Weak saturation of ideals on P κ (λ)

Annals of Pure and Applied Logic

2015

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Let κ be a regular uncountable cardinal, and λ a cardinal greater than κ with cofinality less than κ. We consider a strengthening of the diamond principle ♦ κ,λ that asserts that any subset of some fixed collection of λ + elements of P κ (λ) can be guessed on a stationary set. This new principle, denoted by ♦ κ,λ [λ + ], implies that the nonstationary ideal on P κ (λ) is not 2 (λ + ) -saturated. We establish that if λ is large enough and there are no inner models with fairly large cardinals, then ♦ κ,λ [λ + ] holds. More precisely, it is shown that if 2 (κ ℵ 0 ) ≤ λ + and both Shelah's Strong Hypothesis SSH and the Almost Disjoint Sets principle ADS λ hold, then ♦ κ,λ [λ + ] holds. The paper also contains ZFC results. Suppose for example that 2 κ ≤ λ + , there is a strong limit cardinal τ with cf(λ) < τ ≤ κ, and either κ is a successor cardinal greater than ρ +3 , where ρ is the largest limit cardinal less than κ, or κ is a limit cardinal and σ κ < λ < (σ κ ) +κ for some cardinal σ ≥ 2. Then ♦ κ,λ [λ + ] holds.

“…More negative results concerning A κ,λ (τ, π) and B κ,λ (τ, π) can be found in [22] and [25]. Let us now concentrate on ADS λ .…”

Section: Scalesmentioning

confidence: 92%

“…Now suppose to the contrary that σ < κ. Then by Lemma 4.11 of [22], cf(σ) ≤ τ . Furthermore by Observation 2.12, d(τ, σ) ≥ κ.…”

Section: Scalesmentioning

confidence: 95%

See 1 more Smart Citation

Guessing more sets

Annals of Pure and Applied Logic

2015

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“…The sequence

\overset{f}{true}

is friendly if for any cardinal σ with

| A | < σ < sup (A)

, and any

α < π

with

cf (α) = σ^{+ +}

, there is an order type

σ^{+ +}

closed unbounded subset X of α such that

sup ({f_{β} : β \in X}) \leq_{I} f_{α}

. Fact (Matet ) Let

\vec{f} = ⟨ f_{α} : α < π ⟩

be a friendly, increasing sequence in

(Π A, <_{I})

, σ be a cardinal with

| A | < σ < sup (A)

, and Z be a subset of π of order type

σ^{+ 3}

. Then there is an order type σ subset X of Z such that the sequence

⟨ f_{α} : α \in X ⟩

is strongly increasing.…”

Section: Basic Materialsmentioning

confidence: 99%

Two‐cardinal diamond star

Mathematical Logic Qtrly

2014

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“…Set χ = u(κ, ν) if ν < κ +κ , and χ = ν + otherwise. Then by results of [14] and [13], there is A ∈ NS + κ,χ such that cof κ (I κ,χ |A) ≤ ν. It is simple to see that if B and f are in Proposition 2.6 (ii), then for any cardinal τ , I κ,ν is (weakly) τ -saturated if and only if I κ,χ |B is (weakly) τ -saturated.…”

Section: Proposition 26mentioning

confidence: 99%

Non‐saturation of the non‐stationary ideal onP_κ(λ) with λ of countable cofinality

Mathematical Logic Qtrly

2012

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MSC (2010) 03E05Given a regular uncountable cardinal κ and a cardinal λ > κ of cofinality ω, we show that the restriction of the non-stationary ideal on Pκ (λ) to the set of all a with cf(sup(a ∩ κ)) = ω is not λ + + -saturated and even not 2 (λ < κ ) -saturated in case 2 λ = λ + . We actually prove the stronger result that there is Q ⊆ NG + κ ,λ with |Q| = λ + + such that A ∩ B is a non-cofinal subset of Pκ (λ) for any two distinct members A, B of Q, where NG κ ,λ denotes the game ideal on Pκ (λ). We also remark that for κ > ω1 , adding λ + 3 Cohen subsets of ω1 to L makes NG κ ,λ λ + 3 -saturated.