2010 Mathematics Subject Classification. 03E02, 03E04, 03E55.
The research was supported by Israel Science Foundation Grant no. 1838/19 and Grant no. 2320/23. The second author’s research was supported by the Israel Science Foundation Grant no. 1963/21. This is publication 1249 (the 204th prime number) of the third author, publication 47 (the 15th prime number) of the first author and publication 37 (the 12th prime number) of the second author.
Introduction
Let G = ( V , E ) G=(V,E) G = ( V , E ) λ \lambda λ c : E → θ c:E\to\theta c : E → θ W ⊆ V W\subseteq V W ⊆ V independent if [ W ] 2 ∩ E = ∅ [W]^2\cap E=\varnothing [ W ] 2 ∩ E = ∅ G G G λ \lambda λ λ \lambda λ G G G
The above discussion can be formulated in the language of partition calculus, without mentioning graphs at all. The ordinary partition relation λ → ( κ , ( 3 ) θ ) 2 \lambda\to(\kappa,(3)_\theta)^2 λ → ( κ , ( 3 ) θ ) 2 c : [ λ ] 2 → θ c:[\lambda]^2\to\theta c : [ λ ] 2 → θ A ∈ [ λ ] κ A\in[\lambda]^\kappa A ∈ [ λ ] κ c ′ ′ [ A ] 2 = { 0 } c^{\prime\prime}[A]^2=\{0\} c ′′ [ A ] 2 = { 0 } B ∈ [ λ ] 3 B\in[\lambda]^3 B ∈ [ λ ] 3 γ ∈ ( 0 , θ ) \gamma\in(0,\theta) γ ∈ ( 0 , θ ) c ′ ′ [ B ] 2 = { γ } c^{\prime\prime}[B]^2=\{\gamma\} c ′′ [ B ] 2 = { γ } κ = λ \kappa=\lambda κ = λ γ ∈ ( 0 , θ ) \gamma\in(0,\theta) γ ∈ ( 0 , θ ) λ → ( λ , ( 3 ) θ ) 2 \lambda\to(\lambda,(3)_\theta)^2 λ → ( λ , ( 3 ) θ ) 2 λ \lambda λ
Erdős, Hajnal and Rado investigated this relation in [10 ] . They established several results, and focused in particular on graphs whose size is a successor of a singular cardinal. A good account appears in the monograph [EHMR84], in which the following is phrased and proved:
Theorem 0.1. Assume that λ \lambda λ 2 λ = λ + 2^\lambda=\lambda^+ 2 λ = λ + λ + ↛ ( λ + , ( 3 ) cf ( λ ) ) 2 \lambda^+\nrightarrow(\lambda^+,(3)_{\operatorname{cf}(\lambda)})^2 λ + ↛ ( λ + , ( 3 ) cf ( λ ) ) 2
Actually, they proved something a bit stronger, see [EHMR84], Theorem 20.2. A natural question is whether the assumption 2 λ = λ + 2^\lambda=\lambda^+ 2 λ = λ + 2 cf ( λ ) ≥ λ + 2^{\operatorname{cf}(\lambda)}\geq\lambda^+ 2 cf ( λ ) ≥ λ + λ + ↛ ( λ + , ( 3 ) cf ( λ ) ) 2 \lambda^+\nrightarrow(\lambda^+,(3)_{\operatorname{cf}(\lambda)})^2 λ + ↛ ( λ + , ( 3 ) cf ( λ ) ) 2 2 cf ( λ ) < λ 2^{\operatorname{cf}(\lambda)}<\lambda 2 cf ( λ ) < λ λ \lambda λ λ = ℵ ω \lambda=\aleph_\omega λ = ℵ ω [8 ] , Problem 5, the pertinent question appears as follows:
Question 0.2. Can one prove without assuming GCH that the relation ℵ ω + 1 ↛ ( ℵ ω + 1 , ( 3 ) ℵ 0 ) 2 \aleph_{\omega+1}\nrightarrow(\aleph_{\omega+1},(3)_{\aleph_0})^2 ℵ ω + 1 ↛ ( ℵ ω + 1 , ( 3 ) ℵ 0 ) 2
It appeared, again, in [EHMR84], Problem 20.1.1 λ + ↛ ( λ + , ( 3 ) cf ( λ ) ) 2 \lambda^+ \nrightarrow(\lambda^+,(3)_{\operatorname{cf}(\lambda)})^2 λ + ↛ ( λ + , ( 3 ) cf ( λ ) ) 2 2 λ > λ + 2^\lambda> \lambda^+ 2 λ > λ + λ \lambda λ
Our strategy is to replace the hypothesis 2 λ = λ + 2^\lambda= \lambda^+ 2 λ = λ + λ \lambda λ λ + \lambda^+ λ + λ \lambda λ λ + \lambda^+ λ + 2 λ > λ + 2^\lambda> \lambda^+ 2 λ > λ +
This approach is rendered here twice. Firstly, we get the negative arrow relation λ i + ↛ ( λ i + , ( 3 ) cf ( λ i ) ) 2 \lambda_i^+ \nrightarrow(\lambda_i^+,(3)_{\operatorname{cf}(\lambda_i)})^2 λ i + ↛ ( λ i + , ( 3 ) cf ( λ i ) ) 2 λ i \lambda_i λ i λ \lambda λ λ + \lambda^+ λ + λ \lambda λ λ \lambda λ λ \lambda λ ℵ ω \aleph_\omega ℵ ω λ i \lambda_i λ i ℵ n \aleph_n ℵ n ℵ ω \aleph_\omega ℵ ω
There is another possible approach towards the same negative arrow relation. This approach is based on prediction principles from the tiltan family. Suppose that diamond holds at λ + \lambda^+ λ + 2 λ = λ + 2^\lambda= \lambda^+ 2 λ = λ + ♣ ( λ ) \clubsuit(\lambda) ♣ ( λ ) ( A α ∣ α ∈ λ + ) (A_\alpha\mid\alpha\in\lambda^+) ( A α ∣ α ∈ λ + ) [ λ + ] λ [\lambda^+]^\lambda [ λ + ] λ A ∈ [ λ + ] λ + A \in[\lambda^+]^{\lambda^+} A ∈ [ λ + ] λ + A α ⊆ A A_\alpha\subseteq A A α ⊆ A λ + \lambda^+ λ + 2 λ > λ + 2^\lambda> \lambda^+ 2 λ > λ + ♣ ( λ ) \clubsuit(\lambda) ♣ ( λ ) λ + ↛ ( λ + , ( 3 ) cf ( λ ) ) 2 \lambda^+ \nrightarrow(\lambda^+,(3)_{\operatorname{cf}(\lambda)})^2 λ + ↛ ( λ + , ( 3 ) cf ( λ ) ) 2 ♣ ( λ ) \clubsuit(\lambda) ♣ ( λ ) 2 λ > λ + 2^\lambda> \lambda^+ 2 λ > λ + λ \lambda λ ♣ ( λ ) \clubsuit(\lambda) ♣ ( λ ) λ \lambda λ
The rest of the paper contains three additional sections, and an appendix. In the first section we solve the general problem using simple arguments of pcf theory. As mentioned above, this solution does not cover the case of ℵ ω \aleph_\omega ℵ ω ℵ ω \aleph_\omega ℵ ω λ + \lambda^+ λ +
Our notation is coherent with [EHM R84]. We shall use the Jerusalem forcing notation, namely we force upwards. A function f : E → P ( E ) f:E\to\mathcal{P}(E) f : E → P ( E ) x ∉ f ( x ) x \notin f(x) x ∈ / f ( x ) x ∈ E x \in E x ∈ E X ⊆ E X \subseteq E X ⊆ E f f f f ( y ) ∩ X = ∅ f(y) \cap X = \varnothing f ( y ) ∩ X = ∅ y ∈ X y \in X y ∈ X κ = cf ( κ ) < λ \kappa= \operatorname{cf}(\kappa) < \lambda κ = cf ( κ ) < λ S κ λ = { δ ∈ λ ∣ cf ( δ ) = κ } S^\lambda_\kappa= \{\delta\in\lambda\mid\operatorname{cf}(\delta) = \kappa\} S κ λ = { δ ∈ λ ∣ cf ( δ ) = κ } ℵ 0 < cf ( λ ) \aleph_0 < \operatorname{cf}(\lambda) ℵ 0 < cf ( λ ) S κ λ S^\lambda_\kappa S κ λ λ \lambda λ κ \kappa κ γ \gamma γ j : V → M j : V \to M j : V → M κ = crit ( j ) \kappa= \operatorname{crit}(j) κ = crit ( j ) γ M ⊆ M {}^{\gamma}M \subseteq M γ M ⊆ M κ \kappa κ κ \kappa κ γ \gamma γ γ \gamma γ
Supercompactness can be characterized by the following (global) property. Suppose that κ ≤ λ \kappa\leq\lambda κ ≤ λ U \mathcal{U} U P κ λ \mathcal{P}_\kappa\lambda P κ λ { x ∈ P κ λ ∣ α ∈ x } ∈ U \{x \in\mathcal{P}_\kappa\lambda\mid\alpha\in x\} \in\mathcal{U} { x ∈ P κ λ ∣ α ∈ x } ∈ U α ∈ λ \alpha\in\lambda α ∈ λ U \mathcal{U} U ∈ \in ∈ A A A U \mathcal{U} U B ⊆ A B \subseteq A B ⊆ A B ∈ U B \in\mathcal{U} B ∈ U κ \kappa κ λ \lambda λ P κ λ \mathcal{P}_\kappa\lambda P κ λ κ \kappa κ κ \kappa κ λ \lambda λ λ ≥ κ \lambda\geq\kappa λ ≥ κ
We shall use the idea of indestructibility (at supercompact cardinals) as appeared in the seminal work of Laver, [20 ] . It is shown there that a supercompact cardinal κ \kappa κ κ \kappa κ
For basic background concerning Prikry type forcings we refer to [13 ] , and to the papers of Magidor [21 ] and [22 ] in which the basic method of Prikry forcing with interleaved collapses was introduced. We also refer to [Hay23] in this context. For background in pcf theory we suggest [1 ] , [4 ] and [She94].
A negative relation from simple pcf arguments
In this section we suggest our first approach for proving the negative relation λ + ↛ ( λ + , ( 3 ) cf ( λ ) ) 2 \lambda^{+} \nrightarrow(\lambda^{+},(3)_{\operatorname{cf}(\lambda)})^{2} λ + ↛ ( λ + , ( 3 ) cf ( λ ) ) 2 λ \lambda λ 2 λ > λ + 2^{\lambda} > \lambda^{+} 2 λ > λ + λ \lambda λ λ + \lambda^{+} λ +
Let J \mathcal{J} J θ \theta θ ( λ i ∣ i ∈ θ ) (\lambda_i \mid i \in\theta) ( λ i ∣ i ∈ θ ) ( f α ∣ α ∈ μ ) (f_{\alpha} \mid\alpha\in\mu) ( f α ∣ α ∈ μ ) ∏ i ∈ θ λ i \prod_{i \in\theta}\lambda_i ∏ i ∈ θ λ i J \mathcal{J} J { i ∈ θ ∣ f α ( i ) ≥ f β ( i ) } ∈ J \{i \in\theta\mid f_{\alpha}(i) \ge f_{\beta}(i)\} \in\mathcal{J} { i ∈ θ ∣ f α ( i ) ≥ f β ( i )} ∈ J α < β < μ \alpha< \beta< \mu α < β < μ g ∈ ∏ i ∈ θ λ i g \in\prod_{i \in\theta}\lambda_i g ∈ ∏ i ∈ θ λ i α ∈ μ \alpha\in\mu α ∈ μ g ≤ J f α g \le_{\mathcal{J}} f_{\alpha} g ≤ J f α J \mathcal{J} J θ \theta θ J θ b d \mathcal{J}^{\mathrm{bd}}_{\theta} J θ bd tcf ( ∏ i ∈ θ λ i , J ) = κ \operatorname{tcf}(\prod_{i \in\theta}\lambda_i,\mathcal{J}) = \kappa tcf ( ∏ i ∈ θ λ i , J ) = κ J \mathcal{J} J ∏ i ∈ θ λ i \prod_{i \in\theta}\lambda_i ∏ i ∈ θ λ i κ \kappa κ
Theorem 1.1. Assume that:
(a) μ > cf ( μ ) = θ \mu> \operatorname{cf}(\mu) = \theta μ > cf ( μ ) = θ
(b) μ \mu μ 2
(c) 2 μ > μ + 2^{\mu} > \mu^{+} 2 μ > μ +
(d) ( μ i ∣ i ∈ θ ) (\mu_i \mid i \in\theta) ( μ i ∣ i ∈ θ ) μ = ⋃ i ∈ θ μ i \mu= \bigcup_{i \in\theta}\mu_i μ = ⋃ i ∈ θ μ i
(e) μ i > cf ( μ i ) = θ \mu_i > \operatorname{cf}(\mu_i) = \theta μ i > cf ( μ i ) = θ i ∈ θ i \in\theta i ∈ θ
(f) μ i \mu_i μ i i ∈ θ i \in\theta i ∈ θ
(g) 2 μ i = μ i + 2^{\mu_i} = \mu_i^{+} 2 μ i = μ i + i ∈ θ i \in\theta i ∈ θ
(h) tcf ( ∏ i ∈ θ μ i + , J θ b d ) = μ + \operatorname{tcf}(\prod_{i \in\theta}\mu_i^{+},\mathcal{J}^{\mathrm{bd}}_{\theta}) = \mu^{+} tcf ( ∏ i ∈ θ μ i + , J θ bd ) = μ +
Then μ + ↛ ( μ + , ( 3 ) cf ( μ ) ) 2 \mu^{+} \nrightarrow(\mu^{+},(3)_{\operatorname{cf}(\mu)})^{2} μ + ↛ ( μ + , ( 3 ) cf ( μ ) ) 2
Proof. For every i ∈ θ i \in\theta i ∈ θ c i : [ μ i + ] 2 → θ c_i : [\mu_i^{+}]^{2} \to\theta c i : [ μ i + ] 2 → θ μ i + ↛ ( μ i + , ( 3 ) θ ) 2 \mu_i^{+} \nrightarrow(\mu_i^{+},(3)_{\theta})^{2} μ i + ↛ ( μ i + , ( 3 ) θ ) 2 c : [ μ + ] 2 → θ c : [\mu^{+}]^{2} \to\theta c : [ μ + ] 2 → θ c i c_i c i μ + \mu^{+} μ +
We need two mathematical objects to define our coloring. The first is a scale ( f α ∣ α ∈ μ + ) (f_{\alpha} \mid\alpha\in\mu^{+}) ( f α ∣ α ∈ μ + ) ( ∏ i ∈ θ μ i + , J θ b d ) (\prod_{i \in\theta}\mu_i^{+},\mathcal{J}^{\mathrm{bd}}_{\theta}) ( ∏ i ∈ θ μ i + , J θ bd ) ( h i ∣ i ∈ θ ) (h_i \mid i \in\theta) ( h i ∣ i ∈ θ ) h i ∈ θ θ h_i \in{}^{\theta}\theta h i ∈ θ θ h i ( 0 ) = 0 h_i(0) = 0 h i ( 0 ) = 0 i ∈ θ i \in\theta i ∈ θ i < j < θ i < j < \theta i < j < θ rang ( h i ) ∩ rang ( h j ) = { 0 } \operatorname{rang}(h_i) \cap\operatorname{rang}(h_j) = \{0\} rang ( h i ) ∩ rang ( h j ) = { 0 } α < β < μ + \alpha< \beta< \mu^{+} α < β < μ + i ( α , β ) i(\alpha,\beta) i ( α , β ) j ∈ θ j \in\theta j ∈ θ f α ( j ) ≠ f β ( j ) f_{\alpha}(j) \ne f_{\beta}(j) f α ( j ) = f β ( j ) f α < J θ b d f β f_{\alpha} <_{\mathcal{J}^{\mathrm{bd}}_{\theta}} f_{\beta} f α < J θ bd f β c : [ μ + ] 2 → θ c : [\mu^{+}]^{2} \to\theta c : [ μ + ] 2 → θ α < β < μ + \alpha< \beta< \mu^{+} α < β < μ + i = i ( α , β ) i = i(\alpha,\beta) i = i ( α , β ) c ( α , β ) = h i ( c i ( { f α ( i ) , f β ( i ) } ) ) c(\alpha,\beta) = h_i(c_i(\{f_{\alpha}(i),f_{\beta}(i)\})) c ( α , β ) = h i ( c i ({ f α ( i ) , f β ( i )})) 3 c c c μ + ↛ ( μ + , ( 3 ) cf ( μ ) ) 2 \mu^{+} \nrightarrow(\mu^{+},(3)_{\operatorname{cf}(\mu)})^{2} μ + ↛ ( μ + , ( 3 ) cf ( μ ) ) 2 A ∈ [ μ + ] μ + A \in[\mu^{+}]^{\mu^{+}} A ∈ [ μ + ] μ + i ∈ θ i \in\theta i ∈ θ A i = { f α ( i ) ∣ α ∈ A } A_i=\{f_\alpha(i)\mid\alpha\in A\} A i = { f α ( i ) ∣ α ∈ A } X = { i ∈ θ ∣ μ i + = ⋃ A i } X=\{i\in\theta\mid\mu_i^{+}=\bigcup A_i\} X = { i ∈ θ ∣ μ i + = ⋃ A i } X = θ m o d J θ b d X=\theta\mod J_{\theta}^{\mathrm{bd}} X = θ mod J θ bd i ∈ X i\in X i ∈ X ε ∈ A i \varepsilon\in A_i ε ∈ A i α ε ∈ A \alpha_\varepsilon\in A α ε ∈ A ε < ζ \varepsilon<\zeta ε < ζ α ε < α ζ \alpha_\varepsilon<\alpha_\zeta α ε < α ζ i ( α ε , α ζ ) = i i(\alpha_\varepsilon,\alpha_\zeta)=i i ( α ε , α ζ ) = i A i A_i A i μ i + \mu_i^{+} μ i + ∣ ∏ j ∈ i μ j + ∣ < μ i + \left|\prod_{j\in i}\mu_j^{+}\right|<\mu_i^{+} ∏ j ∈ i μ j + < μ i + B i ∈ [ μ i + ] μ i + B_i\in[\mu_i^{+}]^{\mu_i^{+}} B i ∈ [ μ i + ] μ i + g ∈ ∏ j ∈ i μ j + g\in\prod_{j\in i}\mu_j^{+} g ∈ ∏ j ∈ i μ j + ε < ζ \varepsilon<\zeta ε < ζ B i B_i B i α ε < α ζ \alpha_\varepsilon<\alpha_\zeta α ε < α ζ f α ε ↾ i = g f_{\alpha_\varepsilon}\mathbin{\upharpoonright}i=g f α ε ↾ i = g ε ∈ B i \varepsilon\in B_i ε ∈ B i c i c_i c i μ i + ↛ ( μ i + , ( 3 ) θ ) 2 \mu_i^{+}\nrightarrow(\mu_i^{+},(3)_\theta)^2 μ i + ↛ ( μ i + , ( 3 ) θ ) 2 ε , ζ ∈ B i \varepsilon,\zeta\in B_i ε , ζ ∈ B i ε < ζ \varepsilon<\zeta ε < ζ c i ( ε , ζ ) ≠ 0 c_i(\varepsilon,\zeta)\ne0 c i ( ε , ζ ) = 0 c ( α ε , α ζ ) ≠ 0 c(\alpha_\varepsilon,\alpha_\zeta)\ne0 c ( α ε , α ζ ) = 0
(ℶ) Assume that α < β < γ < μ + \alpha<\beta<\gamma<\mu^{+} α < β < γ < μ + i ( α , β ) ≠ i ( α , γ ) i(\alpha,\beta)\ne i(\alpha,\gamma) i ( α , β ) = i ( α , γ ) i ( α , β ) ≠ i ( β , γ ) i(\alpha,\beta)\ne i(\beta,\gamma) i ( α , β ) = i ( β , γ ) i ( α , γ ) ≠ i ( β , γ ) i(\alpha,\gamma)\ne i(\beta,\gamma) i ( α , γ ) = i ( β , γ ) { α , β , γ } \{\alpha,\beta,\gamma\} { α , β , γ } c c c ξ > 0 \xi>0 ξ > 0 i ≠ j i\ne j i = j rang ( h i ) ∩ rang ( h j ) = { 0 } \operatorname{rang}(h_i)\cap\operatorname{rang}(h_j)=\{0\} rang ( h i ) ∩ rang ( h j ) = { 0 } c c c i ( α , β ) = i ( α , γ ) = i ( β , γ ) = i i(\alpha,\beta)=i(\alpha,\gamma)=i(\beta,\gamma)=i i ( α , β ) = i ( α , γ ) = i ( β , γ ) = i c ↾ [ { α , β , γ } ] 2 = { ξ } c\mathbin{\upharpoonright}[\{\alpha,\beta,\gamma\}]^2=\{\xi\} c ↾ [{ α , β , γ } ] 2 = { ξ } ξ > 0 \xi>0 ξ > 0 c i ↾ [ { f α ( i ) , f β ( i ) , f γ ( i ) } ] 2 = { ξ } c_i\mathbin{\upharpoonright}[\{f_\alpha(i),f_\beta(i),f_\gamma(i)\}]^2=\{\xi\} c i ↾ [{ f α ( i ) , f β ( i ) , f γ ( i )} ] 2 = { ξ } h i h_i h i c i c_i c i
□ 1.1 \square_{1.1} □ 1.1
A corollary to the above theorem gives a partial answer to the question of Erdős and Hajnal, by proving the consistency of the negative relation even if 2 μ > μ + 2^\mu>\mu^{+} 2 μ > μ + Q ˉ μ \bar{\mathbb{Q}}_\mu Q ˉ μ [16 Definition 2.3] . For being self-contained, we unfold the definition of this forcing notion.
Let μ \mu μ μ ˉ = ( μ i ∣ i ∈ μ ) \bar{\mu}=(\mu_i\mid i\in\mu) μ ˉ = ( μ i ∣ i ∈ μ ) 2 ∣ i ∣ < μ i 2^{|i|}<\mu_i 2 ∣ i ∣ < μ i i ∈ μ i\in\mu i ∈ μ p ∈ Q μ ˉ p\in\mathbb{Q}_{\bar{\mu}} p ∈ Q μ ˉ ( η , f ) = ( η p , f p ) (\eta,f)=(\eta^p,f^p) ( η , f ) = ( η p , f p ) ℓ g ( η ) ∈ μ \ell g(\eta)\in\mu ℓ g ( η ) ∈ μ η ∈ ∏ i ∈ ℓ g ( η ) μ i \eta\in\prod_{i\in\ell g(\eta)}\mu_i η ∈ ∏ i ∈ ℓ g ( η ) μ i η \eta η p p p f ∈ ∏ i ∈ μ μ i f\in\prod_{i\in\mu}\mu_i f ∈ ∏ i ∈ μ μ i η ⊲ f \eta\lhd f η ⊲ f p , q ∈ Q μ ˉ p,q\in\mathbb{Q}_{\bar{\mu}} p , q ∈ Q μ ˉ p ≤ q p\le q p ≤ q η p ⊴ η q \eta^p\unlhd\eta^q η p ⊴ η q f p ( j ) ≤ f q ( j ) f^p(j)\le f^q(j) f p ( j ) ≤ f q ( j ) j ∈ μ j\in\mu j ∈ μ 4
Intuitively, Q μ ˉ \mathbb{Q}_{\bar{\mu}} Q μ ˉ h ∈ ∏ i ∈ μ μ i h\in\prod_{i\in\mu}\mu_i h ∈ ∏ i ∈ μ μ i 2 μ = μ + 2^\mu=\mu^{+} 2 μ = μ + Q μ ˉ \mathbb{Q}_{\bar{\mu}} Q μ ˉ μ + \mu^{+} μ + Q μ ˉ \mathbb{Q}_{\bar{\mu}} Q μ ˉ ( < μ ) (<\mu) ( < μ ) Q μ ˉ \mathbb{Q}_{\bar{\mu}} Q μ ˉ μ \mu μ [16 ] . If Υ = cf ( Υ ) > μ \Upsilon=\operatorname{cf}(\Upsilon)>\mu Υ = cf ( Υ ) > μ μ \mu μ tcf ( ∏ i ∈ cf ( μ ) μ ρ i , J cf ( μ ) b d ) = Υ \operatorname{tcf}\left(\prod_{i\in\operatorname{cf}(\mu)}\mu_{\rho_i},J_{\operatorname{cf}(\mu)}^{\mathrm{bd}}\right)=\Upsilon tcf ( ∏ i ∈ cf ( μ ) μ ρ i , J cf ( μ ) bd ) = Υ ( ρ i ∣ i ∈ cf ( μ ) ) (\rho_i\mid i\in\operatorname{cf}(\mu)) ( ρ i ∣ i ∈ cf ( μ )) μ \mu μ [16 Lemma 3.1 and Remark 3.2] .
Corollary 1.2. Assuming the existence of a supercompact cardinal in the ground model, one can force μ + ↛ ( μ + , ( 3 ) cf ( μ ) ) 2 \mu^{+} \nrightarrow(\mu^{+},(3)_{\operatorname{cf}(\mu)})^{2} μ + ↛ ( μ + , ( 3 ) cf ( μ ) ) 2 2 μ > μ + 2^{\mu} > \mu^{+} 2 μ > μ + μ \mu μ
Proof. Our goal is to force the assumptions of Theorem 1.1. Let μ \mu μ ℵ 0 ≤ θ ∈ μ \aleph_{0} \leq\theta\in\mu ℵ 0 ≤ θ ∈ μ μ \mu μ Q μ ˉ \mathbb{Q}_{\bar{\mu}} Q μ ˉ μ \mu μ ( μ i ∣ i ∈ μ ) (\mu_i \mid i \in\mu) ( μ i ∣ i ∈ μ ) cf ( μ i ) = θ \operatorname{cf}(\mu_i) = \theta cf ( μ i ) = θ i ∈ μ i \in\mu i ∈ μ μ = ⋃ i ∈ μ μ i \mu= \bigcup_{i \in\mu} \mu_i μ = ⋃ i ∈ μ μ i 2 μ i = μ i + < μ i + 1 2^{\mu_i} = \mu_i^{+} < \mu_{i+1} 2 μ i = μ i + < μ i + 1 i ∈ μ i \in\mu i ∈ μ 5
We force with Q μ ˉ \mathbb{Q}_{\bar{\mu}} Q μ ˉ θ = cf ( μ ) \theta= \operatorname{cf}(\mu) θ = cf ( μ ) tcf ( ∏ i ∈ θ μ i + , J θ b d ) = μ + \operatorname{tcf}(\prod_{i \in\theta} \mu_i^{+},J^{\mathrm{bd}}_{\theta}) = \mu^{+} tcf ( ∏ i ∈ θ μ i + , J θ bd ) = μ + θ = ℵ 0 \theta= \aleph_{0} θ = ℵ 0 μ + \mu^{+} μ + 2 μ 2^{\mu} 2 μ 2 μ i = μ i + 2^{\mu_i} = \mu_i^{+} 2 μ i = μ i + Q μ ˉ \mathbb{Q}_{\bar{\mu}} Q μ ˉ χ \chi χ χ ∈ μ \chi\in\mu χ ∈ μ
It seems that the above method cannot be applied to ℵ ω \aleph_{\omega} ℵ ω μ \mu μ θ \theta θ θ \theta θ μ + \mu^{+} μ + θ \theta θ ℵ ω \aleph_{\omega} ℵ ω ℵ ω 2 \aleph_{\omega_{2}} ℵ ω 2 ℵ ω 2 \aleph_{\omega_{2}} ℵ ω 2 ω \omega ω
Theorem 1.3. Assuming the existence of a strong cardinal in the ground model, one can force μ + ↛ ( μ + , ( 3 ) cf ( μ ) ) 2 \mu^{+} \nrightarrow(\mu^{+},(3)_{\operatorname{cf}(\mu)})^{2} μ + ↛ ( μ + , ( 3 ) cf ( μ ) ) 2 2 μ > μ + 2^{\mu} > \mu^{+} 2 μ > μ + μ \mu μ μ = ℵ ω 2 \mu= \aleph_{\omega_{2}} μ = ℵ ω 2
Proof. Let μ \mu μ λ ≥ μ + + \lambda\geq\mu^{++} λ ≥ μ ++ E E E ( μ , λ ) (\mu,\lambda) ( μ , λ ) j : V → M ≅ Ult ( V , E ) j: V \to M \cong\operatorname{Ult}(V,E) j : V → M ≅ Ult ( V , E ) M ⊇ V μ + + M \supseteq V_{\mu^{++}} M ⊇ V μ ++ μ \mu μ μ = ℵ ω 2 \mu= \aleph_{\omega_{2}} μ = ℵ ω 2 6 G G G V V V μ \mu μ V [ G ] V[G] V [ G ] 2 μ = μ + + 2^\mu=\mu^{++} 2 μ = μ ++ μ \mu μ μ \mu μ V [ G ] V[G] V [ G ] μ = ℵ ω 2 \mu=\aleph_{\omega_2} μ = ℵ ω 2 V [ G ] V[G] V [ G ]
Let ( ρ n ∣ n ∈ ω ) (\rho_n\mid n\in\omega) ( ρ n ∣ n ∈ ω ) E E E tcf ( ∏ n ∈ ω ρ n + + , J b d ) = μ + + \operatorname{tcf}(\prod_{n\in\omega}\rho_n^{++},J^{\mathrm{bd}})=\mu^{++} tcf ( ∏ n ∈ ω ρ n ++ , J bd ) = μ ++ [13 ] . Moreover, up to a modification of a proper initial segment, this is the only sequence with true cofinality μ + + \mu^{++} μ ++ ( μ n ∣ n ∈ ω ) (\mu_n\mid n\in\omega) ( μ n ∣ n ∈ ω ) μ = ⋃ n ∈ ω μ n \mu=\bigcup_{n\in\omega}\mu_n μ = ⋃ n ∈ ω μ n tcf ( ∏ n ∈ ω μ n + , J b d ) = μ + \operatorname{tcf}(\prod_{n\in\omega}\mu_n^+,J^{\mathrm{bd}})=\mu^+ tcf ( ∏ n ∈ ω μ n + , J bd ) = μ + V [ G ] V[G] V [ G ] Theorem 1.1 hold, and therefore μ + ↛ ( μ + , ( 3 ) cf ( μ ) ) 2 \mu^+\nrightarrow(\mu^+,(3)_{\operatorname{cf}(\mu)})^2 μ + ↛ ( μ + , ( 3 ) cf ( μ ) ) 2 μ = ℵ ω 2 \mu=\aleph_{\omega_2} μ = ℵ ω 2 V [ G ] V[G] V [ G ] ( μ n ∣ n ∈ ω ) (\mu_n\mid n\in\omega) ( μ n ∣ n ∈ ω )
□ 1.3 \square_{1.3} □ 1.3
Notice that the large cardinal assumption here is just a strong cardinal, and in fact one needs only something around a μ + + \mu^{++} μ ++ [15 ] that o ( μ ) = μ + + o(\mu)=\mu^{++} o ( μ ) = μ ++ cf ( μ ) V [ G ] = ω \operatorname{cf}(\mu)^{V[G]}=\omega cf ( μ ) V [ G ] = ω cf ( μ ) \operatorname{cf}(\mu) cf ( μ )
We conclude this section with a question about the possible consistency of the opposite situation. Maybe the most interesting problem which issues from our study is whether the negative relation holds in ZFC. We believe that the positive relation λ + → ( λ + , ( 3 ) cf ( λ ) ) 2 \lambda^+\to(\lambda^+,(3)_{\operatorname{cf}(\lambda)})^2 λ + → ( λ + , ( 3 ) cf ( λ ) ) 2
Question 1.4. Is it consistent that λ \lambda λ λ + → ( λ + , ( 3 ) cf ( λ ) ) 2 \lambda^+\to(\lambda^+,(3)_{\operatorname{cf}(\lambda)})^2 λ + → ( λ + , ( 3 ) cf ( λ ) ) 2 λ = ℵ ω \lambda=\aleph_\omega λ = ℵ ω
It follows from the proofs in this section that a positive answer to the above question must be forced by adding a lot of bounded subsets to λ \lambda λ λ \lambda λ λ + \lambda^+ λ +
More pcf and the first infinite singular cardinal
In this section we prove that the negative relation λ + ↛ ( λ + , ( 3 ) cf ( λ ) ) 2 \lambda^{+} \nrightarrow(\lambda^{+}, (3)_{\operatorname{cf}(\lambda)})^{2} λ + ↛ ( λ + , ( 3 ) cf ( λ ) ) 2 λ \lambda λ 2 λ > λ + 2^{\lambda} > \lambda^{+} 2 λ > λ + λ \lambda λ λ = ℵ ω \lambda= \aleph_{\omega} λ = ℵ ω
The basic idea is similar to that of the proof of Theorem 1.1 in the previous section. Namely, from negative arrow relations on a sequence of cardinals below λ \lambda λ λ + \lambda^{+} λ + λ + \lambda^{+} λ + λ \lambda λ λ + \lambda^{+} λ + ℵ ω \aleph_{\omega} ℵ ω ℵ n \aleph_n ℵ n Theorem 1.1 .
So, first we deal with obtaining those weaker instances of the negative partition relation from local instances of the generalized continuum hypothesis. This is the content of the first claim of this section.
Claim 2.1. Assume that κ < λ \kappa< \lambda κ < λ λ = λ < κ \lambda= \lambda^{<\kappa} λ = λ < κ ( c , D ) (c, \mathcal{D}) ( c , D )
(a) c : [ λ ] 2 → { 0 , 1 } c : [\lambda]^2 \to\{0,1\} c : [ λ ] 2 → { 0 , 1 }
(b) D \mathcal{D} D κ \kappa κ λ \lambda λ
(c) If A ⊆ λ A \subseteq\lambda A ⊆ λ c ↾ [ A ] 2 c \mathbin{\upharpoonright} [A]^2 c ↾ [ A ] 2 A = ∅ m o d D A = \varnothing\bmod\mathcal{D} A = ∅ mod D
(d) If B = { α , β , γ } ∈ [ λ ] 3 B = \{\alpha,\beta,\gamma\} \in[\lambda]^3 B = { α , β , γ } ∈ [ λ ] 3 0 ∈ c ′ ′ [ B ] 2 0 \in c''[B]^2 0 ∈ c ′′ [ B ] 2
Thus there is no 0-monochromatic D \mathcal{D} D c c c
Proof. Enumerate the elements of [ λ ] < κ [\lambda]^{<\kappa} [ λ ] < κ ( u ξ ∣ ξ ∈ λ ) (u_{\xi} \mid\xi\in\lambda) ( u ξ ∣ ξ ∈ λ ) u 0 = ∅ u_0 = \varnothing u 0 = ∅ u ∈ [ λ ] < κ u \in[\lambda]^{<\kappa} u ∈ [ λ ] < κ λ \lambda λ c c c ⋃ α ∈ λ c α \bigcup_{\alpha\in\lambda} c_{\alpha} ⋃ α ∈ λ c α c α c_{\alpha} c α ξ α \xi_{\alpha} ξ α α ∈ λ \alpha\in\lambda α ∈ λ α = 0 \alpha= 0 α = 0 c α = ∅ c_{\alpha} = \varnothing c α = ∅ 0 < α 0 < \alpha 0 < α c α = ⋃ { c β : β ∈ α } c_{\alpha} = \bigcup\{c_{\beta} : \beta\in\alpha\} c α = ⋃ { c β : β ∈ α }
If α = β + 1 \alpha= \beta+ 1 α = β + 1 c α c_{\alpha} c α ξ β \xi_{\beta} ξ β ξ β \xi_{\beta} ξ β ( β + 1 ) (\beta+ 1) ( β + 1 ) ξ β \xi_{\beta} ξ β ξ ∈ λ \xi\in\lambda ξ ∈ λ ξ ∉ { ξ γ ∣ γ ∈ β } ∪ { 0 } \xi\notin\{\xi_{\gamma} \mid\gamma\in\beta\} \cup\{0\} ξ ∈ / { ξ γ ∣ γ ∈ β } ∪ { 0 } c β ↾ [ u ξ ] 2 c_{\beta} \mathbin{\upharpoonright} [u_{\xi}]^2 c β ↾ [ u ξ ] 2 ξ β = 0 \xi_{\beta} = 0 ξ β = 0
Now if η < ζ < β \eta< \zeta< \beta η < ζ < β c α ( { η , ζ } ) = c β ( { η , ζ } ) c_{\alpha}(\{\eta,\zeta\}) = c_{\beta}(\{\eta,\zeta\}) c α ({ η , ζ }) = c β ({ η , ζ }) γ ∈ β \gamma\in\beta γ ∈ β c α ( { γ , β } ) = 1 c_{\alpha}(\{\gamma,\beta\}) = 1 c α ({ γ , β }) = 1 γ ∈ u ξ β \gamma\in u_{\xi_{\beta}} γ ∈ u ξ β c α ( { γ , β } ) = 0 c_{\alpha}(\{\gamma,\beta\}) = 0 c α ({ γ , β }) = 0 γ ∉ u ξ β \gamma\notin u_{\xi_{\beta}} γ ∈ / u ξ β c α c_{\alpha} c α c β c_{\beta} c β u ξ β u_{\xi_{\beta}} u ξ β
Let I \mathcal{I} I κ \kappa κ λ \lambda λ c c c λ \lambda λ A ⊆ λ A \subseteq\lambda A ⊆ λ A ∈ I A \in\mathcal{I} A ∈ I ζ ∈ κ \zeta\in\kappa ζ ∈ κ A ξ ⊆ λ A_{\xi} \subseteq\lambda A ξ ⊆ λ ξ ∈ ζ \xi\in\zeta ξ ∈ ζ c ′ ′ [ A ξ ] 2 = { 0 } c''[A_{\xi}]^2 = \{0\} c ′′ [ A ξ ] 2 = { 0 } ξ ∈ ζ \xi\in\zeta ξ ∈ ζ A − ⋃ { A ξ ∣ ξ ∈ ζ } ∈ [ λ ] < λ A - \bigcup\{A_{\xi} \mid\xi\in\zeta\} \in[\lambda]^{<\lambda} A − ⋃ { A ξ ∣ ξ ∈ ζ } ∈ [ λ ] < λ D \mathcal{D} D { λ − A ∣ A ∈ I } \{\lambda- A \mid A \in\mathcal{I}\} { λ − A ∣ A ∈ I } D \mathcal{D} D κ \kappa κ λ \lambda λ c c c I \mathcal{I} I ε < σ < τ \varepsilon< \sigma< \tau ε < σ < τ c ( ε , τ ) = c ( σ , τ ) = 1 c(\varepsilon,\tau)=c(\sigma,\tau)=1 c ( ε , τ ) = c ( σ , τ ) = 1 ε , σ ∈ u ξ τ \varepsilon,\sigma\in u_{\xi_\tau} ε , σ ∈ u ξ τ c ( ε , σ ) = 0 c(\varepsilon,\sigma)=0 c ( ε , σ ) = 0 D \mathcal{D} D λ ∉ I \lambda\notin\mathcal{I} λ ∈ / I
Assume towards contradiction that λ ∈ I \lambda\in\mathcal{I} λ ∈ I ζ ∈ κ \zeta\in\kappa ζ ∈ κ A ε ⊆ λ A_\varepsilon\subseteq\lambda A ε ⊆ λ ε ∈ ζ \varepsilon\in\zeta ε ∈ ζ c ↾ [ A ε ] 2 c\mathbin{\upharpoonright}[A_\varepsilon]^2 c ↾ [ A ε ] 2 ε ∈ ζ \varepsilon\in\zeta ε ∈ ζ B = λ \ ⋃ ε ∈ ζ A ε ∈ [ λ ] < λ B=\lambda\mathbin{\backslash}\bigcup_{\varepsilon\in\zeta}A_\varepsilon\in[\lambda]^{<\lambda} B = λ \ ⋃ ε ∈ ζ A ε ∈ [ λ ] < λ λ \lambda λ B B B λ \lambda λ ∣ A ε ∣ = λ |A_\varepsilon|=\lambda ∣ A ε ∣ = λ ε ∈ ζ \varepsilon\in\zeta ε ∈ ζ B B B A ε A_\varepsilon A ε λ \lambda λ B B B
We choose a sequence of ordinals ( β α ε ∣ α ∈ λ , ε ∈ ζ ) (\beta_{\alpha\varepsilon}\mid\alpha\in\lambda,\varepsilon\in\zeta) ( β α ε ∣ α ∈ λ , ε ∈ ζ ) β α ε ∈ A ε \beta_{\alpha\varepsilon}\in A_\varepsilon β α ε ∈ A ε α ∈ λ , ε ∈ ζ \alpha\in\lambda,\varepsilon\in\zeta α ∈ λ , ε ∈ ζ ∣ A ε ∣ = λ |A_\varepsilon|=\lambda ∣ A ε ∣ = λ ε ∈ ζ \varepsilon\in\zeta ε ∈ ζ α ∈ λ \alpha\in\lambda α ∈ λ V α = ⋃ { u ξ β α ε ∣ ε ∈ ζ } V_\alpha=\bigcup\{u_{\xi_{\beta_{\alpha\varepsilon}}}\mid\varepsilon\in\zeta\} V α = ⋃ { u ξ β α ε ∣ ε ∈ ζ } W α = { γ ∈ λ ∣ ∃ ε ∈ ζ , β γ ε ∈ V α } W_\alpha=\{\gamma\in\lambda\mid\exists\varepsilon\in\zeta,\ \beta_{\gamma\varepsilon}\in V_\alpha\} W α = { γ ∈ λ ∣ ∃ ε ∈ ζ , β γ ε ∈ V α } V α ∈ [ λ ] < κ V_\alpha\in[\lambda]^{<\kappa} V α ∈ [ λ ] < κ κ \kappa κ W α ∈ [ λ ] < κ W_\alpha\in[\lambda]^{<\kappa} W α ∈ [ λ ] < κ α ∉ W α \alpha\notin W_\alpha α ∈ / W α α ∈ λ \alpha\in\lambda α ∈ λ { W α ∣ α ∈ λ } \{W_\alpha\mid\alpha\in\lambda\} { W α ∣ α ∈ λ } Y ∈ [ λ ] λ Y\in[\lambda]^\lambda Y ∈ [ λ ] λ 7 { α , β } ⊆ Y \{\alpha,\beta\}\subseteq Y { α , β } ⊆ Y α ∉ W β \alpha\notin W_\beta α ∈ / W β
Let f : ζ → Y f:\zeta\to Y f : ζ → Y B ∩ rang ( f ) = ∅ B\cap\operatorname{rang}(f)=\varnothing B ∩ rang ( f ) = ∅ ε ′ < ε < ζ \varepsilon'<\varepsilon<\zeta ε ′ < ε < ζ β f ( ε ′ ) ε ′ < β f ( ε ) ε \beta_{f(\varepsilon')\varepsilon'}<\beta_{f(\varepsilon)\varepsilon} β f ( ε ′ ) ε ′ < β f ( ε ) ε ∣ Y ∣ = λ |Y|=\lambda ∣ Y ∣ = λ u = { β f ( ε ) ε ∣ ε ∈ ζ } u=\{\beta_{f(\varepsilon)\varepsilon}\mid\varepsilon\in\zeta\} u = { β f ( ε ) ε ∣ ε ∈ ζ } u ∈ [ λ ] < κ u\in[\lambda]^{<\kappa} u ∈ [ λ ] < κ c ↾ [ u ] 2 c\mathbin{\upharpoonright}[u]^2 c ↾ [ u ] 2 ε ′ < ε < ζ \varepsilon'<\varepsilon<\zeta ε ′ < ε < ζ c ( β f ( ε ′ ) ε ′ , β f ( ε ) ε ) = 1 c(\beta_{f(\varepsilon')\varepsilon'},\beta_{f(\varepsilon)\varepsilon})=1 c ( β f ( ε ′ ) ε ′ , β f ( ε ) ε ) = 1 β f ( ε ′ ) ε ′ ∈ u ξ β f ( ε ) ε ⊆ V f ( ε ) \beta_{f(\varepsilon')\varepsilon'}\in u_{\xi_{\beta_{f(\varepsilon)\varepsilon}}}\subseteq V_{f(\varepsilon)} β f ( ε ′ ) ε ′ ∈ u ξ β f ( ε ) ε ⊆ V f ( ε ) f ( ε ′ ) ∈ W f ( ε ) f(\varepsilon')\in W_{f(\varepsilon)} f ( ε ′ ) ∈ W f ( ε ) f ( ε ′ ) ∉ W f ( ε ) f(\varepsilon')\notin W_{f(\varepsilon)} f ( ε ′ ) ∈ / W f ( ε ) f ( ε ′ ) f(\varepsilon') f ( ε ′ ) f ( ε ) f(\varepsilon) f ( ε ) Y Y Y
Recall that u u u λ \lambda λ [ λ ] < κ [\lambda]^{<\kappa} [ λ ] < κ ξ α \xi_\alpha ξ α u = u ξ α u=u_{\xi_\alpha} u = u ξ α ξ α \xi_\alpha ξ α c ↾ [ u ξ α ] 2 = { 0 } c\mathbin{\upharpoonright}[u_{\xi_\alpha}]^2=\{0\} c ↾ [ u ξ α ] 2 = { 0 } ξ γ \xi_\gamma ξ γ ξ α \xi_\alpha ξ α ξ α \xi_\alpha ξ α α > ⋃ B \alpha>\bigcup B α > ⋃ B α ∉ B \alpha\notin B α ∈ / B ε ∈ ζ \varepsilon\in\zeta ε ∈ ζ α ∈ A ε \alpha\in A_\varepsilon α ∈ A ε β f ( ε ) ε ∈ A ε \beta_{f(\varepsilon)\varepsilon}\in A_\varepsilon β f ( ε ) ε ∈ A ε c ( { α , β f ( ε ) ε } ) = 0 c(\{\alpha,\beta_{f(\varepsilon)\varepsilon}\})=0 c ({ α , β f ( ε ) ε }) = 0 c ↾ [ A ε ] 2 c\mathbin{\upharpoonright}[A_\varepsilon]^2 c ↾ [ A ε ] 2 β f ( ε ) ε ∈ u = u ξ α \beta_{f(\varepsilon)\varepsilon}\in u=u_{\xi_\alpha} β f ( ε ) ε ∈ u = u ξ α c ( { α , β f ( ε ) ε } ) = 1 c(\{\alpha,\beta_{f(\varepsilon)\varepsilon}\})=1 c ({ α , β f ( ε ) ε }) = 1
□2.1
The above claim is exactly what we need in order to lift negative partition relations to the negative arrow relation over a successor of a singular cardinal. We emphasize that the coloring c c c A A A λ \lambda λ A = ∅ ( m o d D ) A=\varnothing\pmod{\mathcal{D}} A = ∅ ( mod D )
Theorem 2.2. Suppose that:
(A) ( N ) λ > cf ( λ ) = θ (\mathsf{N})\ \lambda>\operatorname{cf}(\lambda)=\theta ( N ) λ > cf ( λ ) = θ
(B) ( ∃ ) λ = ⋃ i ∈ θ κ i = ⋃ i ∈ θ λ i (\exists)\ \lambda=\bigcup_{i\in\theta}\kappa_i=\bigcup_{i\in\theta}\lambda_i ( ∃ ) λ = ⋃ i ∈ θ κ i = ⋃ i ∈ θ λ i
(ג) κ i = cf ( κ i ) < λ i = cf ( λ i ) \kappa_i=\operatorname{cf}(\kappa_i)<\lambda_i=\operatorname{cf}(\lambda_i) κ i = cf ( κ i ) < λ i = cf ( λ i ) λ i < κ i = λ i \lambda_i^{<\kappa_i}=\lambda_i λ i < κ i = λ i i ∈ θ i\in\theta i ∈ θ
(ד) ∣ ∏ i < j λ i ∣ < κ j \left|\prod_{i<j}\lambda_i\right|<\kappa_j ∏ i < j λ i < κ j j ∈ θ j\in\theta j ∈ θ
(B) (א) D i \mathcal{D}_i D i κ i \kappa_i κ i λ i \lambda_i λ i i ∈ θ i\in\theta i ∈ θ
(ב) c i : [ λ i ] 2 → { 0 , 1 } c_i:[\lambda_i]^2\to\{0,1\} c i : [ λ i ] 2 → { 0 , 1 } i ∈ θ i\in\theta i ∈ θ
(ג) If A ⊆ λ i A\subseteq\lambda_i A ⊆ λ i c i ′ ′ [ A ] 2 = { 0 } c_i''[A]^2=\{0\} c i ′′ [ A ] 2 = { 0 } A = ∅ mod D i A=\varnothing\mathbin{\operatorname{mod}}\mathcal{D}_i A = ∅ mod D i
(ד) If α < β < γ < λ i \alpha<\beta<\gamma<\lambda_i α < β < γ < λ i ε = c i ( { α , β } ) = c i ( { α , γ } ) = c i ( { β , γ } ) \varepsilon=c_i(\{\alpha,\beta\})=c_i(\{\alpha,\gamma\})=c_i(\{\beta,\gamma\}) ε = c i ({ α , β }) = c i ({ α , γ }) = c i ({ β , γ }) ε = 0 \varepsilon=0 ε = 0
(C) (א) η ˉ = ( η α ∣ α ∈ λ + ) ⊆ ∏ i ∈ θ λ i \bar{\eta}=(\eta_\alpha\mid\alpha\in\lambda^+)\subseteq\prod_{i\in\theta}\lambda_i η ˉ = ( η α ∣ α ∈ λ + ) ⊆ ∏ i ∈ θ λ i
(ב) If α < β < λ + \alpha<\beta<\lambda^+ α < β < λ + η α ≠ η β \eta_\alpha\ne\eta_\beta η α = η β
(ג) If B i ∈ D i B_i\in\mathcal{D}_i B i ∈ D i i ∈ θ i\in\theta i ∈ θ α 0 ∈ λ + \alpha_0\in\lambda^+ α 0 ∈ λ + α 0 ≤ α ∈ λ + \alpha_0\leq\alpha\in\lambda^+ α 0 ≤ α ∈ λ + i α ∈ θ i_\alpha\in\theta i α ∈ θ η α ( i ) ∈ B i \eta_\alpha(i)\in B_i η α ( i ) ∈ B i i α ≤ i ∈ θ i_\alpha\leq i\in\theta i α ≤ i ∈ θ
Then λ + ↛ ( λ + , ( 3 ) θ ) 2 \lambda^+\nrightarrow(\lambda^+,(3)_\theta)^2 λ + ↛ ( λ + , ( 3 ) θ ) 2
Proof. We define c : [ λ + ] 2 → θ × { 0 , 1 } c:[\lambda^+]^2\to\theta\times\{0,1\} c : [ λ + ] 2 → θ × { 0 , 1 } α < β < λ + \alpha<\beta<\lambda^+ α < β < λ + i α β = lg ( η α ∩ η β ) ∈ θ i_{\alpha\beta}=\lg(\eta_\alpha\cap\eta_\beta)\in\theta i α β = lg ( η α ∩ η β ) ∈ θ ε α β = c i α β ( { η α ( i ) , η β ( i ) } ) \varepsilon_{\alpha\beta}=c_{i_{\alpha\beta}}(\{\eta_\alpha(i),\eta_\beta(i)\}) ε α β = c i α β ({ η α ( i ) , η β ( i )}) i = i α β + 1 i=i_{\alpha\beta}+1 i = i α β + 1 0 < ε α β 0<\varepsilon_{\alpha\beta} 0 < ε α β c ( { α , β } ) = ( i α β , ε α β ) c(\{\alpha,\beta\})=(i_{\alpha\beta},\varepsilon_{\alpha\beta}) c ({ α , β }) = ( i α β , ε α β ) c ( { α , β } ) = ( 0 , 0 ) c(\{\alpha,\beta\})=(0,0) c ({ α , β }) = ( 0 , 0 ) θ × { 0 , 1 } \theta\times\{0,1\} θ × { 0 , 1 } ( 0 , 0 ) (0,0) ( 0 , 0 )
We claim that c c c λ + ↛ ( λ + , ( 3 ) θ ) 2 \lambda^+\nrightarrow(\lambda^+,(3)_\theta)^2 λ + ↛ ( λ + , ( 3 ) θ ) 2 ε = 1 \varepsilon=1 ε = 1 ( λ + , { { α , β } ∣ c ( { α , β } ) = ( i , ε ) } ) (\lambda^+,\{\{\alpha,\beta\}\mid c(\{\alpha,\beta\})=(i,\varepsilon)\}) ( λ + , {{ α , β } ∣ c ({ α , β }) = ( i , ε )}) c ( { α , β } ) = c ( { α , γ } ) = c ( { β , γ } ) = ( i , ε ) c(\{\alpha,\beta\})=c(\{\alpha,\gamma\})=c(\{\beta,\gamma\})=(i,\varepsilon) c ({ α , β }) = c ({ α , γ }) = c ({ β , γ }) = ( i , ε ) { η α ( i ) , η β ( i ) , η γ ( i ) } \{\eta_\alpha(i),\eta_\beta(i),\eta_\gamma(i)\} { η α ( i ) , η β ( i ) , η γ ( i )} ε \varepsilon ε c i c_i c i ε = 1 \varepsilon=1 ε = 1 A ∈ [ λ + ] λ + A\in[\lambda^+]^{\lambda^+} A ∈ [ λ + ] λ + c ↾ [ A ] 2 c\mathbin{\upharpoonright}[A]^2 c ↾ [ A ] 2 ( 0 , 0 ) (0,0) ( 0 , 0 )
To see this, assume towards contradiction that A ∈ [ λ + ] λ + A\in[\lambda^+]^{\lambda^+} A ∈ [ λ + ] λ + i ∈ θ i\in\theta i ∈ θ A i = { η α ( i ) ∣ α ∈ A } ⊆ λ i A_i=\{\eta_\alpha(i)\mid\alpha\in A\}\subseteq\lambda_i A i = { η α ( i ) ∣ α ∈ A } ⊆ λ i A i ∈ D i + A_i\in\mathcal{D}_i^+ A i ∈ D i + i ∈ θ i\in\theta i ∈ θ i i i α β ∈ A \alpha_\beta\in A α β ∈ A β ∈ A i \beta\in A_i β ∈ A i η α β ( i ) = β \eta_{\alpha_\beta}(i)=\beta η α β ( i ) = β ν ∈ ∏ j < i λ j \nu\in\prod_{j<i}\lambda_j ν ∈ ∏ j < i λ j A i ν = { β ∈ A i ∣ η α β ↾ i = ν } A_{i\nu}=\{\beta\in A_i\mid\eta_{\alpha_\beta}\mathbin{\upharpoonright}i=\nu\} A i ν = { β ∈ A i ∣ η α β ↾ i = ν } A i = ⋃ { A i ν ∣ ν ∈ ∏ j < i λ j } A_i=\bigcup\{A_{i\nu}\mid\nu\in\prod_{j<i}\lambda_j\} A i = ⋃ { A i ν ∣ ν ∈ ∏ j < i λ j } D i \mathcal{D}_i D i κ i \kappa_i κ i ∣ ∏ j < i λ j ∣ < κ i \left|\prod_{j<i}\lambda_j\right|<\kappa_i ∏ j < i λ j < κ i ν ∈ ∏ j < i λ j \nu\in\prod_{j<i}\lambda_j ν ∈ ∏ j < i λ j A i ν ∈ D i + A_{i\nu}\in\mathcal{D}_i^+ A i ν ∈ D i + β , β ′ ∈ A i ν \beta,\beta'\in A_{i\nu} β , β ′ ∈ A i ν β ≠ β ′ \beta\ne\beta' β = β ′ c i ( { β , β ′ } ) = ε > 0 c_i(\{\beta,\beta'\})=\varepsilon>0 c i ({ β , β ′ }) = ε > 0 c ( { α β , α β ′ } ) = ( i , ε ) ≠ ( 0 , 0 ) c(\{\alpha_\beta,\alpha_{\beta'}\})=(i,\varepsilon)\ne(0,0) c ({ α β , α β ′ }) = ( i , ε ) = ( 0 , 0 ) α β , α β ′ ∈ A \alpha_\beta,\alpha_{\beta'}\in A α β , α β ′ ∈ A c ( { α β , α β ′ } ) = ( 0 , 0 ) c(\{\alpha_\beta,\alpha_{\beta'}\})=(0,0) c ({ α β , α β ′ }) = ( 0 , 0 )
In the second case, A i = ∅ mod D i A_i=\varnothing\mathbin{\operatorname{mod}}\mathcal{D}_i A i = ∅ mod D i i ∈ θ i\in\theta i ∈ θ ∀ ∗ i < θ \forall^*i<\theta ∀ ∗ i < θ i ∈ θ i\in\theta i ∈ θ θ \theta θ B = { α ∈ λ ∣ ( ∀ ∗ i < θ ) ( η α ( i ) ∈ A i ) } B=\{\alpha\in\lambda\mid(\forall^*i<\theta)(\eta_\alpha(i)\in A_i)\} B = { α ∈ λ ∣ ( ∀ ∗ i < θ ) ( η α ( i ) ∈ A i )} B ∈ [ λ ] < λ B\in[\lambda]^{<\lambda} B ∈ [ λ ] < λ B i = λ i − A i ∈ D i B_i=\lambda_i-A_i\in\mathcal{D}_i B i = λ i − A i ∈ D i α ∈ A \alpha\in A α ∈ A η α ( i ) ∈ A i \eta_\alpha(i)\in A_i η α ( i ) ∈ A i i ∈ θ . H e n c e i\in\theta. Hence i ∈ θ . H e n ce ∣ A ∣ < λ |A|<\lambda ∣ A ∣ < λ
□ 2.2 \square_{2.2} □ 2.2
In order to utilize the above theorem one has to show that the assumptions there hold (or can be forced) under the assumptions of ( A ) (A) ( A ) λ \lambda λ ( B ) (B) ( B ) Claim 2.1 . Thus the only challenge is ( C ) (C) ( C )
Lemma 2.3. Let λ \lambda λ R ⊆ λ R \subseteq\lambda R ⊆ λ ζ ∈ R \zeta\in R ζ ∈ R D ζ \mathcal{D}_{\zeta} D ζ ζ \zeta ζ ζ + \zeta^{+} ζ + ( < λ ) (<\lambda) ( < λ ) λ + \lambda^{+} λ + D \mathbb{D} D D \mathbb{D} D
( ℵ ) (\aleph) ( ℵ ) ( f α ∣ α ∈ λ + ) (f_{\alpha} \mid\alpha\in\lambda^{+}) ( f α ∣ α ∈ λ + ) ∏ ζ ∈ R ζ + \prod_{\zeta\in R} \zeta^{+} ∏ ζ ∈ R ζ +
( ℶ ) (\beth) ( ℶ ) ( A ζ ∣ ζ ∈ R ) ∈ ∏ ζ ∈ R D ζ (A_{\zeta} \mid\zeta\in R) \in\prod_{\zeta\in R} \mathcal{D}_{\zeta} ( A ζ ∣ ζ ∈ R ) ∈ ∏ ζ ∈ R D ζ α 0 ∈ λ + \alpha_{0} \in\lambda^{+} α 0 ∈ λ + α 0 ≤ α ∈ λ + \alpha_{0} \leq\alpha\in\lambda^{+} α 0 ≤ α ∈ λ + i α ∈ λ i_{\alpha} \in\lambda i α ∈ λ i α ≤ ζ ∈ R i_{\alpha} \leq\zeta\in R i α ≤ ζ ∈ R f α ( ζ ) ∈ A ζ f_{\alpha}(\zeta) \in A_{\zeta} f α ( ζ ) ∈ A ζ
Proof. Let ( D α , H β ∣ β ∈ λ + , α ≤ λ + ) (\mathbb{D}_{\alpha}, \mathbb{H}_{\beta} \mid\beta\in\lambda^{+}, \alpha\leq\lambda^{+}) ( D α , H β ∣ β ∈ λ + , α ≤ λ + ) ( < λ ) (<\lambda) ( < λ ) λ + \lambda^{+} λ + H β \mathbb{H}_{\beta} H β D β \mathbb{D}_{\beta} D β p ∈ H β p \in\mathbb{H}_{\beta} p ∈ H β ( s , A ˉ ) = ( s p , A ˉ p ) (s,\bar{A})=(s_{p},\bar{A}_{p}) ( s , A ˉ ) = ( s p , A ˉ p ) s s s dom ( s ) = R ∩ ξ s \operatorname{dom}(s)=R\cap\xi_{s} dom ( s ) = R ∩ ξ s ξ s ∈ λ \xi_{s}\in\lambda ξ s ∈ λ A ˉ \bar{A} A ˉ ( A ζ ∣ ζ ∈ R − dom ( s ) ) (A_{\zeta}\mid\zeta\in R-\operatorname{dom}(s)) ( A ζ ∣ ζ ∈ R − dom ( s )) ζ ∈ dom ( s ) \zeta\in\operatorname{dom}(s) ζ ∈ dom ( s ) s ( ζ ) ∈ ζ + s(\zeta)\in\zeta^{+} s ( ζ ) ∈ ζ + ζ ∈ R − dom ( s ) \zeta\in R-\operatorname{dom}(s) ζ ∈ R − dom ( s ) A ζ ∈ D ζ A_{\zeta}\in\mathcal{D}_{\zeta} A ζ ∈ D ζ p , q ∈ H β p,q\in\mathbb{H}_{\beta} p , q ∈ H β p ≤ q p\leq q p ≤ q s p ⊇ s q s_{p}\supseteq s_{q} s p ⊇ s q A ζ q ⊆ A ζ p A_{\zeta}^{q}\subseteq A_{\zeta}^{p} A ζ q ⊆ A ζ p ζ ∈ R − dom ( s q ) \zeta\in R-\operatorname{dom}(s_{q}) ζ ∈ R − dom ( s q ) ζ ∈ dom ( s q ) − dom ( s p ) \zeta\in\operatorname{dom}(s_{q})-\operatorname{dom}(s_{p}) ζ ∈ dom ( s q ) − dom ( s p ) s q ( ζ ) ∈ A ζ p s_{q}(\zeta)\in A_{\zeta}^{p} s q ( ζ ) ∈ A ζ p H β \mathbb{H}_{\beta} H β ∏ ζ ∈ R ζ + \prod_{\zeta\in R}\zeta^{+} ∏ ζ ∈ R ζ + s p s_{p} s p p p p A ˉ p \bar{A}_{p} A ˉ p p p p
Observe that H β \mathbb{H}_{\beta} H β ( < λ ) (<\lambda) ( < λ ) γ ∈ λ \gamma\in\lambda γ ∈ λ γ + \gamma^{+} γ + T γ = { p ∈ H β ∣ ∣ dom ( s p ) ∣ > γ } T_{\gamma}=\{p\in\mathbb{H}_{\beta}\mid\lvert\operatorname{dom}(s_{p})\rvert>\gamma\} T γ = { p ∈ H β ∣ ∣ dom ( s p )∣ > γ } { p i ∣ i < γ } \{p_{i}\mid i<\gamma\} { p i ∣ i < γ } s p = ⋃ i < γ s p i s_{p}=\bigcup_{i<\gamma}s_{p_{i}} s p = ⋃ i < γ s p i ζ ∈ R − dom ( s p ) \zeta\in R-\operatorname{dom}(s_{p}) ζ ∈ R − dom ( s p ) A ζ p = ⋂ i < γ A ζ p i A_{\zeta}^{p}=\bigcap_{i<\gamma}A_{\zeta}^{p_{i}} A ζ p = ⋂ i < γ A ζ p i A ζ p ∈ D ζ A_{\zeta}^{p}\in\mathcal{D}_{\zeta} A ζ p ∈ D ζ ζ > γ \zeta>\gamma ζ > γ D ζ \mathcal{D}_{\zeta} D ζ ζ \zeta ζ p = ( s p , A ˉ p ) p=(s_{p},\bar{A}_{p}) p = ( s p , A ˉ p ) { p i ∣ i < γ } \{p_{i}\mid i<\gamma\} { p i ∣ i < γ } H β \mathbb{H}_{\beta} H β λ \lambda λ p , q ∈ H β p,q\in\mathbb{H}_{\beta} p , q ∈ H β s p = s q s_{p}=s_{q} s p = s q p ∥ q p\mathbin{\|}q p ∥ q λ \lambda λ λ \lambda λ H β \mathbb{H}_{\beta} H β λ \lambda λ
Let H ⊆ H β H\subseteq\mathbb{H}_{\beta} H ⊆ H β f β = ⋃ { s ∣ ( s , A ˉ ) ∈ H } f_{\beta}=\bigcup\{s\mid(s,\bar{A})\in H\} f β = ⋃ { s ∣ ( s , A ˉ ) ∈ H } H H H f β f_{\beta} f β dom ( f β ) = R \operatorname{dom}(f_{\beta})=R dom ( f β ) = R f β ( ζ ) ∈ ζ + f_{\beta}(\zeta)\in\zeta^{+} f β ( ζ ) ∈ ζ + ζ ∈ R \zeta\in R ζ ∈ R D = D λ + \mathbb{D}=\mathbb{D}_{\lambda^{+}} D = D λ + D \mathbb{D} D ( < λ ) (<\lambda) ( < λ ) λ + \lambda^{+} λ + D \mathbb{D} D ( < λ ) (<\lambda) ( < λ ) ( < λ ) (<\lambda) ( < λ ) λ \lambda λ
Fix a V V V G ⊆ D G\subseteq\mathbb{D} G ⊆ D V [ G ] V[G] V [ G ]
As noted above, ( f α ∣ α ∈ λ + ) ⊆ ∏ ζ ∈ R ζ + (f_{\alpha}\mid\alpha\in\lambda^{+})\subseteq\prod_{\zeta\in R}\zeta^{+} ( f α ∣ α ∈ λ + ) ⊆ ∏ ζ ∈ R ζ + D \mathbb{D} D ( < λ ) (<\lambda) ( < λ ) λ \lambda λ V [ G ] V[G] V [ G ] λ \lambda λ A ˉ = ( A ˉ ζ ∣ ζ ∈ R ) \bar{A}=(\bar{A}_{\zeta}\mid\zeta\in R) A ˉ = ( A ˉ ζ ∣ ζ ∈ R ) A ˉ [ G ] ∈ ∏ ζ ∈ R D ζ ∩ V [ G ] \bar{A}[G]\in\prod_{\zeta\in R}\mathcal{D}_{\zeta}\cap V[G] A ˉ [ G ] ∈ ∏ ζ ∈ R D ζ ∩ V [ G ] D \mathbb{D} D λ + \lambda^{+} λ + γ ∈ λ + \gamma\in\lambda^{+} γ ∈ λ + A ˉ ∈ V D γ \bar{A}\in V^{\mathbb{D}_{\gamma}} A ˉ ∈ V D γ γ ≤ β ∈ λ + \gamma\leq\beta\in\lambda^{+} γ ≤ β ∈ λ + i β ∈ λ i_\beta\in\lambda i β ∈ λ f β ( ζ ) ∈ A ˉ ζ f_\beta(\zeta) \in\bar{A}_\zeta f β ( ζ ) ∈ A ˉ ζ i β ≤ ζ ∈ λ i_\beta\leq\zeta\in\lambda i β ≤ ζ ∈ λ
To see this, let p p p A ˉ \bar{A} A ˉ γ \gamma γ β ~ ∈ [ γ , λ + ) \tilde{\beta} \in[\gamma,\lambda^+) β ~ ∈ [ γ , λ + ) q ∈ D q \in\mathbb{D} q ∈ D p ≤ q p \leq q p ≤ q β ∈ dom ( q ) \beta\in\operatorname{dom}(q) β ∈ dom ( q ) q ( β ) = ( t q , B ˉ q ) q(\beta)=(t_q,\bar{B}_q) q ( β ) = ( t q , B ˉ q ) q ↾ β ⊩ B ˉ ξ q ⊆ A ˉ ξ q\mathbin{\upharpoonright}\beta\Vdash\bar{B}^{q}_{\xi} \subseteq\bar{A}_{\xi} q ↾ β ⊩ B ˉ ξ q ⊆ A ˉ ξ ξ ∈ dom ( B ˉ q ) \xi\in\operatorname{dom}(\bar{B}_q) ξ ∈ dom ( B ˉ q ) A ˉ ξ \bar{A}_{\xi} A ˉ ξ D γ \mathbb{D}_{\gamma} D γ D β \mathbb{D}_{\beta} D β i β = ⋃ dom ( t q ) i_\beta=\bigcup\operatorname{dom}(t_q) i β = ⋃ dom ( t q ) i β ≤ ζ ∈ λ i_\beta\leq\zeta\in\lambda i β ≤ ζ ∈ λ q ⊩ f β ( ζ ) ∈ B ζ q ⊆ A ζ q\Vdash f_\beta(\zeta)\in B^{q}_{\zeta}\subseteq A_{\zeta} q ⊩ f β ( ζ ) ∈ B ζ q ⊆ A ζ p p p □ \square □
We remark that for densely many conditions p ∈ D p\in\mathbb{D} p ∈ D α ∈ dom ( p ) \alpha\in\operatorname{dom}(p) α ∈ dom ( p ) p ( α ) p(\alpha) p ( α ) D \mathbb{D} D
By the previous lemma, we can force our pcf assumption using a ( < λ ) ({<}\lambda) ( < λ ) λ \lambda λ λ \lambda λ
As our forcing is not sufficiently directed-closed, we need a weaker property that would imply the existence of a master condition for lifting certain supercompact embeddings.
Lemma 2.4. Let R R R D \mathbb{D} D ζ ∈ R \zeta\in R ζ ∈ R F ⊆ D F\subseteq\mathbb{D} F ⊆ D ∣ F ∣ < ζ |F|<\zeta ∣ F ∣ < ζ α ∈ ⋃ { dom ( p ) ∣ p ∈ F } \alpha\in\bigcup\{\operatorname{dom}(p)\mid p\in F\} α ∈ ⋃ { dom ( p ) ∣ p ∈ F } ⋃ { dom ( s p ( α ) ) ∣ p ∈ F , α ∈ dom ( p ) } ≥ ζ \bigcup\{\operatorname{dom}(s_p(\alpha))\mid p\in F,\alpha\in\operatorname{dom}(p)\}\geq\zeta ⋃ { dom ( s p ( α )) ∣ p ∈ F , α ∈ dom ( p )} ≥ ζ m ∈ D m\in\mathbb{D} m ∈ D p ≤ m p\leq m p ≤ m p ∈ F p\in F p ∈ F
Proof. We commence with letting dom ( m ) = ⋃ { dom ( p ) ∣ p ∈ F } \operatorname{dom}(m)=\bigcup\{\operatorname{dom}(p)\mid p\in F\} dom ( m ) = ⋃ { dom ( p ) ∣ p ∈ F } α ∈ dom ( m ) \alpha\in\operatorname{dom}(m) α ∈ dom ( m ) s m ( α ) = ⋃ { s p ( α ) ∣ p ∈ F , α ∈ dom ( p ) } s_m(\alpha)=\bigcup\{s_p(\alpha)\mid p\in F,\alpha\in\operatorname{dom}(p)\} s m ( α ) = ⋃ { s p ( α ) ∣ p ∈ F , α ∈ dom ( p )} A ˉ ξ m ( α ) = ⋂ { A ˉ ξ p ( α ) ∣ p ∈ F , α ∈ dom ( p ) } \bar{A}^{m(\alpha)}_{\xi}=\bigcap\{\bar{A}^{p(\alpha)}_{\xi}\mid p\in F,\alpha\in\operatorname{dom}(p)\} A ˉ ξ m ( α ) = ⋂ { A ˉ ξ p ( α ) ∣ p ∈ F , α ∈ dom ( p )} ξ ∈ R − ⋃ dom ( s m ( α ) ) \xi\in R-\bigcup\operatorname{dom}(s_m(\alpha)) ξ ∈ R − ⋃ dom ( s m ( α )) A ˉ ξ m ( α ) \bar{A}^{m(\alpha)}_{\xi} A ˉ ξ m ( α ) D ξ \mathcal{D}_{\xi} D ξ D ξ \mathcal{D}_{\xi} D ξ δ ∈ λ + \delta\in\lambda^+ δ ∈ λ + m ↾ δ m\mathbin{\upharpoonright}\delta m ↾ δ p ↾ δ ≤ m ↾ δ p\mathbin{\upharpoonright}\delta\leq m\mathbin{\upharpoonright}\delta p ↾ δ ≤ m ↾ δ p ∈ F p\in F p ∈ F
For δ = 0 \delta=0 δ = 0 δ \delta δ ∣ dom ( m ) ∣ < λ |\operatorname{dom}(m)|<\lambda ∣ dom ( m ) ∣ < λ δ = α + 1 ∈ λ + \delta=\alpha+1\in\lambda^+ δ = α + 1 ∈ λ + m ↾ α m\mathbin{\upharpoonright}\alpha m ↾ α A ˉ p ( α ) \bar{A}^{p(\alpha)} A ˉ p ( α ) p ∈ F p\in F p ∈ F m ↾ δ m\mathbin{\upharpoonright}\delta m ↾ δ p ↾ δ ≤ m ↾ δ p\mathbin{\upharpoonright}\delta\leq m\mathbin{\upharpoonright}\delta p ↾ δ ≤ m ↾ δ p ∈ F p\in F p ∈ F
Fix p ∈ F p\in F p ∈ F s p ( α ) ⊴ s m ( α ) s_{p(\alpha)}\trianglelefteq s_{m(\alpha)} s p ( α ) ⊴ s m ( α ) A ˉ ξ m ( α ) ⊆ A ˉ ξ p ( α ) \bar{A}^{m(\alpha)}_{\xi}\subseteq\bar{A}^{p(\alpha)}_{\xi} A ˉ ξ m ( α ) ⊆ A ˉ ξ p ( α ) ξ \xi ξ s m ( α ) s_{m(\alpha)} s m ( α ) s p ( α ) s_{p(\alpha)} s p ( α ) r ∈ F r \in F r ∈ F ξ ∈ dom ( s r ( α ) ) \xi\in\operatorname{dom}(s_{r(\alpha)}) ξ ∈ dom ( s r ( α ) ) s m ( α ) ( ξ ) = s r ( α ) ( ξ ) s_{m(\alpha)}(\xi)=s_{r(\alpha)}(\xi) s m ( α ) ( ξ ) = s r ( α ) ( ξ ) F F F t ∈ F t \in F t ∈ F r , p ≤ t r,p \leq t r , p ≤ t s t ( α ) ( ξ ) = s r ( α ) ( ξ ) s_{t(\alpha)}(\xi)=s_{r(\alpha)}(\xi) s t ( α ) ( ξ ) = s r ( α ) ( ξ ) t ↾ α ⊩ m ( α ) ( ξ ) = s t ( α ) ( ξ ) ∈ A ˉ ξ p ( α ) t\mathbin{\upharpoonright}\alpha\Vdash m(\alpha)(\xi)=s_{t(\alpha)}(\xi)\in\bar{A}^{p(\alpha)}_{\xi} t ↾ α ⊩ m ( α ) ( ξ ) = s t ( α ) ( ξ ) ∈ A ˉ ξ p ( α ) 2.4 _{2.4} 2.4
As mentioned before, we need the above lemma in order to prove a version of Laver’s indestructibility. In the work of Laver, a supercompact cardinal λ \lambda λ λ \lambda λ λ \lambda λ λ \lambda λ R R R D \mathbb{D} D λ \lambda λ
Lemma 2.5. Let λ \lambda λ L \mathbb{L} L L \mathbb{L} L R ⊆ λ R \subseteq\lambda R ⊆ λ 8 D ζ \mathcal{D}_{\zeta} D ζ ζ \zeta ζ ζ + \zeta^{+} ζ + ζ ∈ R \zeta\in R ζ ∈ R D \mathbb{D} D λ \lambda λ
Proof. Let ℓ : λ → V λ \ell:\lambda\to V_{\lambda} ℓ : λ → V λ 9 L \mathbb{L} L ( L α , Q ˙ β ∣ β < λ , α ≤ λ ) (\mathbb{L}_{\alpha},\dot{\mathbb{Q}}_{\beta}\mid\beta<\lambda,\alpha\leq\lambda) ( L α , Q ˙ β ∣ β < λ , α ≤ λ ) Q ˙ β \dot{\mathbb{Q}}_{\beta} Q ˙ β L β \mathbb{L}_{\beta} L β β \beta β ℓ ( α ) ∈ V β \ell(\alpha)\in V_{\beta} ℓ ( α ) ∈ V β α ∈ β \alpha\in\beta α ∈ β ℓ ( β ) \ell(\beta) ℓ ( β ) ( γ , τ ) (\gamma,\tau) ( γ , τ ) τ \tau τ L β \mathbb{L}_{\beta} L β ( < β ) (<\beta) ( < β )
One can verify that L \mathbb{L} L λ \lambda λ β \beta β ℓ \ell ℓ β \beta β L \mathbb{L} L Q ˙ β \dot{\mathbb{Q}}_{\beta} Q ˙ β L \mathbb{L} L V V V G ⊆ L G\subseteq\mathbb{L} G ⊆ L V [ G ] V[G] V [ G ] V [ G ] V[G] V [ G ] g ⊆ D g\subseteq\mathbb{D} g ⊆ D λ \lambda λ V [ G ] [ g ] V[G][g] V [ G ] [ g ]
To see this, suppose that μ \mu μ j : V → M j:V\to M j : V → M μ \mu μ j ( ℓ ) ( λ ) = ( μ , D ) j(\ell)(\lambda)=(\mu,\mathbb{D}) j ( ℓ ) ( λ ) = ( μ , D ) μ \mu μ μ = cf ( μ ) ≥ λ + \mu=\operatorname{cf}(\mu)\geq\lambda^{+} μ = cf ( μ ) ≥ λ + j j j P λ μ P_{\lambda}\mu P λ μ
By elementarity, j ( L ) = L ∗ D ˙ ∗ L ˙ t a i l j(\mathbb{L})=\mathbb{L}*\dot{\mathbb{D}}*\dot{\mathbb{L}}_{\mathrm{tail}} j ( L ) = L ∗ D ˙ ∗ L ˙ tail L t a i l \mathbb{L}_{\mathrm{tail}} L tail μ + \mu^{+} μ + V [ G ] [ g ] V[G][g] V [ G ] [ g ] F = { j ( q ) ∣ q ∈ g } F=\{j(q)\mid q\in g\} F = { j ( q ) ∣ q ∈ g } M M M j ↾ D ∈ M j\mathbin{\upharpoonright}\mathbb{D}\in M j ↾ D ∈ M F ∈ M [ G ] [ g ] F\in M[G][g] F ∈ M [ G ] [ g ] F F F R R R min ( R − λ ) > λ + \min(R-\lambda)>\lambda^{+} min ( R − λ ) > λ + ∣ F ∣ = λ + |F| = \lambda^{+} ∣ F ∣ = λ + m m m F F F
By GCH, which holds in V [ G ] [ g ] V[G][g] V [ G ] [ g ] L t a i l L_{\mathrm{tail}} L tail ∣ j ( 2 λ + ) ∣ V = ∣ P μ λ ( λ + + ) ∣ V = μ + |j(2^{\lambda^{+}})|^{V} = |P^{\lambda}_{\mu}(\lambda^{++})|^{V} = \mu^{+} ∣ j ( 2 λ + ) ∣ V = ∣ P μ λ ( λ ++ ) ∣ V = μ + L t a i l L_{\mathrm{tail}} L tail μ + \mu^{+} μ + M [ G ] [ g ] M[G][g] M [ G ] [ g ] h h h m m m m m m j ′ ′ g j''g j ′′ g h h h j : V → M j : V \to M j : V → M j + : V [ G ] [ g ] → M [ H ] [ h ] j^{+} : V[G][g] \to M[H][h] j + : V [ G ] [ g ] → M [ H ] [ h ] λ \lambda λ μ \mu μ V [ G ] [ g ] V[G][g] V [ G ] [ g ] μ \mu μ λ \lambda λ V [ G ] [ g ] V[G][g] V [ G ] [ g ] □ 2.5 \square_{2.5} □ 2.5
The above setting is rendered in a universe satisfying GCH. Our goal, however, is to force the negative arrow relation when 2 λ > λ + 2^{\lambda} > \lambda^{+} 2 λ > λ + λ = ℵ ω \lambda= \aleph_{\omega} λ = ℵ ω 2 λ 2^{\lambda} 2 λ ℵ ω \aleph_{\omega} ℵ ω [23 ] .
Let W W W L L L R ⊆ λ R \subseteq\lambda R ⊆ λ R R R ρ + 4 \rho^{+4} ρ + 4 ρ \rho ρ R R R λ \lambda λ ρ + 4 ∈ R \rho^{+4} \in R ρ + 4 ∈ R c ρ : [ ρ + 4 ] 2 → { 0 , 1 } c_{\rho} : [\rho^{+4}]^{2} \to\{0,1\} c ρ : [ ρ + 4 ] 2 → { 0 , 1 } c ρ c_{\rho} c ρ A ⊆ ρ + 4 A \subseteq\rho^{+4} A ⊆ ρ + 4 c ρ c_{\rho} c ρ A = ∅ m o d D ρ A = \varnothing\bmod\mathcal{D}_{\rho} A = ∅ mod D ρ D ρ \mathcal{D}_{\rho} D ρ ρ + 3 \rho^{+3} ρ + 3 ρ + 4 \rho^{+4} ρ + 4 I ρ I_{\rho} I ρ I ρ I_{\rho} I ρ ρ + 3 \rho^{+3} ρ + 3 ρ + 4 \rho^{+4} ρ + 4 c ρ c_{\rho} c ρ
Let D \mathbb{D} D R R R V V V W W W D \mathbb{D} D ( f α ∣ α ∈ λ + ) (f_{\alpha} \mid\alpha\in\lambda^{+}) ( f α ∣ α ∈ λ + ) ( λ , λ + + ) (\lambda,\lambda^{++}) ( λ , λ ++ ) E E E V V V j : V → M j : V \to M j : V → M U 0 \mathcal{U}_{0} U 0 E E E i : V → Ult ( V , U 0 ) ≅ N i : V \to\operatorname{Ult}(V,\mathcal{U}_{0}) \cong N i : V → Ult ( V , U 0 ) ≅ N K 0 K_{0} K 0 N N N Col N ( ( λ + 5 ) N , < i ( λ ) ) \operatorname{Col}^{N}((\lambda^{+5})^{N}, < i(\lambda)) Col N (( λ + 5 ) N , < i ( λ )) k : N → M k : N \to M k : N → M K K K k [ K 0 ] k[K_{0}] k [ K 0 ]
V → j M i ↓ ↑ k Ult ( V , U 0 ) ≅ N \begin{CD}
V @>j>> M \\
@V i VV @AA k A \\
\operatorname{Ult}(V,\mathcal{U}_{0}) \cong N
\end{CD} V i ↓ ⏐ Ult ( V , U 0 ) ≅ N j M ⏐ ↑ k Claim 2.6. K K K M M M Col M ( ( λ + 5 ) M , < j ( λ ) ) \operatorname{Col}^{M}((\lambda^{+5})^{M}, < j(\lambda)) Col M (( λ + 5 ) M , < j ( λ ))
Proof. Let D D D Col M ( ( λ + 5 ) M , < j ( λ ) ) \operatorname{Col}^{M}((\lambda^{+5})^{M}, < j(\lambda)) Col M (( λ + 5 ) M , < j ( λ )) D ∈ M D \in M D ∈ M D ∩ K ≠ ∅ D \cap K \ne\varnothing D ∩ K = ∅ j j j a ∈ [ λ + + ] n a \in[\lambda^{++}]^{n} a ∈ [ λ ++ ] n
for some n ∈ ω n \in\omega n ∈ ω f : [ λ ] n → V f:[\lambda]^n \to V f : [ λ ] n → V j f ( a ) = D jf(a)=D j f ( a ) = D D ′ = ⋂ { j f ( z ) ∣ z ∈ [ ( λ + + ) N ] n , j f ( z ) is a dense open subset of Col N ( ( λ + 5 ) N , < j ( λ ) ) } D'=\bigcap\{jf(z)\mid z\in[(\lambda^{++})^N]^n,\ jf(z)\text{ is a dense open subset of }\operatorname{Col}^N((\lambda^{+5})^N,<j(\lambda))\} D ′ = ⋂ { j f ( z ) ∣ z ∈ [( λ ++ ) N ] n , j f ( z ) is a dense open subset of Col N (( λ + 5 ) N , < j ( λ ))} D ′ ∈ N D'\in N D ′ ∈ N Col N ( ( λ + 5 ) N , < j ( λ ) ) \operatorname{Col}^N((\lambda^{+5})^N,<j(\lambda)) Col N (( λ + 5 ) N , < j ( λ )) ( λ + 5 ) N (\lambda^{+5})^N ( λ + 5 ) N D ′ D' D ′ ( λ + ) N (\lambda^+)^N ( λ + ) N D ′ D' D ′ D ′ ∩ K ≠ ∅ D'\cap K\ne\varnothing D ′ ∩ K = ∅ k ( D ′ ) ∩ K ≠ ∅ k(D')\cap K\ne\varnothing k ( D ′ ) ∩ K = ∅ D = k ( j f ) ( a ) D=k(jf)(a) D = k ( j f ) ( a ) a ∈ k ( [ ( λ + + ) N ] n ) a\in k([( \lambda^{++})^N]^n) a ∈ k ([( λ ++ ) N ] n ) D ⊇ k ( D ′ ) D\supseteq k(D') D ⊇ k ( D ′ ) D ∩ K ≠ ∅ D\cap K\ne\varnothing D ∩ K = ∅
Working in V V V P \mathbb{P} P E E E K K K [23 ] , for every d ∈ [ λ + + ] ≤ λ d\in[\lambda^{++}]^{\le\lambda} d ∈ [ λ ++ ] ≤ λ E ( d ) = { x ∈ V λ ∣ ( j ↾ d ) − 1 ∈ j ( x ) } E(d)=\{x\in V_\lambda\mid(j\mathbin{\upharpoonright}d)^{-1}\in j(x)\} E ( d ) = { x ∈ V λ ∣ ( j ↾ d ) − 1 ∈ j ( x )} E ( d ) E(d) E ( d ) λ \lambda λ V λ V_\lambda V λ ( j ↾ d ) − 1 (j\mathbin{\upharpoonright}d)^{-1} ( j ↾ d ) − 1 j ( d ) j(d) j ( d ) d d d E ( d ) E(d) E ( d ) A ∈ E ( d ) A\in E(d) A ∈ E ( d ) A A A d d d
We define our forcing notion P \mathbb{P} P p ∈ P p\in\mathbb{P} p ∈ P ( c ˉ p , f p , A p , F p ) (\bar c^p,f^p,A^p,F^p) ( c ˉ p , f p , A p , F p )
(a) f p f^p f p λ + + − λ \lambda^{++}-\lambda λ ++ − λ λ < ω \lambda^{<\omega} λ < ω
(b) d p = dom ( f p ) d^p=\operatorname{dom}(f^p) d p = dom ( f p ) ∣ d p ∣ ≤ λ |d^p|\le\lambda ∣ d p ∣ ≤ λ
(c) λ ∈ d p \lambda\in d^p λ ∈ d p p p p f p ( λ ) = ( ρ i p ∣ i ∈ n ) f^p(\lambda)=(\rho_i^p\mid i\in n) f p ( λ ) = ( ρ i p ∣ i ∈ n )
(d) c ˉ p ∈ ∏ i ≤ n Col ( ( ρ i − 1 p ) + 5 , < ρ i p ) \bar c^p\in\prod_{i\le n}\operatorname{Col}((\rho_{i-1}^p)^{+5},<\rho_i^p) c ˉ p ∈ ∏ i ≤ n Col (( ρ i − 1 p ) + 5 , < ρ i p ) ρ − 1 p = ω \rho_{-1}^p=\omega ρ − 1 p = ω ρ n p = λ \rho_n^p=\lambda ρ n p = λ
(e) A p ∈ E ( d p ) A^p\in E(d^p) A p ∈ E ( d p )
(f) If η ∈ A p \eta\in A^p η ∈ A p η ( λ ) > ⋃ dom ( c n p ) \eta(\lambda)>\bigcup\operatorname{dom}(c_n^p) η ( λ ) > ⋃ dom ( c n p ) η ( λ ) > ρ n − 1 p \eta(\lambda)>\rho_{n-1}^p η ( λ ) > ρ n − 1 p
(g) F p F^p F p A p A^p A p V λ V_\lambda V λ
(h) For every η ∈ A p \eta\in A^p η ∈ A p F p ( η ) ∈ Col ( η ( λ ) + 5 , < λ ) F^p(\eta)\in\operatorname{Col}(\eta(\lambda)^{+5},<\lambda) F p ( η ) ∈ Col ( η ( λ ) + 5 , < λ )
(i) j ( F p ) ( ( j ↾ d p ) − 1 ) ∈ K j(F^p)((j\mathbin{\upharpoonright}d^p)^{-1})\in K j ( F p ) (( j ↾ d p ) − 1 ) ∈ K
We have to define the forcing order. To this end, we define the direct order ≤ ∗ \le^* ≤ ∗
Assume, therefore, that p , q ∈ P p,q\in\mathbb{P} p , q ∈ P q q q p p p p ≤ ∗ q p\le^*q p ≤ ∗ q
(α \alpha α f p ⊆ f q f^p\subseteq f^q f p ⊆ f q lg ( f p ( λ ) ) = lg ( f q ( λ ) ) \lg(f^p(\lambda))=\lg(f^q(\lambda)) lg ( f p ( λ )) = lg ( f q ( λ ))
(β \beta β A q ⊆ { η ∣ η ↾ d p ∈ A p } A^q\subseteq\{\eta\mid\eta\mathbin{\upharpoonright}d^p\in A^p\} A q ⊆ { η ∣ η ↾ d p ∈ A p }
(γ \gamma γ c i q c_i^q c i q c i p c_i^p c i p i ≤ n i\le n i ≤ n 10
(δ \delta δ F q ( η ) ≥ F p ( η ↾ d p ) F^q(\eta)\ge F^p(\eta\mathbin{\upharpoonright}d^p) F q ( η ) ≥ F p ( η ↾ d p ) η ∈ A q \eta\in A^q η ∈ A q ≥ \ge ≥
Assume now that p ∈ P p\in\mathbb{P} p ∈ P η ∈ A p \eta\in A^p η ∈ A p p ⌢ η p^\frown\eta p ⌢ η r r r
(α \alpha α f r ( α ) = f p ( α ) f^r(\alpha)=f^p(\alpha) f r ( α ) = f p ( α ) α ∉ dom ( η ) \alpha\notin\operatorname{dom}(\eta) α ∈ / dom ( η ) f r ( α ) = f p ( α ) ⌢ η ( α ) f^r(\alpha)=f^p(\alpha)^\frown\eta(\alpha) f r ( α ) = f p ( α ) ⌢ η ( α ) α ∈ dom ( η ) \alpha\in\operatorname{dom}(\eta) α ∈ dom ( η )
(β \beta β c ˉ r = c ˉ p ⌢ F p ( η ) \bar c^r=\bar c^p{}^\frown F^p(\eta) c ˉ r = c ˉ p ⌢ F p ( η )
(γ \gamma γ A r = A p − { ρ ∈ A p ∣ ρ ( λ ) ≤ max { η ( λ ) , ⋃ dom ( F p ( η ) ) } } A^r=A^p-\{\rho\in A^p\mid\rho(\lambda)\le\max\{\eta(\lambda),\bigcup\operatorname{dom}(F^p(\eta))\}\} A r = A p − { ρ ∈ A p ∣ ρ ( λ ) ≤ max { η ( λ ) , ⋃ dom ( F p ( η ))}}
( δ ) F r = F p ↾ A r . (\delta)\quad F^r=F^p\upharpoonright A^r. ( δ ) F r = F p ↾ A r .
Finally, q q q p p p q q q p p p p ≤ P q p\leq_{\mathbb{P}}q p ≤ P q q q q p p p
Notice that ( P , ≤ ∗ ) (\mathbb{P},\leq^*) ( P , ≤ ∗ ) σ \sigma σ E ( d ) E(d) E ( d ) λ \lambda λ ℵ 1 \aleph_1 ℵ 1 P \mathbb{P} P λ \lambda λ λ \lambda λ λ = ℵ ω \lambda=\aleph_\omega λ = ℵ ω λ \lambda λ 2 ℵ ω = ℵ ω + 2 2^{\aleph_\omega}=\aleph_{\omega+2} 2 ℵ ω = ℵ ω + 2
Claim 2.7. Let ρ ˉ = ( ρ n ∣ n ∈ ω ) \bar{\rho}=(\rho_n\mid n\in\omega) ρ ˉ = ( ρ n ∣ n ∈ ω ) p ∈ P p\in\mathbb{P} p ∈ P τ \tau τ P \mathbb{P} P ρ n + 5 \rho_n^{+5} ρ n + 5 q ∈ P q\in\mathbb{P} q ∈ P p ≤ ∗ q p\leq^*q p ≤ ∗ q g g g dom ( g ) = λ \operatorname{dom}(g)=\lambda dom ( g ) = λ g ( α ) ∈ [ α + 5 ] ≤ α + + g(\alpha)\in[\alpha^{+5}]^{\leq\alpha^{++}} g ( α ) ∈ [ α + 5 ] ≤ α ++ α ∈ λ \alpha\in\lambda α ∈ λ q ⊩ τ ∈ g ( ρ n ) q\Vdash\tau\in g(\rho_n) q ⊩ τ ∈ g ( ρ n )
Proof. Let D D D τ \tau τ n n n q q q p ≤ ∗ q p\leq^*q p ≤ ∗ q q q q n n n D D D
Recall that d p d^p d p F p F^p F p η 0 , … , η n − 1 \eta_0,\ldots,\eta_{n-1} η 0 , … , η n − 1 n n n q q q τ \tau τ q q q η i \eta_i η i
Since d p ∈ [ λ + + ] ≤ λ d^p\in[\lambda^{++}]^{\leq\lambda} d p ∈ [ λ ++ ] ≤ λ ( j ↾ d p ) − 1 ∈ j ( A q ) (j\upharpoonright d^p)^{-1}\in j(A^q) ( j ↾ d p ) − 1 ∈ j ( A q ) A q A^q A q η ∈ A q \eta\in A^q η ∈ A q α ∈ dom ( η ) ⊆ d p \alpha\in\operatorname{dom}(\eta)\subseteq d^p α ∈ dom ( η ) ⊆ d p η ( α ) ≤ η ( λ ) + + \eta(\alpha)\leq\eta(\lambda)^{++} η ( α ) ≤ η ( λ ) ++ h : λ → d p h:\lambda\to d^p h : λ → d p A q A^q A q dom ( η ) = h ′ ′ η ( λ ) \operatorname{dom}(\eta)=h^{\prime\prime}\eta(\lambda) dom ( η ) = h ′′ η ( λ ) ∣ { η ∈ A q ∣ α = η ( λ ) } ∣ ≤ ∣ [ α + + ] α ∣ = α + + |\{\eta\in A^q\mid\alpha=\eta(\lambda)\}|\leq|[\alpha^{++}]^\alpha|=\alpha^{++} ∣ { η ∈ A q ∣ α = η ( λ )} ∣ ≤ ∣ [ α ++ ] α ∣ = α ++ α ∈ λ \alpha\in\lambda α ∈ λ g ( α ) g(\alpha) g ( α ) τ \tau τ η 0 , … , η n − 1 \eta_0,\ldots,\eta_{n-1} η 0 , … , η n − 1 η n − 1 ( λ ) = α \eta_{n-1}(\lambda)=\alpha η n − 1 ( λ ) = α q ⊩ τ ∈ g ( ρ n ) q\Vdash\tau\in g(\rho_n) q ⊩ τ ∈ g ( ρ n )
For every n ∈ ω n\in\omega n ∈ ω D n \mathcal{D}_n D n ρ n + 3 \rho_n^{+3} ρ n + 3 ρ n + 4 \rho_n^{+4} ρ n + 4 I n \mathcal{I}_n I n D n \mathcal{D}_n D n P \mathbb{P} P λ \lambda λ P \mathbb{P} P
Claim 2.8. Let c : [ ρ n + 4 ] 2 → { 0 , 1 } c:[\rho_n^{+4}]^2\to\{0,1\} c : [ ρ n + 4 ] 2 → { 0 , 1 } I ˙ \dot{I} I ˙ − m o n o c h r o m a t i c s e t u n d e r -monochromatic set under − m o n oc h r o ma t i cse t u n d er I ˙ \dot{I} I ˙ ρ n + 3 \rho_n^{+3} ρ n + 3 0 0 0 I ˙ ⊆ I \dot{I}\subseteq I I ˙ ⊆ I I ∈ V I\in V I ∈ V I = ∅ m o d D I=\varnothing\bmod\mathcal{D} I = ∅ mod D D \mathcal{D} D
Proof. Fix an enumeration of P V ( ρ n + 4 ) \mathbb{P}^V(\rho_n^{+4}) P V ( ρ n + 4 ) n ∈ ω n\in\omega n ∈ ω ρ n + 5 \rho_n^{+5} ρ n + 5 p p p I ˙ \dot{I} I ˙ 0 0 0 f p ( λ ) f^{p}(\lambda) f p ( λ ) ℓ g ( f p ( λ ) ) > n \ell g(f^{p}(\lambda))>n ℓ g ( f p ( λ )) > n ρ n \rho_n ρ n p p p
Let C = ∏ i ≤ n Col ( ρ i − 1 + 4 , < ρ i ) C=\prod_{i\leq n} \operatorname{Col}(\rho_{i-1}^{+4},<\rho_i) C = ∏ i ≤ n Col ( ρ i − 1 + 4 , < ρ i ) C C C n + 1 n+1 n + 1 p p p ∣ C ∣ = ρ n |C|=\rho_n ∣ C ∣ = ρ n ρ n + 5 \rho_{n}^{+5} ρ n + 5 ( ( s α , ξ α ) ∣ α ∈ ρ n + 4 ) ((s_\alpha,\xi_\alpha)\mid\alpha\in\rho_n^{+4}) (( s α , ξ α ) ∣ α ∈ ρ n + 4 ) C × ρ n + 4 C\times\rho_n^{+4} C × ρ n + 4 α ∈ ρ n + 4 \alpha\in\rho_n^{+4} α ∈ ρ n + 4 p α p_\alpha p α
( ℵ ) (\aleph) ( ℵ ) ( p α ∣ α ∈ ρ n + 4 ) (p_\alpha\mid\alpha\in\rho_n^{+4}) ( p α ∣ α ∈ ρ n + 4 ) ≤ ∗ \leq^* ≤ ∗
( ℶ ) (\beth) ( ℶ ) c ˉ p α ↾ n + 1 \bar c^{p_\alpha}\mathbin{\upharpoonright} n+1 c ˉ p α ↾ n + 1
( ℷ ) (\gimel) ( ℷ ) r r r p α p_\alpha p α p α p_\alpha p α c ˉ r ↾ n + 2 = s α ⌢ c n + 1 r \bar c^r\mathbin{\upharpoonright} n+2=s_\alpha^\frown c_{n+1}^r c ˉ r ↾ n + 2 = s α ⌢ c n + 1 r r r r ξ ˇ α ∈ I ˙ \check{\xi}_\alpha\in\dot I ξ ˇ α ∈ I ˙ p α p_\alpha p α
Let q q q ( p α ∣ α ∈ ρ n + 4 ) (p_\alpha\mid\alpha\in\rho_n^{+4}) ( p α ∣ α ∈ ρ n + 4 ) q [ s ] q[s] q [ s ] q q q c ˉ q \bar c^q c ˉ q s ⌢ ( c ˉ q ↾ [ n , ℓ g ( c ˉ q ) ) ) s^\frown(\bar c^q\mathbin{\upharpoonright}[n,\ell g(\bar c^q))) s ⌢ ( c ˉ q ↾ [ n , ℓ g ( c ˉ q )))
Let r 0 = c ˉ q ↾ n + 1 r_0=\bar c^q\mathbin{\upharpoonright} n+1 r 0 = c ˉ q ↾ n + 1 s ∈ C s\in C s ∈ C I s = { α ∈ ρ n + 4 ∣ q [ s ] ⊩ α ˇ ∈ I ˙ } I_s=\{\alpha\in\rho_n^{+4}\mid q[s]\Vdash\check{\alpha}\in\dot I\} I s = { α ∈ ρ n + 4 ∣ q [ s ] ⊩ α ˇ ∈ I ˙ } I s ∈ V I_s\in V I s ∈ V ⊩ P I ˙ ⊆ ⋃ { I s ∣ r 0 ≤ C s } \Vdash_{\mathbb P}\dot I\subseteq\bigcup\{I_s\mid r_0\leq_C s\} ⊩ P I ˙ ⊆ ⋃ { I s ∣ r 0 ≤ C s } I s I_s I s c c c s s s r 0 ≤ C s r_0\leq_Cs r 0 ≤ C s I s ∈ I n I_s\in\mathcal I_n I s ∈ I n I n \mathcal I_n I n D n \mathcal D_n D n I ˙ \dot I I ˙ I n \mathcal I_n I n
□ 2.8 \square_{2.8} □ 2.8
Let R = { ρ n + 4 ∣ n ∈ ω } R=\{\rho_n^{+4}\mid n\in\omega\} R = { ρ n + 4 ∣ n ∈ ω } ( f α ↾ R ∣ α ∈ λ + ) (f_\alpha\mathbin{\upharpoonright}R\mid\alpha\in\lambda^+) ( f α ↾ R ∣ α ∈ λ + ) Theorem 2.2 . Specifically, we must show that for every sequence of sets ( A n ∣ n ∈ ω ) (A_n\mid n\in\omega) ( A n ∣ n ∈ ω ) A n ∈ D n A_n\in\mathcal D_n A n ∈ D n n ∈ ω n\in\omega n ∈ ω γ ∈ λ + \gamma\in\lambda^+ γ ∈ λ + γ ≤ β ∈ λ + \gamma\leq\beta\in\lambda^+ γ ≤ β ∈ λ + n β ∈ ω n_\beta\in\omega n β ∈ ω f β ( n ) ∈ A n f_\beta(n)\in A_n f β ( n ) ∈ A n n β ≤ n ∈ ω n_\beta\leq n\in\omega n β ≤ n ∈ ω
By Claim 2.8 we may assume that A n ∈ V A_n\in V A n ∈ V n ∈ ω n\in\omega n ∈ ω ( A n ∣ n ∈ ω ) (A_n\mid n\in\omega) ( A n ∣ n ∈ ω ) V V V p p p P \mathbb P P Claim 2.7 repeatedly and using the enumeration of the ground model sets, we construct a ≤ ∗ \leq^* ≤ ∗ ( q n ∣ n ∈ ω ) (q_n\mid n\in\omega) ( q n ∣ n ∈ ω ) q 0 ≥ p q_0\geq p q 0 ≥ p q n ⊩ A n ∈ A ˙ n ( ρ n ) q_n\Vdash A_n\in\dot{\mathcal A}_n(\rho_n) q n ⊩ A n ∈ A ˙ n ( ρ n ) ∣ A n ( γ ) ∣ ≤ γ + + |\mathcal A_n(\gamma)|\leq\gamma^{++} ∣ A n ( γ ) ∣ ≤ γ ++ γ ∈ λ \gamma\in\lambda γ ∈ λ
Since ( P , ≤ ∗ ) (\mathbb P,\leq^*) ( P , ≤ ∗ ) σ \sigma σ q q q q n ≤ q q_n\leq q q n ≤ q n ∈ ω n\in\omega n ∈ ω ζ ∈ R \zeta\in R ζ ∈ R B ζ = ⋂ { E n ∣ n ∈ ω , E n ∈ A n ( ζ ) } B_\zeta=\bigcap\{E_n\mid n\in\omega,E_n\in\mathcal A_n(\zeta)\} B ζ = ⋂ { E n ∣ n ∈ ω , E n ∈ A n ( ζ )} D ζ \mathcal D_\zeta D ζ B ζ ∈ D ζ B_\zeta\in\mathcal D_\zeta B ζ ∈ D ζ ( B ζ ∣ ζ ∈ R ) (B_\zeta\mid\zeta\in R) ( B ζ ∣ ζ ∈ R ) γ ∈ λ + \gamma\in\lambda^+ γ ∈ λ + γ ≤ β ∈ λ + \gamma\leq\beta\in\lambda^+ γ ≤ β ∈ λ + ζ β ∈ λ \zeta_\beta\in\lambda ζ β ∈ λ f β ( ζ ) ∈ B ζ f_\beta(\zeta)\in B_\zeta f β ( ζ ) ∈ B ζ ζ β ≤ ζ ∈ R \zeta_\beta\leq\zeta\in R ζ β ≤ ζ ∈ R B ρ n + 4 ⊆ A n B_{\rho_n^{+4}}\subseteq A_n B ρ n + 4 ⊆ A n n ∈ ω n\in\omega n ∈ ω
In order to force the failure of SCH at λ \lambda λ κ \kappa κ o ( κ ) = κ + + o(\kappa)=\kappa^{++} o ( κ ) = κ ++ [11 ] and in [12 ] . In our constructions we started from a supercompact cardinal in the ground model. The gap between these large cardinals invites the following:
Question 2.9. Let λ \lambda λ
(א) What is the consistency strength of the negative arrow relation λ + ↛ ( λ + , 3 ) cf ( λ ) 2 \lambda^+ \not\to(\lambda^+,3)_{\operatorname{cf}(\lambda)}^2 λ + → ( λ + , 3 ) cf ( λ ) 2 2 λ > λ + 2^\lambda> \lambda^+ 2 λ > λ +
(ב) What is the consistency strength of the same negative relation with 2 λ > λ + 2^\lambda> \lambda^+ 2 λ > λ + λ = ℵ ω \lambda= \aleph_\omega λ = ℵ ω
(ג) What is the consistency strength of the negative relation at every strong limit singular cardinal λ \lambda λ 2 λ > λ + 2^\lambda> \lambda^+ 2 λ > λ + λ \lambda λ
A negative relation from stick
In the previous sections we used pcf assumptions in order to cope with the problem of Erdős and Hajnal. We move now to the second approach in which prediction principles play an important role. The prediction principle that we need for the combinatorial proof is called stick . The idea of stick as a prediction principle is well-articulated in [2 Chapter 4(12)] : “It consults its stick, its rod directs it”. Here we need the mathematical incarnation of this idea. We commence with the following definition.
Definition 3.1. Suppose that κ < λ \kappa<\lambda κ < λ
(ℵ) ♣ ( κ , λ ) = min { ∣ F ∣ ∣ F ⊆ [ λ ] κ ∧ ∀ y ∈ [ λ ] λ ∃ x ∈ F , x ⊆ y } \clubsuit(\kappa,\lambda)=\min\{|{\cal F}|\mid{\cal F}\subseteq[\lambda]^\kappa\land\forall y\in[\lambda]^\lambda\exists x\in{\cal F},x\subseteq y\} ♣ ( κ , λ ) = min { ∣ F ∣ ∣ F ⊆ [ λ ] κ ∧ ∀ y ∈ [ λ ] λ ∃ x ∈ F , x ⊆ y }
(ℶ) Denote ♣ ( λ , λ + ) \clubsuit(\lambda,\lambda^+) ♣ ( λ , λ + ) ♣ ( λ ) \clubsuit(\lambda) ♣ ( λ )
The stick principle is closely related to the club principle, but no stationary sets are involved in the prediction. This fact makes ♣ ( λ ) \clubsuit(\lambda) ♣ ( λ ) λ \lambda λ 11 tiltan ), which appeared for the first time in [24 ] . If κ = cf ( κ ) > ℵ 0 \kappa=\operatorname{cf}(\kappa)>\aleph_0 κ = cf ( κ ) > ℵ 0 S ⊆ κ S\subseteq\kappa S ⊆ κ ( T δ ∣ δ ∈ S ) (T_\delta\mid\delta\in S) ( T δ ∣ δ ∈ S ) T δ T_\delta T δ δ \delta δ δ ∈ S , \delta\in S, δ ∈ S , 12 A ∈ [ κ ] κ A\in[\kappa]^\kappa A ∈ [ κ ] κ S A = { δ ∈ S ∣ T δ ⊆ A } S_A=\{\delta\in S\mid T_\delta\subseteq A\} S A = { δ ∈ S ∣ T δ ⊆ A } ♣ S \clubsuit_S ♣ S
We proceed to the combinatorial result. Our goal is to prove the relation λ + ↛ ( λ + , ( 3 ) cf ( λ ) ) 2 \lambda^+\nrightarrow(\lambda^+,(3)^{\operatorname{cf}(\lambda)})^2 λ + ↛ ( λ + , ( 3 ) cf ( λ ) ) 2 ♣ ( λ ) \clubsuit(\lambda) ♣ ( λ ) κ \kappa κ ♣ ( κ ) \clubsuit(\kappa) ♣ ( κ ) [5 ] . Here we apply a similar idea to successors of singular cardinals. We need the following lemma about free sets from [18 ] . The lemma and its proof also appear in [EHMR84, Lemma 20.3].
Lemma 3.2. Let κ \kappa κ E = ⋃ α ∈ κ E α E=\bigcup_{\alpha\in\kappa}E_\alpha E = ⋃ α ∈ κ E α ∣ E α ∣ > κ |E_\alpha|>\kappa ∣ E α ∣ > κ α ∈ κ \alpha\in\kappa α ∈ κ f : E → P ( E ) f:E\to\mathcal{P}(E) f : E → P ( E ) ∣ f ( x ) ∩ E α ∣ < κ |f(x)\cap E_\alpha|<\kappa ∣ f ( x ) ∩ E α ∣ < κ x ∈ E , α ∈ κ x\in E,\alpha\in\kappa x ∈ E , α ∈ κ X X X f f f X ∩ E α ≠ ∅ X\cap E_\alpha\ne\varnothing X ∩ E α = ∅ α ∈ κ \alpha\in\kappa α ∈ κ
We can state now the following:
Theorem 3.3. Suppose that θ = cf ( λ ) < λ \theta=\operatorname{cf}(\lambda)<\lambda θ = cf ( λ ) < λ ♣ ( λ ) \clubsuit(\lambda) ♣ ( λ ) λ + ↛ ( λ + , ( 3 ) cf ( λ ) ) 2 \lambda^+\nrightarrow(\lambda^+,(3)^{\operatorname{cf}(\lambda)})^2 λ + ↛ ( λ + , ( 3 ) cf ( λ ) ) 2
Proof. Let ( κ i ∣ 1 ≤ i ∈ θ ) (\kappa_i\mid1\leq i\in\theta) ( κ i ∣ 1 ≤ i ∈ θ ) λ = ⋃ i ∈ θ κ i \lambda= \bigcup_{i\in\theta}\kappa_i λ = ⋃ i ∈ θ κ i κ 1 \kappa_1 κ 1 P = ( P i ∣ i ∈ θ ) \mathcal{P}=(P_i\mid i\in\theta) P = ( P i ∣ i ∈ θ ) [ λ + ] 2 [\lambda^+]^2 [ λ + ] 2 13 α < β < λ + \alpha<\beta<\lambda^+ α < β < λ + c ( α , β ) = i c(\alpha,\beta)=i c ( α , β ) = i { α , β } ∈ P i \{\alpha,\beta\}\in P_i { α , β } ∈ P i 14 P \mathcal{P} P i ∈ ( 0 , θ ) i\in(0,\theta) i ∈ ( 0 , θ ) f i : λ + → [ λ + ] ≤ κ i f_i:\lambda^+\to[\lambda^+]^{\leq\kappa_i} f i : λ + → [ λ + ] ≤ κ i f i ( α ) ⊆ α f_i(\alpha)\subseteq\alpha f i ( α ) ⊆ α i ∈ θ i\in\theta i ∈ θ α ∈ λ + \alpha\in\lambda^+ α ∈ λ + P i = { { α , β } ∣ α < β < λ + , α ∈ f i ( β ) } P_i=\{\{\alpha,\beta\}\mid\alpha<\beta<\lambda^+,\alpha\in f_i(\beta)\} P i = {{ α , β } ∣ α < β < λ + , α ∈ f i ( β )} P i P_i P i i > 0 i>0 i > 0 P 0 = [ λ + ] 2 − ⋃ { P i ∣ 1 ≤ i < θ } P_0=[\lambda^+]^2-\bigcup\{P_i\mid1\leq i<\theta\} P 0 = [ λ + ] 2 − ⋃ { P i ∣ 1 ≤ i < θ }
The construction of each f i f_i f i α ∈ λ + \alpha\in\lambda^+ α ∈ λ + α \alpha α f i ( α ) f_i(\alpha) f i ( α ) i ∈ ( 0 , θ ) i\in(0,\theta) i ∈ ( 0 , θ ) α ∈ λ + \alpha\in\lambda^+ α ∈ λ + f i ( γ ) f_i(\gamma) f i ( γ ) γ ∈ α \gamma\in\alpha γ ∈ α i ∈ θ i\in\theta i ∈ θ ( T η ∣ η ∈ λ + ) (T_\eta\mid\eta\in\lambda^+) ( T η ∣ η ∈ λ + ) λ \lambda λ T η ∈ [ λ + ] λ T_\eta\in[\lambda^+]^\lambda T η ∈ [ λ + ] λ η ∈ λ + \eta\in\lambda^+ η ∈ λ + S α = { T η ∣ η ∈ α , T η ⊆ α } S_\alpha=\{T_\eta\mid\eta\in\alpha,T_\eta\subseteq\alpha\} S α = { T η ∣ η ∈ α , T η ⊆ α } ∣ S α ∣ ≤ ∣ α ∣ ≤ λ |S_\alpha|\leq|\alpha|\leq\lambda ∣ S α ∣ ≤ ∣ α ∣ ≤ λ S α = ⋃ { S i α ∣ 1 ≤ i ∈ θ } S_\alpha=\bigcup\{S_i^\alpha\mid1\leq i\in\theta\} S α = ⋃ { S i α ∣ 1 ≤ i ∈ θ } i < j ⇒ S i α ∩ S j α = ∅ i<j\Rightarrow S_i^\alpha\cap S_j^\alpha=\varnothing i < j ⇒ S i α ∩ S j α = ∅ ∣ S i α ∣ ≤ κ i |S_i^\alpha|\leq\kappa_i ∣ S i α ∣ ≤ κ i i ∈ ( 0 , θ ) i\in(0,\theta) i ∈ ( 0 , θ )
In order to define f i ( α ) f_i(\alpha) f i ( α ) i ∈ ( 0 , θ ) i\in(0,\theta) i ∈ ( 0 , θ ) i i i κ i \kappa_i κ i κ \kappa κ f i ↾ α f_i\mathbin{\upharpoonright}\alpha f i ↾ α f f f ∣ f i ( γ ) ∣ ≤ κ i |f_i(\gamma)|\leq\kappa_i ∣ f i ( γ ) ∣ ≤ κ i γ ∈ α \gamma\in\alpha γ ∈ α X = X α i X=X_\alpha^i X = X α i f i ↾ α f_i\mathbin{\upharpoonright}\alpha f i ↾ α X ∩ T ≠ ∅ X\cap T\ne\varnothing X ∩ T = ∅ T ∈ S i α T\in S_i^\alpha T ∈ S i α X X X ∣ X ∣ ≤ ∣ S i α ∣ ≤ κ i |X|\leq|S_i^\alpha|\leq\kappa_i ∣ X ∣ ≤ ∣ S i α ∣ ≤ κ i f i ( α ) = X = X α i f_i(\alpha)=X=X_\alpha^i f i ( α ) = X = X α i P \mathcal{P} P [ λ + ] 2 [\lambda^+]^2 [ λ + ] 2
We define, at this stage, the coloring c : [ λ + ] 2 → θ c:[\lambda^+]^2\to\theta c : [ λ + ] 2 → θ c ( α , β ) = i c(\alpha,\beta)=i c ( α , β ) = i i ∈ θ i\in\theta i ∈ θ { α , β } ∈ P i \{\alpha,\beta\}\in P_i { α , β } ∈ P i c c c α < β < δ < λ + \alpha<\beta<\delta<\lambda^+ α < β < δ < λ + i ∈ θ i\in\theta i ∈ θ c ( α , β ) = c ( α , δ ) = c ( β , δ ) = i c(\alpha,\beta)=c(\alpha,\delta)=c(\beta,\delta)=i c ( α , β ) = c ( α , δ ) = c ( β , δ ) = i i > 0 i>0 i > 0 α < β < δ < λ + \alpha<\beta<\delta<\lambda^+ α < β < δ < λ + c ( α , δ ) = c ( β , δ ) = i c(\alpha,\delta)=c(\beta,\delta)=i c ( α , δ ) = c ( β , δ ) = i { α , δ } , { β , δ } ∈ P i \{\alpha,\delta\},\{\beta,\delta\}\in P_i { α , δ } , { β , δ } ∈ P i α , β ∈ f i ( δ ) = X \alpha,\beta\in f_i(\delta)=X α , β ∈ f i ( δ ) = X X X X f i ↾ δ f_i\mathbin{\upharpoonright}\delta f i ↾ δ β ∈ X \beta\in X β ∈ X f i ( β ) ∩ X = ∅ f_i(\beta)\cap X=\varnothing f i ( β ) ∩ X = ∅ α ∈ X \alpha\in X α ∈ X α ∉ f i ( β ) \alpha\notin f_i(\beta) α ∈ / f i ( β ) c ( α , β ) ≠ i c(\alpha,\beta)\ne i c ( α , β ) = i
Secondly, we argue that there is no 0 0 0 λ + \lambda^+ λ + λ + \lambda^+ λ + A ∈ [ λ + ] λ + A\in[\lambda^+]^{\lambda^+} A ∈ [ λ + ] λ + η ∈ λ + \eta\in\lambda^+ η ∈ λ + T η ⊆ A T_\eta\subseteq A T η ⊆ A ξ > η \xi>\eta ξ > η ξ > sup ( T η ) \xi>\sup(T_\eta) ξ > sup ( T η ) T η ∈ S ξ T_\eta\in S_\xi T η ∈ S ξ A A A λ + \lambda^+ λ + ξ > η , sup ( T η ) \xi>\eta,\sup(T_\eta) ξ > η , sup ( T η ) ξ ∈ A \xi\in A ξ ∈ A S ξ = ⋃ { S i ξ ∣ 1 ≤ i ∈ θ } S_\xi=\bigcup\{S_i^\xi\mid1\leq i\in\theta\} S ξ = ⋃ { S i ξ ∣ 1 ≤ i ∈ θ } T η ∈ S i ξ T_\eta\in S_i^\xi T η ∈ S i ξ i ∈ ( 0 , θ ) i\in(0,\theta) i ∈ ( 0 , θ )
By the choice of f i ( ξ ) f_i(\xi) f i ( ξ ) T η ∩ f i ( ξ ) ≠ ∅ T_\eta\cap f_i(\xi)\ne\varnothing T η ∩ f i ( ξ ) = ∅ α ∈ T η ∩ f i ( ξ ) \alpha\in T_\eta\cap f_i(\xi) α ∈ T η ∩ f i ( ξ ) α ∈ f i ( ξ ) \alpha\in f_i(\xi) α ∈ f i ( ξ ) { α , ξ } ∈ P i \{\alpha,\xi\}\in P_i { α , ξ } ∈ P i c ( α , ξ ) ≠ 0 c(\alpha,\xi) \ne0 c ( α , ξ ) = 0 α ∈ T η ⊆ A \alpha\in T_\eta\subseteq A α ∈ T η ⊆ A ξ ∈ A \xi\in A ξ ∈ A c ′ ′ [ A ] 2 ≠ { 0 } c^{\prime\prime}[A]^2 \ne\{0\} c ′′ [ A ] 2 = { 0 } □ 3.3 \square_{3.3} □ 3.3
We would like to emphasize an important aspect of the proof. The size of each T η T_\eta T η λ \lambda λ ♣ ( λ ) = ♣ ( λ , λ + ) \clubsuit(\lambda)=\clubsuit(\lambda,\lambda^+) ♣ ( λ ) = ♣ ( λ , λ + ) ♣ ( λ ) \clubsuit(\lambda) ♣ ( λ ) ♣ ( κ , λ + ) \clubsuit(\kappa,\lambda^+) ♣ ( κ , λ + ) ♣ S κ + \clubsuit_{S_{\kappa^+}} ♣ S κ + κ < λ \kappa<\lambda κ < λ ♣ ( λ ) \clubsuit(\lambda) ♣ ( λ ) λ \lambda λ X X X E α E_\alpha E α E α E_\alpha E α ∣ T η ∣ > κ i \lvert T_\eta\rvert>\kappa_i ∣ T η ∣ > κ i κ i < λ \kappa_i<\lambda κ i < λ T η T_\eta T η E α E_\alpha E α κ i \kappa_i κ i κ i \kappa_i κ i ( κ i ∣ i ∈ θ ) (\kappa_i\mid i\in\theta) ( κ i ∣ i ∈ θ ) T η T_\eta T η λ \lambda λ
Though we do not know how to force ♣ ( λ ) \clubsuit(\lambda) ♣ ( λ ) λ \lambda λ λ \lambda λ U \mathcal U U λ \lambda λ Gal ( U , λ + , λ + ) \operatorname{Gal}(\mathcal U,\lambda^+,\lambda^+) Gal ( U , λ + , λ + ) { A α ∣ α ∈ λ } ⊆ U \{A_\alpha\mid\alpha\in\lambda\}\subseteq\mathcal U { A α ∣ α ∈ λ } ⊆ U { A α i ∣ i ∈ λ + } \{A_{\alpha_i}\mid i\in\lambda^+\} { A α i ∣ i ∈ λ + } ⋂ i ∈ λ + A α i ∈ U \bigcap_{i\in\lambda^+}A_{\alpha_i}\in\mathcal U ⋂ i ∈ λ + A α i ∈ U λ \lambda λ [3 ] . If one forces Prikry through such an ultrafilter then every new set of size λ + \lambda^+ λ + λ + \lambda^+ λ +
Indeed, let A ˙ \dot A A ˙ λ + \lambda^+ λ + p ∈ P p\in\mathbb P p ∈ P α ∈ A ˙ [ G ] \alpha\in\dot A[G] α ∈ A ˙ [ G ] p α = ( s α , A α ) p_\alpha=(s_\alpha,A_\alpha) p α = ( s α , A α ) p α ⊩ α ˇ ∈ A ˙ p_\alpha\Vdash\check\alpha\in\dot A p α ⊩ α ˇ ∈ A ˙ A 0 ∈ [ A ˙ ] λ + A_0\in[\dot A]^{\lambda^+} A 0 ∈ [ A ˙ ] λ + s α = s s_\alpha=s s α = s α ∈ A 0 \alpha\in A_0 α ∈ A 0 Gal ( U , λ + , λ + ) \operatorname{Gal}(\mathcal U,\lambda^+,\lambda^+) Gal ( U , λ + , λ + ) B ∈ U B\in\mathcal U B ∈ U A 1 ∈ [ A 0 ] λ + A_1\in[A_0]^{\lambda^+} A 1 ∈ [ A 0 ] λ + B ⊆ A α B\subseteq A_\alpha B ⊆ A α α ∈ A 1 \alpha\in A_1 α ∈ A 1 q = ( s , B ) q=(s,B) q = ( s , B ) p α ≤ q p_\alpha\leq q p α ≤ q α ∈ A 1 \alpha\in A_1 α ∈ A 1 A 1 ∈ V A_1\in V A 1 ∈ V q ⊩ A 1 ⊆ A ˙ q\Vdash A_1\subseteq\dot A q ⊩ A 1 ⊆ A ˙ [14 ] .
Thus if stick or tiltan hold at λ + \lambda^+ λ + λ \lambda λ 2 λ 2^\lambda 2 λ λ + \lambda^+ λ +
Question 3.4. Is it consistent that λ \lambda λ U \mathcal U U λ \lambda λ Gal ( U , λ + , λ + ) \operatorname{Gal}(\mathcal U,\lambda^+,\lambda^+) Gal ( U , λ + , λ + ) ♣ ( λ ) \clubsuit(\lambda) ♣ ( λ ) 2 λ > λ + 2^\lambda>\lambda^+ 2 λ > λ +
Appendix
In this section we describe the extender-based Prikry forcing with interleaved collapses, and we prove some basic facts with respect to this forcing notion. The main statement to be proved is that this forcing notion enjoys the strong Prikry property. By and large, the results presented in this section are known. However, some of the statements and proofs are folklore, and apparently there is no citable source for them. The presentation of the extender-based Prikry forcing in this section follows the conventions set by Merimovich in various articles.
Definition 4.1 (Strong cardinal). Let κ \kappa κ κ \kappa κ β \beta β j : V → M j: V \to M j : V → M M M M κ \kappa κ crit j = κ \operatorname{crit} j = \kappa crit j = κ V β ⊆ M V_\beta\subseteq M V β ⊆ M κ \kappa κ κ \kappa κ β \beta β β \beta β
Thus, the concept of a strong cardinal depicts a global property. This property comes in handy while trying to use extenders. Fix an elementary embedding j : V → M j: V \to M j : V → M κ \kappa κ α ≤ j ( κ ) \alpha\leq j(\kappa) α ≤ j ( κ ) E E E d ∈ [ α ∖ κ ] ≤ κ d \in[\alpha\setminus\kappa]^{\leq\kappa} d ∈ [ α ∖ κ ] ≤ κ E d = { X ⊆ ( d × κ ) < κ ∣ ( j ↾ d ) − 1 ∈ j ( X ) } E_d = \{X \subseteq(d \times\kappa)^{<\kappa} \mid(j \mathbin{\upharpoonright} d)^{-1} \in j(X)\} E d = { X ⊆ ( d × κ ) < κ ∣ ( j ↾ d ) − 1 ∈ j ( X )} d = ∅ d = \varnothing d = ∅ E d E_d E d
Let d ⊆ e d \subseteq e d ⊆ e [ α ∖ κ ] ≤ κ [\alpha\setminus\kappa]^{\leq\kappa} [ α ∖ κ ] ≤ κ π e , d \pi_{e,d} π e , d E e E_e E e E d E_d E d x ↦ x ↾ d x \mapsto x \mathbin{\upharpoonright} d x ↦ x ↾ d E E E ⟨ ⟨ E d ∣ d ∈ [ α ∖ κ ] ≤ κ ⟩ , ⟨ π e , d ∣ d ⊆ e ⊆ α , ∣ e ∣ ≤ κ ⟩ ⟩ \langle\langle E_d \mid d \in[\alpha\setminus\kappa]^{\leq\kappa}\rangle,\langle\pi_{e,d} \mid d \subseteq e \subseteq\alpha,\ |e| \leq\kappa\rangle\rangle ⟨⟨ E d ∣ d ∈ [ α ∖ κ ] ≤ κ ⟩ , ⟨ π e , d ∣ d ⊆ e ⊆ α , ∣ e ∣ ≤ κ ⟩⟩ E E E ( κ , α ) (\kappa,\alpha) ( κ , α ) j j j κ \kappa κ κ + 2 \kappa^{+2} κ + 2 ( κ , κ + + ) (\kappa,\kappa^{++}) ( κ , κ ++ )
Given an extender one can take the ultrapower by the extender in order to derive an elementary embedding, from which the extender can be recovered. Note that the assumption that the extender is derived from an elementary embedding with critical point κ \kappa κ
Lemma 4.2. Let E E E
(ℵ) For every d d d x ↦ x ( κ ) x \mapsto x(\kappa) x ↦ x ( κ ) ( d × κ ) < κ (d \times\kappa)^{<\kappa} ( d × κ ) < κ E d E_d E d
(ℶ) Let d ∈ [ α ∖ κ ] κ d \in[\alpha\setminus\kappa]^\kappa d ∈ [ α ∖ κ ] κ g : κ → d g: \kappa\to d g : κ → d B ∈ E d B \in E_d B ∈ E d x ∈ B x \in B x ∈ B g ↾ x ( κ ) g \mathbin{\upharpoonright} x(\kappa) g ↾ x ( κ ) x ( κ ) x(\kappa) x ( κ ) dom x \operatorname{dom} x dom x
Proof. Both parts follow directly from the definition. The second part plays an important role in [23 ] .
While the ultrafilters E d E_d E d
Lemma 4.3. Let A ∗ = ⟨ A η ∣ η ∈ B ⟩ A_*=\langle A_\eta\mid\eta\in B\rangle A ∗ = ⟨ A η ∣ η ∈ B ⟩ B ∈ E d B\in E_d B ∈ E d B ⊆ [ d × κ ] < κ B\subseteq[d\times\kappa]^{<\kappa} B ⊆ [ d × κ ] < κ d d d
△ π A η = { ζ ∈ B ∣ ∀ η ∈ B , sup range η < ζ ( κ ) ⇒ ζ ∈ A η } . \triangle^\pi A_\eta=\{\zeta\in B\mid\forall\eta\in B,\sup\operatorname{range}\eta<\zeta(\kappa)\Rightarrow\zeta\in A_\eta\}. △ π A η = { ζ ∈ B ∣ ∀ η ∈ B , sup range η < ζ ( κ ) ⇒ ζ ∈ A η } . Then, △ π A η ∈ E d \triangle^\pi A_\eta\in E_d △ π A η ∈ E d
Proof. Let us verify that the seed of E d E_d E d j E − 1 ↾ ( j ‘ ‘ d ) j_E^{-1}\mathbin{\upharpoonright}(j``d) j E − 1 ↾ ( j ‘‘ d ) j E ( A ∗ ) j_E(A_*) j E ( A ∗ )
{ η ∈ j E ( B ) ∣ sup range η < j E − 1 ↾ ( j ‘ ‘ d ) ( κ ) = κ } = j ‘ ‘ B . \{\eta\in j_E(B)\mid\sup\operatorname{range}\eta<j_E^{-1}\mathbin{\upharpoonright}(j``d)(\kappa)=\kappa\}=j``B. { η ∈ j E ( B ) ∣ sup range η < j E − 1 ↾ ( j ‘‘ d ) ( κ ) = κ } = j ‘‘ B . Indeed, every such element must be smaller than κ \kappa κ j ( g ) ‘ ‘ κ = j ‘ ‘ d j(g)``\kappa=j``d j ( g ) ‘‘ κ = j ‘‘ d E d E_d E d
Given an extender E E E [23 ] . We would like to incorporate collapses as well, and in the general case there are some subtleties, which are pcf-related, that limit the general possibilities.
Let us assume from now on that E E E ( κ , κ + + ) (\kappa,\kappa^{++}) ( κ , κ ++ ) h : κ → κ h:\kappa\to\kappa h : κ → κ α \alpha α h ( α ) > α + + h(\alpha)>\alpha^{++} h ( α ) > α ++ h ( α ) h(\alpha) h ( α ) h h h
cf κ = ω \operatorname{cf}\kappa=\omega cf κ = ω ⟨ ρ n ∣ n < ω ⟩ \langle\rho_n\mid n<\omega\rangle ⟨ ρ n ∣ n < ω ⟩
2 κ ≥ κ + 2^\kappa\ge\kappa^+ 2 κ ≥ κ +
Every cardinal in the open intervals ( h ( ρ n ) , ρ n + 1 ) (h(\rho_n),\rho_{n+1}) ( h ( ρ n ) , ρ n + 1 ) ( h ( ω ) , ρ 0 ) (h(\omega),\rho_0) ( h ( ω ) , ρ 0 )
If we start with a model of GCH, then GCH holds below κ \kappa κ
Lemma 4.4. Assume 2 κ = κ + 2^\kappa=\kappa^+ 2 κ = κ + j : V → M j:V\to M j : V → M E E E ι : V → N \iota:V\to N ι : V → N j j j k k k N N N K ~ ⊆ Col N ( ι ( h ) ( κ ) , < ι ( κ ) ) \widetilde K\subseteq\operatorname{Col}^N(\iota(h)(\kappa),<\iota(\kappa)) K ⊆ Col N ( ι ( h ) ( κ ) , < ι ( κ )) K = k ‘ ‘ K ~ K=k``\widetilde K K = k ‘‘ K M M M
Proof. See [7 Lemma 15.3] .
Lemma 4.5. Let K K K ⟨ K , j E ( K ) , … , j E m ( K ) ⟩ \langle K,j_E(K),\ldots,j_E^m(K)\rangle ⟨ K , j E ( K ) , … , j E m ( K )⟩ m + 1 m+1 m + 1 E E E
Proof. First notice that the filters K ~ , ι ( K ~ ) , … , ι m ( K ~ ) \widetilde K,\iota(\widetilde K),\ldots,\iota^m(\widetilde K) K , ι ( K ) , … , ι m ( K ) [6 Fact 2] . Second, by induction, let us lift those filters to the iterated ultrapower by E E E
Note that the filter ⟨ K , j E ( K ) , … , j E m ( K ) ⟩ \langle K,j_E(K),\ldots,j_E^m(K)\rangle ⟨ K , j E ( K ) , … , j E m ( K )⟩ ⟨ j E ( C ) ( κ ) , j C 2 ( j E ( κ ) ) , … , j E m ( C ) ( j C m − 1 ( κ ) ) ⟩ \langle j_E(C)(\kappa),j_C^2(j_E(\kappa)),\ldots,j_E^m(C)(j_C^{m-1}(\kappa))\rangle ⟨ j E ( C ) ( κ ) , j C 2 ( j E ( κ )) , … , j E m ( C ) ( j C m − 1 ( κ ))⟩
p = ⟨ f , A , c 0 , … , c n , C ⟩ p=\langle f,A,c_0,\ldots,c_n,C\rangle p = ⟨ f , A , c 0 , … , c n , C ⟩ where:
f f f κ + + ∖ κ \kappa^{++}\setminus\kappa κ ++ ∖ κ κ < ω \kappa^{<\omega} κ < ω ∣ dom f ∣ < κ |\operatorname{dom}f|<\kappa ∣ dom f ∣ < κ κ ∈ dom f \kappa\in\operatorname{dom}f κ ∈ dom f 15
Let d = dom f d=\operatorname{dom}f d = dom f A ∈ E d A\in E_d A ∈ E d h ∈ A h\in A h ∈ A h h h h ( κ ) h(\kappa) h ( κ ) f ( α ) f(\alpha) f ( α ) α ∈ d \alpha\in d α ∈ d
n = dom f ( κ ) n=\operatorname{dom}f(\kappa) n = dom f ( κ ) ⟨ ρ i ∣ i < n ⟩ = f ( κ ) \langle\rho_i\mid i<n\rangle=f(\kappa) ⟨ ρ i ∣ i < n ⟩ = f ( κ ) ρ − 1 = ω \rho_{-1}=\omega ρ − 1 = ω ρ n = κ \rho_n=\kappa ρ n = κ
For every i ≤ n i\leq n i ≤ n c i ∈ Col ( h ( ρ i − 1 ) , < ρ i ) c_i\in\operatorname{Col}(h(\rho_{i-1}),<\rho_i) c i ∈ Col ( h ( ρ i − 1 ) , < ρ i )
C C C A A A x , y ∈ A x,y\in A x , y ∈ A x ( κ ) = y ( κ ) x(\kappa)=y(\kappa) x ( κ ) = y ( κ ) C ( x ) = C ( y ) C(x)=C(y) C ( x ) = C ( y ) [ C , d ] E ∈ K [C,d]_E\in K [ C , d ] E ∈ K
Given conditions p = ⟨ f p , A p , c 0 p , … , c n p , C p ⟩ p=\langle f^p,A^p,c_0^p,\ldots,c_n^p,C^p\rangle p = ⟨ f p , A p , c 0 p , … , c n p , C p ⟩ q = ⟨ f q , A q , c 0 q , … , c m q , C q ⟩ q=\langle f^q,A^q,c_0^q,\ldots,c_m^q,C^q\rangle q = ⟨ f q , A q , c 0 q , … , c m q , C q ⟩ p ≤ ∗ q p\leq^*q p ≤ ∗ q q q q p p p
n = m n=m n = m
f q ⊇ f p f^q\supseteq f^p f q ⊇ f p
π d q , d p ( A q ) ⊆ A p \pi_{d^q,d^p}(A^q)\subseteq A^p π d q , d p ( A q ) ⊆ A p
For every i ≤ n i\leq n i ≤ n c i q ⊇ c i p c_i^q\supseteq c_i^p c i q ⊇ c i p
For every x ∈ A q x\in A^q x ∈ A q C q ( x ) ⊇ C p ( π d q , d p ( x ) ) C^q(x)\supseteq C^p(\pi_{d^q,d^p}(x)) C q ( x ) ⊇ C p ( π d q , d p ( x ))
Let p = ⟨ f p , A p , c 0 p , … , c n p , C p ⟩ p=\langle f^p,A^p,c_0^p,\ldots,c_n^p,C^p\rangle p = ⟨ f p , A p , c 0 p , … , c n p , C p ⟩ η ∈ A p \eta\in A^p η ∈ A p q = p ⌢ η q=p^\frown\eta q = p ⌢ η
dom f q = dom f p \operatorname{dom}f^q=\operatorname{dom}f^p dom f q = dom f p α ∈ dom η \alpha\in\operatorname{dom}\eta α ∈ dom η f q ( α ) = f p ( α ) ⌢ ⟨ η ( α ) ⟩ f^q(\alpha)=f^p(\alpha)^\frown\langle\eta(\alpha)\rangle f q ( α ) = f p ( α ) ⌢ ⟨ η ( α )⟩ f q ( α ) = f p ( α ) f^q(\alpha)=f^p(\alpha) f q ( α ) = f p ( α )
In particular, the length of q q q n + 1 n+1 n + 1
A q = { x ∈ A p ∣ x ( κ ) > sup range η } A^q=\{x\in A^p\mid x(\kappa)>\sup\operatorname{range}\eta\} A q = { x ∈ A p ∣ x ( κ ) > sup range η }
For every i ≤ n i\leq n i ≤ n c i q = c i p c_i^q=c_i^p c i q = c i p
c n + 1 q = C p ( η ) c_{n+1}^q=C^p(\eta) c n + 1 q = C p ( η )
C q = C p ↾ A q C^q=C^p\mathbin{\upharpoonright}A^q C q = C p ↾ A q
We say that q q q p p p
Let us define recursively that q q q n n n p p p p p p q q q n n n p p p ( n − 1 ) (n-1) ( n − 1 ) p p p p ≤ q p\leq q p ≤ q p p p q q q
Lemma 4.6. ≤ ∗ \leq^* ≤ ∗
Lemma 4.7. p ≤ q p\leq q p ≤ q r r r k k k p p p r ≤ ∗ q r\leq^*q r ≤ ∗ q
Proof. Let us prove that given conditions p p p q q q r r r p ≤ ∗ r p \leq^{*} r p ≤ ∗ r q = r ⌢ η q = r^\frown\eta q = r ⌢ η r r r r ′ r' r ′ r ′ r' r ′ r r r r ′ ≤ ∗ q r' \leq^{*} q r ′ ≤ ∗ q
Let n n n f q ( κ ) f^{q}(\kappa) f q ( κ ) f r ′ = f r ↾ ( d − dom η ) ∪ f q ↾ ( dom η ) f^{r'} = f^{r} \mathord{\upharpoonright} (d-\operatorname{dom}\eta) \cup f^{q} \mathord{\upharpoonright} (\operatorname{dom}\eta) f r ′ = f r ↾ ( d − dom η ) ∪ f q ↾ ( dom η ) A r ′ = A r ∪ { η } A^{r'} = A^{r} \cup\{\eta\} A r ′ = A r ∪ { η } c i r ′ = c i r c_i^{r'} = c_i^{r} c i r ′ = c i r i ≤ n i \leq n i ≤ n C r ′ = C r ∪ { ⟨ η , c n + 1 r ⟩ } C^{r'} = C^{r} \cup\{\langle\eta,c_{n+1}^{r}\rangle\} C r ′ = C r ∪ {⟨ η , c n + 1 r ⟩} x ∈ A r x \in A^{r} x ∈ A r x ( κ ) x(\kappa) x ( κ ) η ( κ ) \eta(\kappa) η ( κ )
□ \square □
This forcing is a typical example of a Prikry type forcing: it carries two types of partial orders, the direct order which satisfies better closure properties and the actual order with which we will force. In order to gain better control over the properties of the extension we would like to decompose the direct order ≤ ∗ \leq^{*} ≤ ∗
We say that p ≤ ∗ ∗ q p \leq^{**} q p ≤ ∗∗ q p ≤ ∗ q p \leq^{*} q p ≤ ∗ q c i p = c i q c_i^{p} = c_i^{q} c i p = c i q i i i
We say that p ≤ c o l q p \leq^{\mathrm{col}} q p ≤ col q p ≤ ∗ q p \leq^{*} q p ≤ ∗ q f p = f q f_p = f_q f p = f q A p = A q A^{p} = A^{q} A p = A q C p = C q C^{p} = C^{q} C p = C q
Our next goal is to prove the strong Prikry property of the above defined forcing notion. In fact, we shall prove the complete Prikry property , defined in [PRS21]. In that paper, it was illustrated that this property gives a cleaner way to understand the properties of a Prikry type forcing. It was also demonstrated in general that the complete Prikry property implies the strong Prikry property.
Lemma 4.8. Let D ⊆ P D \subseteq\mathbb{P} D ⊆ P p p p q q q n n n
p ≤ ∗ ∗ q p \leq^{**} q p ≤ ∗∗ q
There is a dense open set in ≤ c o l \leq^{\mathrm{col}} ≤ col q q q D c o l D_{\mathrm{col}} D col r ∈ D c o l r \in D_{\mathrm{col}} r ∈ D col n n n r r r D D D
Moreover, the Complete Prikry Property holds: for every ≤ ∗ \leq^{*} ≤ ∗ U U U q q q p ≤ ∗ ∗ q p \leq^{**} q p ≤ ∗∗ q n n n
either for every n n n r r r q q q ≤ c o l \leq^{\mathrm{col}} ≤ col r r r U U U
there is a dense subset in ≤ c o l \leq^{\mathrm{col}} ≤ col q q q q ′ ∈ D c o l q' \in D_{\mathrm{col}} q ′ ∈ D col n n n q ′ q' q ′ r r r
Proof. Let U U U ≤ ∗ \leq^{*} ≤ ∗ χ \chi χ ( 2 κ ) + + (2^{\kappa})^{++} ( 2 κ ) ++ p p p M ≺ H ( χ ) M \prec H(\chi) M ≺ H ( χ ) p p p U U U P \mathbb{P} P κ \kappa κ < κ <\kappa < κ
Let us enumerate all dense open sets in the forcing Add ( κ , κ + + ) \operatorname{Add}(\kappa,\kappa^{++}) Add ( κ , κ ++ ) M M M ⟨ D ~ α ∣ α < κ ⟩ \langle\widetilde{D}_{\alpha} \mid\alpha< \kappa\rangle ⟨ D α ∣ α < κ ⟩ f α f_{\alpha} f α f 0 = f p f_{0}=f^{p} f 0 = f p 0 < α < κ 0<\alpha<\kappa 0 < α < κ f α = ⋃ β < α f β f_{\alpha}=\bigcup_{\beta<\alpha}f_{\beta} f α = ⋃ β < α f β f α f_{\alpha} f α d α = dom f α d_{\alpha}=\operatorname{dom}f_{\alpha} d α = dom f α f α + 1 ∈ D ~ α ∩ M f_{\alpha+1}\in\widetilde{D}_{\alpha}\cap M f α + 1 ∈ D α ∩ M f α f_{\alpha} f α f κ = ⋃ α < κ f α f_{\kappa}=\bigcup_{\alpha<\kappa}f_{\alpha} f κ = ⋃ α < κ f α d κ = dom f κ d_{\kappa}=\operatorname{dom}f_{\kappa} d κ = dom f κ α ≤ κ \alpha\le\kappa α ≤ κ
p α = ⟨ f α , π d α , d 0 − 1 ( A p ) , c 0 p , … , c n p , C p ∘ π d α , d 0 ⟩ p_\alpha= \langle f_\alpha, \pi^{-1}_{d_\alpha,d_0}(A^p), c_0^p, \ldots, c_n^p, C^p \circ\pi_{d_\alpha,d_0}\rangle p α = ⟨ f α , π d α , d 0 − 1 ( A p ) , c 0 p , … , c n p , C p ∘ π d α , d 0 ⟩
let A α = π d α , d 0 − 1 ( A p ) A^\alpha= \pi^{-1}_{d_\alpha,d_0}(A^p) A α = π d α , d 0 − 1 ( A p )
Let η ⃗ = ⟨ η 0 , … , η m − 1 ⟩ \vec{\eta} = \langle\eta_0, \ldots, \eta_{m-1}\rangle η = ⟨ η 0 , … , η m − 1 ⟩ π d α , d 0 − 1 ( A p ) \pi^{-1}_{d_\alpha,d_0}(A^p) π d α , d 0 − 1 ( A p ) m m m p α p_\alpha p α η ⃗ \vec{\eta} η g g g f α f_\alpha f α r r r p α η ⃗ p_\alpha\vec{\eta} p α η U U U f r = g ⌢ η ⃗ f^r = g^\frown\vec{\eta} f r = g ⌢ η g g g r ∈ U r \in U r ∈ U M M M f κ f_\kappa f κ η ⃗ \vec{\eta} η p κ η ⃗ p_\kappa\vec{\eta} p κ η U U U f p κ f^{p_\kappa} f p κ
Let us now deal with the collapse parts. Let D D D [ C ] E ≤ [ C κ ] E [C]_E \le[C^\kappa]_E [ C ] E ≤ [ C κ ] E η ∈ A κ \eta\in A^\kappa η ∈ A κ p κ η ≤ ∗ r p_\kappa^\eta\le^* r p κ η ≤ ∗ r c n + 1 r = C ( η ) c^r_{n+1} = C(\eta) c n + 1 r = C ( η ) r ≤ ∗ s r \le^* s r ≤ ∗ s U U U D D D [ C ] E ∈ K ∩ D [C]_E \in K \cap D [ C ] E ∈ K ∩ D [ C κ ] E [C^\kappa]_E [ C κ ] E D D D p κ p^\kappa p κ C κ C_\kappa C κ C C C A κ A^\kappa A κ
Now, for every η ∈ A κ \eta\in A^\kappa η ∈ A κ n n n p κ η p_\kappa^\eta p κ η U U U A κ A^\kappa A κ 16 p κ p_\kappa p κ
Let us assume that for every step up to m m m C C C m m m D m D_m D m ⟨ C 0 , C 1 , … , C m ⟩ \langle C_0, C_1, \ldots, C_m\rangle ⟨ C 0 , C 1 , … , C m ⟩ dom C i = [ A κ ] i + 1 \operatorname{dom} C_i = [A^\kappa]^{i+1} dom C i = [ A κ ] i + 1 A κ A^\kappa A κ η ⃗ = ⟨ η 0 , … , η m ⟩ \vec{\eta} = \langle\eta_0, \ldots, \eta_m\rangle η = ⟨ η 0 , … , η m ⟩ 17 p κ η ⃗ ≤ ∗ r p_\kappa\vec{\eta} \le^* r p κ η ≤ ∗ r C 0 ( η 0 ) , C 1 ( η 0 , η 1 ) , … , C ( η 0 , … , η n − 1 ) C_0(\eta_0), C_1(\eta_0,\eta_1), \ldots, C(\eta_0,\ldots,\eta_{n-1}) C 0 ( η 0 ) , C 1 ( η 0 , η 1 ) , … , C ( η 0 , … , η n − 1 ) U U U [ C ] ∈ K [C] \in K [ C ] ∈ K [ C κ ] [C^\kappa] [ C κ ] C i C_i C i C ( η i ( κ ) ) C(\eta_i(\kappa)) C ( η i ( κ )) D D D [ C κ ] [C^\kappa] [ C κ ] C C C m m m C C C σ \sigma σ ω \omega ω U U U
Let us assume that at step m m m C C C C C C K K K C C C D m D_m D m C C C U U U [ C κ ] [C^\kappa] [ C κ ] K K K
Look at all extensions of the first n n n U U U η \eta η π \pi π
From the strong Prikry property one can derive the following version of the Prikry property.
Lemma 4.9. Let Φ \Phi Φ p p p
Then, there is a condition q q q p ≤ ∗ ∗ q p \leq^{**} q p ≤ ∗∗ q D c o l D_{\mathrm{col}} D col ≤ c o l \leq^{\mathrm{col}} ≤ col q q q Φ \Phi Φ
These properties can be used to compute which cardinals are collapsed in the generic extension.
Lemma 4.10. Assume that 2 κ = κ + 2^\kappa=\kappa^+ 2 κ = κ + G ⊆ P G\subseteq\mathbb{P} G ⊆ P V V V g : κ → κ g:\kappa\to\kappa g : κ → κ V V V α \alpha α g ( α ) ≥ α g(\alpha)\geq\alpha g ( α ) ≥ α V [ G ] V[G] V [ G ] g ( ρ n ) g(\rho_n) g ( ρ n ) n n n g ( α ) ≤ h ( α ) g(\alpha)\leq h(\alpha) g ( α ) ≤ h ( α ) α \alpha α g ( α ) > α + + g(\alpha)>\alpha^{++} g ( α ) > α ++
tcf ∏ g ( ρ n ) = κ + . \operatorname{tcf}\prod g(\rho_n)=\kappa^+. tcf ∏ g ( ρ n ) = κ + . Proof. Let ℓ \ell ℓ ∏ g ( ρ n ) \prod g(\rho_n) ∏ g ( ρ n ) p p p σ \sigma σ q q q n n n m m m D n D_n D n m m m ≤ c o l \leq^{\mathrm{col}} ≤ col q q q D n D_n D n ℓ ( n ) \ell(n) ℓ ( n )
Without loss of generality, p p p 0 0 0 σ \sigma σ q q q D n D_n D n n n n ⊩ ℓ ( n ) ≤ h ( ρ n ) < ρ n + 1 \Vdash\ell(n)\leq h(\rho_n)<\rho_{n+1} ⊩ ℓ ( n ) ≤ h ( ρ n ) < ρ n + 1 m = n m=n m = n d = dom q d=\operatorname{dom}q d = dom q ℓ ( n ) \ell(n) ℓ ( n ) n n n E d E_d E d
Let us now show that there is an E d E_d E d B B B ρ < κ \rho<\kappa ρ < κ { y ∈ B ∣ y ( κ ) = ρ } \{y\in B\mid y(\kappa)=\rho\} { y ∈ B ∣ y ( κ ) = ρ } ρ + + \rho^{++} ρ ++ E d E_d E d s d = j − 1 ‘ ‘ j ‘ ‘ d s_d=j^{-1}``j``d s d = j − 1 ‘‘ j ‘‘ d κ + + = s d ( j ( κ ) ) + + \kappa^{++}=s_d(j(\kappa))^{++} κ ++ = s d ( j ( κ ) ) ++ ∣ ( ρ + + ) ρ ∣ = ρ + + \lvert(\rho^{++})^\rho\rvert=\rho^{++} ∣( ρ ++ ) ρ ∣ = ρ ++
Thus, one can bound ℓ \ell ℓ ⟨ ρ n ∣ n < ω ⟩ \langle\rho_n\mid n<\omega\rangle ⟨ ρ n ∣ n < ω ⟩ f ℓ : κ → κ f_\ell:\kappa\to\kappa f ℓ : κ → κ q ⊩ ∀ n , ℓ ( n ) ≤ f ℓ ( ρ n ) q\Vdash\forall n,\ell(n)\leq f_\ell(\rho_n) q ⊩ ∀ n , ℓ ( n ) ≤ f ℓ ( ρ n ) 2 κ = κ + 2^\kappa=\kappa^+ 2 κ = κ + κ + \kappa^+ κ + κ + \kappa^+ κ +
Acknowledgements
The authors would like to thank the anonymous referee for their useful suggestions and corrections that greatly improved this manuscript.
References
Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel
Email address: shimon.garty@mail.huji.ac.il
Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel
Email address: yair.hayut@mail.huji.ac.il
Institute of Mathematics The Hebrew University of Jerusalem, Jerusalem 91904, Israel and Department of Mathematics Rutgers University New Brunswick, NJ 08854, USA
Email address: shelah@math.huji.ac.il
URL: http://www.math.rutgers.edu/~shelah
