The tree property and the failure of SCH at uncountable cofinality

Sinapova, Dima

doi:10.1007/s00153-012-0281-z

Cited by 6 publications

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“…She also proved similar result for a singular cardinal of uncountable cofinality [11]. In this section we prove Theorem 1.2, which extends the above results of Neeman and Sinapova.…”

Section: Tree Property and The Failure Of Schsupporting

confidence: 80%

“…The reason for giving such a proof is that, unlike the second proof, the method is more flexible and allows us to add the failure of SCH into our conclusion. To be more precise, we uss the method to prove the following theorem, which extends results of Neeman [6] and Sinapova [11], [12]. Theorem 1.2.…”

mentioning

confidence: 80%

“…We now define our main forcing construction. The forcing defined below is motivated from [9] and [11], where we refer to them for more details. The following lemma can be proved as in [9], Lemma 3.3 and [11], Proposition 2.…”

Section: Diagonal Magidor Forcing With Interleaved Collapsesmentioning

confidence: 99%

“…This is enough to show that the tree property is not sufficiently strong to separate the first limit of measurable cardinals from the first limit of strongly compact cardinals. The main advantage of the above theorem is that it works for all cofinalities and also the proofs of the theorem allow us to get more results about tree property.In this paper we will present two different proofs of the above theorem, the fist one uses diagonal Magidor forcing with interleaved collapses and the second one uses Levy collapses.The diagonal Magidor forcing with interleaved collapses was introduced by Sinapova [9] (see also [11]), where she used the forcing to get the failure of SCH at ℵ ω 2 1 together with the existence of a very good scale and a bad scale at ℵ ω 2 1 +1 . Our fist proof of Theorem 1.1 uses the above forcing notion and is much more complicated than the second proof which uses Levy collapses.…”

mentioning

confidence: 99%

“…The diagonal Magidor forcing with interleaved collapses was introduced by Sinapova [9] (see also [11]), where she used the forcing to get the failure of SCH at ℵ ω 2 1 together with the existence of a very good scale and a bad scale at ℵ ω 2 1 +1 . Our fist proof of Theorem 1.1 uses the above forcing notion and is much more complicated than the second proof which uses Levy collapses.…”

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confidence: 99%

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The tree property at the successor of a singular limit of measurable cardinals

Golshani

2017

Arch. Math. Logic

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Abstract. Assume λ is a singular limit of η supercompact cardinals, where η ≤ λ is a limit ordinal. We present two forcing methods for making λ + the successor of the limit of the first η measurable cardinals while the tree property holding at λ + . The first method is then used to get, from the same assumptions, tree property at ℵ η 2 +1 with the failure of SCH at ℵ η 2 . This extends results of Neeman and Sinapova. The second method is also used to get tree property at successor of an arbitrary singular cardinal, which extends some results of Magidor-Shelah, Neeman and Sinapova.

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“…She also proved similar result for a singular cardinal of uncountable cofinality [11]. In this section we prove Theorem 1.2, which extends the above results of Neeman and Sinapova.…”

Section: Tree Property and The Failure Of Schsupporting

confidence: 80%

mentioning

confidence: 80%

Section: Diagonal Magidor Forcing With Interleaved Collapsesmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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The tree property at the successor of a singular limit of measurable cardinals

Golshani

2017

Arch. Math. Logic

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The tree property at first and double successors of singular cardinals with an arbitrary gap

Poveda

2020

Annals of Pure and Applied Logic

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Let µ < κ be a regular cardinal and a supercompact cardinal, respectively. Assume that there is an increasing continuous sequence of cardinals κ ξ : ξ < µ with κ 0 = κ such that for all ξ < µ, κ ξ+1 is a supercompact cardinal. Furthermore, assume that there is a weakly compact cardinal λ > sup ξ<µ κ ξ . Let γ ≥ λ be a cardinal with cof(γ) > κ. Assuming the GCH, we construct a generic extension where κ is strong limit, cof(κ) = µ, 2 κ = γ and TP(κ + ) and TP(κ ++ ) hold. Further, in this model there is a very good and a bad scale at κ. This result generalizes the main theorems of [Sin16a] and [FHS18].

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