Further cardinal arithmetic

Shelah, Saharon

doi:10.1007/bf02761035

Cited by 43 publications

(31 citation statements)

References 30 publications

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“…(2) Note that our results are for µ = µ ℵ0 only; to remove this we need to improve the theorem on pp = cov (i.e. to prove cf(λ) = ℵ 0 < λ ⇒ pp(λ) = cov(λ, λ, ℵ 1 , 2) (or sup{pp(µ): cf µ = ℵ 0 < µ < λ} = cf(S ≤ℵ0 (λ), ⊆) (see [Sh-g], [Sh430,§1]), which seems to me a very serious open problem (see Analitic guide,14]). (4) We can approach 3.15 differently, by 3.20-3.23 below.…”

Section: Discussion 319mentioning

confidence: 99%

The PCF Theorem Revisited

Shelah

2013

The Mathematics of Paul Erdős II

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The pcf theorem (of the possible cofinability theory) was proved for reduced products i<κ λ i /I, where κ < min i<κ λ i . Here we prove this theorem under weaker assumptions such as wsat(I) < min i<κ λ i , where wsat(I) is the minimal θ such that κ cannot be delivered to θ sets / ∈ I (or even slightly weaker condition). We also look at the existence of exact upper bounds relative to < I (< I −eub) as well as cardinalities of reduced products and the cardinals T D (λ). Finally we apply this to the problem of the depth of ultraproducts (and reduced products) of Boolean algebras.

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Section: Discussion 319mentioning

confidence: 99%

The PCF Theorem Revisited

Shelah

2013

The Mathematics of Paul Erdős II

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“…Then λ < j(κ). By [Sh430], then in a generic ultrapower N cov(λ, ℵ 1 , ℵ 1 , 2) < j(κ). However, in V λ + ≤ cov(λ, κ, ℵ 1 , 2) ≤ cov(λ, ℵ 1 ℵ 1 , 2)) N < j(κ).…”

Section: Proofmentioning

confidence: 98%

More on real-valued measurable cardinals and forcing with ideals

Gitik

Shelah

2001

Isr. J. Math.

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1) It is shown that if c is real-valued measurable then the Maharam type of (c, P(c), σ) is 2 c . This answers a question of D. Fremlin [Fr,(P2f)].(2) A different construction of a model with a real-valued measurable cardinal is given from that of R. Solovay [So]. This answers a question of D. Fremlin [Fr,(P1)].(3) The forcing with a κ-complete ideal over a set X, |X| ≥ κ cannot be isomorphic to Random×Cohen or Cohen×Random. The result for X = κ was proved in [Gi-Sh1] but as was pointed out to us by M. Burke the application of it in [Gi-Sh2] requires dealing with any X.The last inequality holds since N is obtained by a c.c.c. forcing and so every countable set of ordinals in N can be covered by a countable set of V . By Shelah [Sh430]Fix a function f ∈ κ κ representing κ is a generic ultrapower and restrict everything to a condition forcing this.Proof: Otherwise, in a generic ultrapower N . j(S)↾κ = S will be bounded. I.e. there will be some t ⊆ κ countable such that for every s ∈ S s ⊇ t. Using c.c.c. of the forcing we find a countable subset of κ in V , t * ⊇ t. Since S is unbounded in V some s ∈ S contains t * . Contradiction. of the claim.Let N be a generic ultrapower. By [Gi-Sh1] there are in N at least κ Cohen (or random) reals over V .Claim 3. There exists a sequence r α | α < κ of reals in V so that(1) every real of V appears in r α | α < κ .(2) for almost all α(mod I) r α+i | i < f (α) are f (α)-Cohen (random) generic overProof: Construct r α | α < κ by induction. On nonlimit stages add reals in order to satisfy (1). For limit α ′ s with S↾f (α) unbounded in [f (α)] ≤ℵ 0 , add f (α)-Cohen (or random) reals. It is possible since there are at least κ candidates in a generic ultrapower by .of the claim.where r α | α < κ is a sequence given by Claim 3.Then, using Claim 3 in N we can find some α * < j(κ) satisfying (2) of Claim 3 such that j(S)↾α * is unbounded in [α * ] ≤ℵ 0 and j(f )(α * ) ≥ (λ + ) V . It is possible since by Claim 1, (λ + ) V < j(κ) and, in V the range of f restricted to a set not in I is unbounded in κ.The following will provide the contradiction and complete the proof of the theorem.

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“…An additional property that we need to show in the present situation is that d <ℵ 1 (κ) cannot be below λ. But this follows by [Sh430,5.3,5.4] and the pcf structure of the models of [Git-Mag] or just directly using the correspondence established in Lemma 2.9 between basic clopen sets of κ 2 of V [G] and κ n 2 of M n . Since already κ 1 2 cannot have a dense set of cardinality less than λ because λ + 2 embeds it and GCH holds.…”

Section: Some Generalizationsmentioning

confidence: 98%