Easton collapses and a strongly saturated filter
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Mathematical Logic
Easton collapses and a strongly saturated filter Masahiro Shioya1 Received: 3 August 2016 / Accepted: 15 April 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We introduce the Easton collapse and show that the two-stage iteration of Easton collapses gives a model in which the successor of a regular cardinal carries a strongly saturated filter. This allows one to get a model in which many successor cardinals carry saturated filters just by iterating Easton collapses. Keywords Easton collapse · Strongly saturated filter · Almost huge cardinal Mathematics Subject Classification 03E05 · 03E35 · 03E55
1 Introduction In [10] Laver established the following Theorem (Laver [10]) Suppose κ is huge and μ < κ is regular. Then there is a forcing extension in which κ = μ+ carries a strongly saturated normal filter. See §2 for the definition of a strongly saturated filter. Laver used a poset of the ˙ ˙ form P ∗ L(κ, j(κ)), where j : V → M witnesses that κ is huge and L(κ, j(κ)) is the P-name for the Laver collapse. The poset P is constructed recursively so that P ˙ j(κ)) can be completely embedded into j(P). The forces κ = μ+ and that P ∗ L(κ, construction of P is essentially due to Kunen [9], who called it the universal collapse. Kunen’s method has been useful in constructing models with a saturated filter on the successor of a regular cardinal. Moreover, it is possible to iterate Kunen’s method and to get a model in which many successor cardinals carry saturated filters. However, this requires a rather complicated construction of the universal collapse. See [6] for a comprehensive survey of Kunen’s method. In this paper we introduce the Easton collapse and show that the two-stage iteration of Easton collapses gives a model with a strongly saturated filter. More specifically, we prove
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Masahiro Shioya [email protected] Institute of Mathematics, University of Tsukuba, Tsukuba 305-8571, Japan
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M. Shioya
Theorem 1 Suppose κ is almost huge, as witnessed by j : V → M, and j(κ) is Mahlo. Let μ < κ ≤ λ < j(κ) with μ and λ both regular. Then the iteration ˙ E(μ, κ) ∗ E(λ, j(κ)) of Easton collapses forces that κ = μ+ , j(κ) = λ+ and Pκ λ carries a strongly saturated normal filter. The key claim of the proof is that j(E(μ, κ)) can be projected onto E(μ, κ) ∗ ˙ E(λ, j(κ)) in some nice way (see Proposition 1). Strong saturation is not the strongest saturation property that a normal filter on Pκ λ (with κ the successor of a regular cardinal) can have. Extending a result of Woodin (see [6]), Eskew [2] constructed a model in which Pκ λ carries a λ-dense normal filter. It is not known, however, how to iterate Eskew’s construction. On the contrary, Theorem 1 (in the special case where κ = λ) allows one to get a model in which many successor cardinals carry saturated filters just by iterating Easton collapses. This in turn gives a simplified proof of Theorem (Foreman [5]) Suppose κ is huge. Then there is a forcing extension in which κ = ωω and every successor cardinal ≤ ωω+1 ca
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