Large cardinals and covering numbers

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.

“…Then ADS λ fails. (ii) (See [20].) Suppose that cf(λ) < κ and there is a cf(λ)-saturated, κ-complete ideal on P κ (λ).…”

Section: Scalesmentioning

confidence: 99%

“…85-86].) Suppose that [20].) If ρ 3 = cf(ρ 3 ) and ρ 4 ≥ ω, then either cf(cov(ρ 1 , ρ 2 , ρ 3 , ρ 4 )) < ρ 4 , or cf(cov(ρ 1 , ρ 2 ,…”

Section: Pcf Related Notionsmentioning

confidence: 99%

Guessing more sets

Annals of Pure and Applied Logic

2015

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“…Suppose that (a)

u (ϑ^{+}, ϱ) \leq ϱ^{+ +}

for every infinite cardinal

ϑ < ϱ

, and (b)

cov (ν, ν, ω_{1}, 2) = ν^{+}

for every cardinal ν such that

ϱ < ν < σ

and

cf (ν) = ω

. Then for any infinite cardinal

τ < σ

d (τ, σ) = \{\begin{matrix} right trueprefixmax (d (τ, τ), σ) & right if & right cf (σ) \neq cf (τ), \\ right trueprefixmax (d (τ, τ), σ^{+}) & right if & right cf (σ) = cf (τ) > ω, \\ right σ^{ℵ_{0}} & right if & right cf (σ) = cf (τ) = ω . \end{matrix}

Proof By (the proof of) [, Corollary 5.7], Facts and and Proposition .

□

Corollary Let ϱ and σ be two cardinals such that

ω < ϱ = ϱ^{< ϱ} \leq σ

, and ϱ is mildly

ν^{+}

‐ineffable for every cardinal ν such that

ϱ < ν < σ

and

cf (ν) = ω

.…”

Section: Density Numbersmentioning

confidence: 99%

Two‐cardinal diamond star

Mathematical Logic Qtrly

2014

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“…Recall that by Corollary 3.7, A κ,λ (π, μ) implies that μ ≤ u(κ, λ). Proposition 3.10 ( [14], Corollary 5.8) Let μ ≥ λ be a cardinal such that A κ,λ (κ, μ) holds. Then u(κ, λ) = u(κ, μ).…”

Section: Proposition 33mentioning

confidence: 99%

“…For a cardinal μ, A κ,λ (μ) asserts the existence of X ⊆ P κ (λ) with |X| = μ such that |X ∩ P (b)| < κ for all b ∈ P κ (λ). It is simple to see [14] that if κ is a successor cardinal and A κ,λ (μ) holds, then no ideal on P κ (λ) is weakly μ-saturated. The paper investigates the validity of A κ,λ (μ).…”

Section: Introductionmentioning

confidence: 99%

Weak saturation of ideals onP_κ(λ)

Mathematical Logic Quarterly

2011

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