∩ Γ,

(b) C is club in ν,

(c) o.t.(C) ≤ κ,

By we mean . If Γ = (κ, κ+), then we write for and for .

These weak square principles were introduced in [Sch2, 5.1]. They generalize Jensen's principles □κ and , which are equivalent to and respectively. Jensen's global □ principle implies □κ for all κ.

Theorem 1.2. Suppose that is a core model. Assume that every countable premouse M which elementarily embeds into a level of is (ω1 + 1)-iterable. Then, for every κ, holds in .

The minimal non-1-small mouse is essentially a sharp for an inner model with a Woodin cardinal. We originally proved Theorem 1.2 under the assumption that is 1-small, building on [MiSt] and [Sch2]. Some generalizations followed by combining our methods with those of [St2] and [SchSt2]. (For example, the tame countably certified core model Kc satisfies .) In order to eliminate the smallness assumption all together, one replaces our use of the Dodd-Jensen lemma in proofs of condensation properties for with the weak Dodd-Jensen lemma of [NSt].

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 64 , Issue 3 , September 1999 , pp. 1087 - 1110
Copyright
Copyright © Association for Symbolic Logic 1999

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References

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