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Feb 18, 2023 334 likes •868 views

ss r st rr r ts ts trst rs

SLIDE 1

❇❧❛ss✕❙❤❡❧❛❤ ❋♦r❝✐♥❣ ❘❡✈✐s✐t❡❞

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❋♦r❝✐♥❣ ❛♥❞ ■ts ❆♣♣❧✐❝❛t✐♦♥s ❘❡tr♦s♣❡❝t✐✈❡ ❲♦r❦s❤♦♣ ❋✐❡❧❞s ■♥st✐t✉t❡✱ ▼❛r❝❤ ✸✶✱ ✷✵✶✺

✶ ✴ ✸✵

SLIDE 2

❆ ♣❛rt ♦❢ ❛ ❧❛r❣❡r ♣r♦❥❡❝t

❋✶✹✷✵ ❜② ❇❧❛ss✱ ▼✐❧❞❡♥❜❡r❣❡r✱ ❙❤❡❧❛❤ ❆ ❙✐♠♣❧❡ Pℵ1✲P♦✐♥t ❛♥❞ ❛ ❙✐♠♣❧❡ Pℵ2✲P♦✐♥t

✷ ✴ ✸✵

SLIDE 3

Pr❡s❡r✈✐♥❣ ❛♥ ✉❧tr❛✜❧t❡r

❉❡✜♥✐t✐♦♥

▲❡t P ❜❡ ❛ ♥♦t✐♦♥ ♦❢ ❢♦r❝✐♥❣✳ ▲❡t U ❜❡ ❛♥ ✉❧tr❛✜❧t❡r ♦✈❡r I✳ ❲❡ s❛② P ♣r❡s❡r✈❡s U ✐❢ P ✏(∀X ⊆ I)(∃Y ∈ U)(Y ⊆ X ∨ Y ⊆ I X)✑✳

❘❡♠❛r❦

Pr❡s❡r✈❛t✐♦♥ ♦❢ ✲♣♦✐♥ts✳ ❙♦♠❡ ❞❡✈❡❧♦♣♠❡♥t ✳ ✳ ✳

❚❤❡♦r❡♠✱ ❙❤❡❧❛❤ ✶✾✾✹

❆♥② ❢♦r❝✐♥❣ ❛❞❞✐♥❣ ❛ r❡❛❧ ❞❡str♦②s ❛♥ ✉❧tr❛✜❧t❡r ♦✈❡r ✳

✸ ✴ ✸✵

SLIDE 4

Pr❡s❡r✈✐♥❣ ❛♥ ✉❧tr❛✜❧t❡r

❉❡✜♥✐t✐♦♥

▲❡t P ❜❡ ❛ ♥♦t✐♦♥ ♦❢ ❢♦r❝✐♥❣✳ ▲❡t U ❜❡ ❛♥ ✉❧tr❛✜❧t❡r ♦✈❡r I✳ ❲❡ s❛② P ♣r❡s❡r✈❡s U ✐❢ P ✏(∀X ⊆ I)(∃Y ∈ U)(Y ⊆ X ∨ Y ⊆ I X)✑✳

❘❡♠❛r❦

Pr❡s❡r✈❛t✐♦♥ ♦❢ P✲♣♦✐♥ts✳ ❙♦♠❡ ❞❡✈❡❧♦♣♠❡♥t ✳ ✳ ✳

❚❤❡♦r❡♠✱ ❙❤❡❧❛❤ ✶✾✾✹

❆♥② ❢♦r❝✐♥❣ ❛❞❞✐♥❣ ❛ r❡❛❧ ❞❡str♦②s ❛♥ ✉❧tr❛✜❧t❡r ♦✈❡r ✳

✸ ✴ ✸✵

SLIDE 5

Pr❡s❡r✈✐♥❣ ❛♥ ✉❧tr❛✜❧t❡r

❉❡✜♥✐t✐♦♥

▲❡t P ❜❡ ❛ ♥♦t✐♦♥ ♦❢ ❢♦r❝✐♥❣✳ ▲❡t U ❜❡ ❛♥ ✉❧tr❛✜❧t❡r ♦✈❡r I✳ ❲❡ s❛② P ♣r❡s❡r✈❡s U ✐❢ P ✏(∀X ⊆ I)(∃Y ∈ U)(Y ⊆ X ∨ Y ⊆ I X)✑✳

❘❡♠❛r❦

Pr❡s❡r✈❛t✐♦♥ ♦❢ P✲♣♦✐♥ts✳ ❙♦♠❡ ❞❡✈❡❧♦♣♠❡♥t ✳ ✳ ✳

❚❤❡♦r❡♠✱ ❙❤❡❧❛❤ ✶✾✾✹

❆♥② ❢♦r❝✐♥❣ ❛❞❞✐♥❣ ❛ r❡❛❧ ❞❡str♦②s ❛♥ ✉❧tr❛✜❧t❡r ♦✈❡r ω✳

✸ ✴ ✸✵

SLIDE 6

❈♦✉♥t❛❜❧❡ s✉♣♣♦rt ✐t❡r❛t✐♦♥

❚❤❡♦r❡♠✱ ❇❧❛ss✱ ❙❤❡❧❛❤

▲❡t E ❜❡ ❛ P✲♣♦✐♥t✳ ▲❡t Pα, Qβ : β < γ, α ≤ γ ❜❡ ❛ ❝♦✉♥t❛❜❧❡ s✉♣♣♦rt ✐t❡r❛t✐♦♥ s✉❝❤ t❤❛t ❡❛❝❤ Pα ✐s ♣r♦♣❡r✳ ■❢ ❡❛❝❤ Pα✱ α < γ✱ ♣r❡s❡r✈❡s E✱ t❤❡♥ ❛❧s♦ Pγ ♣r❡s❡r✈❡s E✳

✹ ✴ ✸✵

SLIDE 7

❆ r❡q✉❡st t♦ ❞❡str♦② ❛ P✲♣♦✐♥t

❙✉♣♣♦s❡✿ ✭✶✮ Pα, Qβ : β < ω2, α ≤ ω2 ✐s ❛ ❝♦✉♥t❛❜❧❡ s✉♣♣♦rt ✐t❡r❛t✐♦♥ ♦❢ ♣r♦♣❡r ✐t❡r❛♥❞s✱ ❛♥❞ ✭✷✮ ✐♥ V Pω2 t❤❡r❡ ✐s ❛ s✐♠♣❧❡ Pℵ2✲♣♦✐♥t U✳ ❚❤❡♥ t❤❡r❡ ✐s ❛♥ ✲❝❧✉❜ ♦❢ st❛❣❡s ❛t ✇❤✐❝❤ ✐s ❛ ✲♣♦✐♥t✳ ❲❡ t❛❦❡ s✉❝❤ ❛ st❛❣❡ ✱ ❛♥❞ ❝♦♥s✐❞❡r t❤❡ ❧❡❛st s✉❝❤ t❤❛t t❤❡r❡ ✐s ✳ ✐s ❞❡str♦②❡❞ ❜② ✭❛♥❞ ❝♦♠♣❧❡♠❡♥t❡❞✮ ❧❛t❡r ✐♥ t❤❡ ✐t❡r❛t✐♦♥✳ ❙♦ ❛ ❢♦r❝✐♥❣ ❞❡str♦②✐♥❣ s♦♠❡ ✲♣♦✐♥ts ❛♥❞ ❦❡❡♣✐♥❣ ♦t❤❡rs ✐s r❡q✉❡st❡❞✳

✺ ✴ ✸✵

SLIDE 8

❆ r❡q✉❡st t♦ ❞❡str♦② ❛ P✲♣♦✐♥t

❙✉♣♣♦s❡✿ ✭✶✮ Pα, Qβ : β < ω2, α ≤ ω2 ✐s ❛ ❝♦✉♥t❛❜❧❡ s✉♣♣♦rt ✐t❡r❛t✐♦♥ ♦❢ ♣r♦♣❡r ✐t❡r❛♥❞s✱ ❛♥❞ ✭✷✮ ✐♥ V Pω2 t❤❡r❡ ✐s ❛ s✐♠♣❧❡ Pℵ2✲♣♦✐♥t U✳ ❚❤❡♥ t❤❡r❡ ✐s ❛♥ ω1✲❝❧✉❜ ♦❢ st❛❣❡s ❛t ✇❤✐❝❤ U ∩ V Pα ✐s ❛ P✲♣♦✐♥t✳ ❲❡ t❛❦❡ s✉❝❤ ❛ st❛❣❡ ✱ ❛♥❞ ❝♦♥s✐❞❡r t❤❡ ❧❡❛st s✉❝❤ t❤❛t t❤❡r❡ ✐s ✳ ✐s ❞❡str♦②❡❞ ❜② ✭❛♥❞ ❝♦♠♣❧❡♠❡♥t❡❞✮ ❧❛t❡r ✐♥ t❤❡ ✐t❡r❛t✐♦♥✳ ❙♦ ❛ ❢♦r❝✐♥❣ ❞❡str♦②✐♥❣ s♦♠❡ ✲♣♦✐♥ts ❛♥❞ ❦❡❡♣✐♥❣ ♦t❤❡rs ✐s r❡q✉❡st❡❞✳

✺ ✴ ✸✵

SLIDE 9

❆ r❡q✉❡st t♦ ❞❡str♦② ❛ P✲♣♦✐♥t

❙✉♣♣♦s❡✿ ✭✶✮ Pα, Qβ : β < ω2, α ≤ ω2 ✐s ❛ ❝♦✉♥t❛❜❧❡ s✉♣♣♦rt ✐t❡r❛t✐♦♥ ♦❢ ♣r♦♣❡r ✐t❡r❛♥❞s✱ ❛♥❞ ✭✷✮ ✐♥ V Pω2 t❤❡r❡ ✐s ❛ s✐♠♣❧❡ Pℵ2✲♣♦✐♥t U✳ ❚❤❡♥ t❤❡r❡ ✐s ❛♥ ω1✲❝❧✉❜ ♦❢ st❛❣❡s ❛t ✇❤✐❝❤ U ∩ V Pα ✐s ❛ P✲♣♦✐♥t✳ ❲❡ t❛❦❡ s✉❝❤ ❛ st❛❣❡ α✱ ❛♥❞ ❝♦♥s✐❞❡r t❤❡ ❧❡❛st β > α s✉❝❤ t❤❛t t❤❡r❡ ✐s X ∈ U \ V Pα✳ U ∩ V P<β ✐s ❞❡str♦②❡❞ ❜② Pβ ✭❛♥❞ ❝♦♠♣❧❡♠❡♥t❡❞✮ ❧❛t❡r ✐♥ t❤❡ ✐t❡r❛t✐♦♥✳ ❙♦ ❛ ❢♦r❝✐♥❣ ❞❡str♦②✐♥❣ s♦♠❡ ✲♣♦✐♥ts ❛♥❞ ❦❡❡♣✐♥❣ ♦t❤❡rs ✐s r❡q✉❡st❡❞✳

✺ ✴ ✸✵

SLIDE 10

❆ r❡q✉❡st t♦ ❞❡str♦② ❛ P✲♣♦✐♥t

❙✉♣♣♦s❡✿ ✭✶✮ Pα, Qβ : β < ω2, α ≤ ω2 ✐s ❛ ❝♦✉♥t❛❜❧❡ s✉♣♣♦rt ✐t❡r❛t✐♦♥ ♦❢ ♣r♦♣❡r ✐t❡r❛♥❞s✱ ❛♥❞ ✭✷✮ ✐♥ V Pω2 t❤❡r❡ ✐s ❛ s✐♠♣❧❡ Pℵ2✲♣♦✐♥t U✳ ❚❤❡♥ t❤❡r❡ ✐s ❛♥ ω1✲❝❧✉❜ ♦❢ st❛❣❡s ❛t ✇❤✐❝❤ U ∩ V Pα ✐s ❛ P✲♣♦✐♥t✳ ❲❡ t❛❦❡ s✉❝❤ ❛ st❛❣❡ α✱ ❛♥❞ ❝♦♥s✐❞❡r t❤❡ ❧❡❛st β > α s✉❝❤ t❤❛t t❤❡r❡ ✐s X ∈ U \ V Pα✳ U ∩ V P<β ✐s ❞❡str♦②❡❞ ❜② Pβ ✭❛♥❞ ❝♦♠♣❧❡♠❡♥t❡❞✮ ❧❛t❡r ✐♥ t❤❡ ✐t❡r❛t✐♦♥✳ ❙♦ ❛ ❢♦r❝✐♥❣ ❞❡str♦②✐♥❣ s♦♠❡ P✲♣♦✐♥ts ❛♥❞ ❦❡❡♣✐♥❣ ♦t❤❡rs ✐s r❡q✉❡st❡❞✳

✺ ✴ ✸✵

SLIDE 11

❙✉❜❢♦r❝✐♥❣s ♦❢ ▼❛t❡t ❢♦r❝✐♥❣

❚❤❡ ❘✉❞✐♥✕❇❧❛ss ♦r❞❡r

▲❡t H, H′ ⊆ [ω]ω ❜❡ ❝❧♦s❡❞ ✉♥❞❡r ❛❧♠♦st s✉♣❡rs❡ts✳ ❲❡ ✇r✐t❡ H ≤RB H′ ❛♥❞ s❛② H ✐s ❘✉❞✐♥✲❇❧❛ss✲❜❡❧♦✇ H′ ✐✛ t❤❡r❡ ✐s ❛ ✜♥✐t❡✲t♦✲♦♥❡ f s✉❝❤ t❤❛t f(H) ⊆ f(H′)✳ ❍❡r❡ f(H) = {X : f−1[X] ∈ H}✳

❚❤❡♦r❡♠✱ ❊✐s✇♦rt❤✱ ✷✵✵✷

▲❡t ❜❡ ❛ st❛❜❧❡ ♦r❞❡r❡❞✲✉♥✐♦♥ ✉❧tr❛✜❧t❡r ♦✈❡r t❤❡ s❡t ♦❢ ❜❧♦❝❦s✳ ❚❤❡ ▼❛t❡t ❢♦r❝✐♥❣ ♣r❡s❡r✈❡s ✐✛ ✳ ❍♦✇❡✈❡r✱ ▼❛t❡t ❢♦r❝✐♥❣ ♥♦t ❛❞❞ ❛♥ ✉♥s♣❧✐t r❡❛❧✳

❉❡✜♥✐t✐♦♥

❆ r❡❛❧ ✐s ❝❛❧❧❡❞ ❛♥ ✉♥s♣❧✐t r❡❛❧ ✐❢ ❡✈❡r② ✇❡ ❤❛✈❡

✻ ✴ ✸✵

SLIDE 12

❙✉❜❢♦r❝✐♥❣s ♦❢ ▼❛t❡t ❢♦r❝✐♥❣

❚❤❡ ❘✉❞✐♥✕❇❧❛ss ♦r❞❡r

▲❡t H, H′ ⊆ [ω]ω ❜❡ ❝❧♦s❡❞ ✉♥❞❡r ❛❧♠♦st s✉♣❡rs❡ts✳ ❲❡ ✇r✐t❡ H ≤RB H′ ❛♥❞ s❛② H ✐s ❘✉❞✐♥✲❇❧❛ss✲❜❡❧♦✇ H′ ✐✛ t❤❡r❡ ✐s ❛ ✜♥✐t❡✲t♦✲♦♥❡ f s✉❝❤ t❤❛t f(H) ⊆ f(H′)✳ ❍❡r❡ f(H) = {X : f−1[X] ∈ H}✳

❚❤❡♦r❡♠✱ ❊✐s✇♦rt❤✱ ✷✵✵✷

▲❡t U ❜❡ ❛ st❛❜❧❡ ♦r❞❡r❡❞✲✉♥✐♦♥ ✉❧tr❛✜❧t❡r ♦✈❡r t❤❡ s❡t ♦❢ ❜❧♦❝❦s✳ ❚❤❡ ▼❛t❡t ❢♦r❝✐♥❣ M(U) ♣r❡s❡r✈❡s E ✐✛ Φ(U) ≤RB E✳ ❍♦✇❡✈❡r✱ ▼❛t❡t ❢♦r❝✐♥❣ ♥♦t ❛❞❞ ❛♥ ✉♥s♣❧✐t r❡❛❧✳

❉❡✜♥✐t✐♦♥

❆ r❡❛❧ ✐s ❝❛❧❧❡❞ ❛♥ ✉♥s♣❧✐t r❡❛❧ ✐❢ ❡✈❡r② ✇❡ ❤❛✈❡

✻ ✴ ✸✵

SLIDE 13

❙✉❜❢♦r❝✐♥❣s ♦❢ ▼❛t❡t ❢♦r❝✐♥❣

❚❤❡ ❘✉❞✐♥✕❇❧❛ss ♦r❞❡r

▲❡t H, H′ ⊆ [ω]ω ❜❡ ❝❧♦s❡❞ ✉♥❞❡r ❛❧♠♦st s✉♣❡rs❡ts✳ ❲❡ ✇r✐t❡ H ≤RB H′ ❛♥❞ s❛② H ✐s ❘✉❞✐♥✲❇❧❛ss✲❜❡❧♦✇ H′ ✐✛ t❤❡r❡ ✐s ❛ ✜♥✐t❡✲t♦✲♦♥❡ f s✉❝❤ t❤❛t f(H) ⊆ f(H′)✳ ❍❡r❡ f(H) = {X : f−1[X] ∈ H}✳

❚❤❡♦r❡♠✱ ❊✐s✇♦rt❤✱ ✷✵✵✷

▲❡t U ❜❡ ❛ st❛❜❧❡ ♦r❞❡r❡❞✲✉♥✐♦♥ ✉❧tr❛✜❧t❡r ♦✈❡r t❤❡ s❡t ♦❢ ❜❧♦❝❦s✳ ❚❤❡ ▼❛t❡t ❢♦r❝✐♥❣ M(U) ♣r❡s❡r✈❡s E ✐✛ Φ(U) ≤RB E✳ ❍♦✇❡✈❡r✱ ▼❛t❡t ❢♦r❝✐♥❣ ♥♦t ❛❞❞ ❛♥ ✉♥s♣❧✐t r❡❛❧✳

❉❡✜♥✐t✐♦♥

❆ r❡❛❧ X ∈ V [G] V ✐s ❝❛❧❧❡❞ ❛♥ ✉♥s♣❧✐t r❡❛❧ ✐❢ ❡✈❡r② Y ∈ [ω]ω ∩ V ✇❡ ❤❛✈❡ Y ⊆∗ X ∨ Y ⊆∗ ω X.

✻ ✴ ✸✵

SLIDE 14

❙❡ts ♦❢ ♣♦ss✐❜✐❧✐t✐❡s

❙♦ ✇❡ ✇♦r❦ ✇✐t❤ s✉❜♦r❞❡rs ♦❢ ❇❧❛ss✲❙❤❡❧❛❤ ❢♦r❝✐♥❣✱ ❢♦r ❡①❛♠♣❧❡ t❤❡ ♦♥❡ ♦❢ ❬❇s❙❤✿✷✹✷❪✳ ❋♦r t♦❞❛② ✇❡ t❛❦❡ ❛ ❢♦r❣❡t❢✉❧ ✈❡rs✐♦♥✳

❉❡✜♥✐t✐♦♥

✭✶✮ ❆ ✜♥✐t❡ ω ✐s ❝❛❧❧❡❞ ❛ ❜❧♦❝❦✳ ❆ s❡t ♦❢ ♣♦ss✐❜✐❧✐t✐❡s ✐s ❛ s✉❜s❡t ♦❢ t❤❡ ♣♦✇❡r s❡t ♦❢ ❛ ❜❧♦❝❦ t❤❛t ❝♦♥t❛✐♥s t❤❡ ❡♠♣t② s❡t✳ ❲❡ ❞❡♥♦t❡ ❜② P t❤❡ s❡t ♦❢ ❛❧❧ s❡ts ♦❢ ♣♦ss✐❜✐❧✐t✐❡s✳ ❚②♣✐❝❛❧❧② ✇❡ ✉s❡ ✈❛r✐❛❜❧❡s s✱ t ❢♦r ❜❧♦❝❦s ❛♥❞ a✱ b✱ c ❢♦r s❡ts ♦❢ ♣♦ss✐❜✐❧✐t✐❡s✳ ✭✷✮ ▲❡t a ❜❡ ❛ s❡t ♦❢ ♣♦ss✐❜✐❧✐t✐❡s ❛♥❞ Y ⊆ ω✳ ❲❡ ❧❡t a ↾ Y = {s ∩ Y : s ∈ a}✳

✼ ✴ ✸✵

SLIDE 15

❆ ❝✉t ❛♥❞ ❝❤♦♦s❡ ❣❛♠❡ ❛♥❞ ❛ ♥♦r♠

▲❡t a ❜❡ ❛ s✉❜s❡t ♦❢ t❤❡ ♣♦✇❡r s❡t ♦❢ ❛ ✜♥✐t❡ s❡t✱ ∅ ∈ a✳ ❲❡ ❞❡✜♥❡ ❛ ♥♦r♠✿ ✭❛✮ nor(a) ≥ 0✱ ❛❧✇❛②s✱ ✭❜✮ ✐✛ ✱ ✭❝✮ ✐✛ ✇❤❡♥❡✈❡r t❤❡♥ ✳ ■❢ ✱ t❤❡♥ ❝♦♥t❛✐♥s ❛ ♥♦♥✲❡♠♣t② s❡t✳

✽ ✴ ✸✵

SLIDE 16

❆ ❝✉t ❛♥❞ ❝❤♦♦s❡ ❣❛♠❡ ❛♥❞ ❛ ♥♦r♠

▲❡t a ❜❡ ❛ s✉❜s❡t ♦❢ t❤❡ ♣♦✇❡r s❡t ♦❢ ❛ ✜♥✐t❡ s❡t✱ ∅ ∈ a✳ ❲❡ ❞❡✜♥❡ ❛ ♥♦r♠✿ ✭❛✮ nor(a) ≥ 0✱ ❛❧✇❛②s✱ ✭❜✮ nor(a) ≥ 1 ✐✛ |a| > 1✱ ✭❝✮ ✐✛ ✇❤❡♥❡✈❡r t❤❡♥ ✳ ■❢ ✱ t❤❡♥ ❝♦♥t❛✐♥s ❛ ♥♦♥✲❡♠♣t② s❡t✳

✽ ✴ ✸✵

SLIDE 17

❆ ❝✉t ❛♥❞ ❝❤♦♦s❡ ❣❛♠❡ ❛♥❞ ❛ ♥♦r♠

▲❡t a ❜❡ ❛ s✉❜s❡t ♦❢ t❤❡ ♣♦✇❡r s❡t ♦❢ ❛ ✜♥✐t❡ s❡t✱ ∅ ∈ a✳ ❲❡ ❞❡✜♥❡ ❛ ♥♦r♠✿ ✭❛✮ nor(a) ≥ 0✱ ❛❧✇❛②s✱ ✭❜✮ nor(a) ≥ 1 ✐✛ |a| > 1✱ ✭❝✮ nor(a) ≥ k + 1 ✐✛ ✇❤❡♥❡✈❡r a = Y1 ∪ Y2 t❤❡♥ max(nor(a ↾ Y1), nor(a ↾ Y2)) ≥ k✳ ■❢ ✱ t❤❡♥ ❝♦♥t❛✐♥s ❛ ♥♦♥✲❡♠♣t② s❡t✳

✽ ✴ ✸✵

SLIDE 18

❆ ❝✉t ❛♥❞ ❝❤♦♦s❡ ❣❛♠❡ ❛♥❞ ❛ ♥♦r♠

▲❡t a ❜❡ ❛ s✉❜s❡t ♦❢ t❤❡ ♣♦✇❡r s❡t ♦❢ ❛ ✜♥✐t❡ s❡t✱ ∅ ∈ a✳ ❲❡ ❞❡✜♥❡ ❛ ♥♦r♠✿ ✭❛✮ nor(a) ≥ 0✱ ❛❧✇❛②s✱ ✭❜✮ nor(a) ≥ 1 ✐✛ |a| > 1✱ ✭❝✮ nor(a) ≥ k + 1 ✐✛ ✇❤❡♥❡✈❡r a = Y1 ∪ Y2 t❤❡♥ max(nor(a ↾ Y1), nor(a ↾ Y2)) ≥ k✳ ■❢ nor(a) ≥ 1✱ t❤❡♥ a ❝♦♥t❛✐♥s ❛ ♥♦♥✲❡♠♣t② s❡t✳

✽ ✴ ✸✵

SLIDE 19

❈♦♥❞✐t✐♦♥s ✐♥ t❤❡ ♣✉r❡ ♣❛rt ♦❢ ❛ ❢♦r❝✐♥❣ ♦r❞❡r

❉❡✜♥✐t✐♦♥

✭✶✮ ❋♦r a, b ∈ P ✇✐t❤ a, b = ∅ ✇❡ ✇r✐t❡ a < b ✐❢ (∀n ∈ a)(∀m ∈ b)(n < m)✳ ✭✷✮ ❆ s❡q✉❡♥❝❡ ♦❢ ♠❡♠❜❡rs ♦❢ ✐s ❝❛❧❧❡❞ ✉♥♠❡s❤❡❞ ✐❢ ❢♦r ❛❧❧ ✱ ✳ ✭✸✮ ❇② ✇❡ ❞❡♥♦t❡ t❤❡ s❡t ♦❢ ✉♥♠❡s❤❡❞ s❡q✉❡♥❝❡s s✉❝❤ t❤❛t ✳ ✭✹✮ ▲❡t ✳ ❲❡ ✇r✐t❡ ❢♦r ✳

✾ ✴ ✸✵

SLIDE 20

❈♦♥❞✐t✐♦♥s ✐♥ t❤❡ ♣✉r❡ ♣❛rt ♦❢ ❛ ❢♦r❝✐♥❣ ♦r❞❡r

❉❡✜♥✐t✐♦♥

✭✶✮ ❋♦r a, b ∈ P ✇✐t❤ a, b = ∅ ✇❡ ✇r✐t❡ a < b ✐❢ (∀n ∈ a)(∀m ∈ b)(n < m)✳ ✭✷✮ ❆ s❡q✉❡♥❝❡ ¯ a = an : n ∈ ω ♦❢ ♠❡♠❜❡rs ♦❢ P ✐s ❝❛❧❧❡❞ ✉♥♠❡s❤❡❞ ✐❢ ❢♦r ❛❧❧ n✱ an < an+1✳ ✭✸✮ ❇② ✇❡ ❞❡♥♦t❡ t❤❡ s❡t ♦❢ ✉♥♠❡s❤❡❞ s❡q✉❡♥❝❡s s✉❝❤ t❤❛t ✳ ✭✹✮ ▲❡t ✳ ❲❡ ✇r✐t❡ ❢♦r ✳

✾ ✴ ✸✵

SLIDE 21

❈♦♥❞✐t✐♦♥s ✐♥ t❤❡ ♣✉r❡ ♣❛rt ♦❢ ❛ ❢♦r❝✐♥❣ ♦r❞❡r

❉❡✜♥✐t✐♦♥

✭✶✮ ❋♦r a, b ∈ P ✇✐t❤ a, b = ∅ ✇❡ ✇r✐t❡ a < b ✐❢ (∀n ∈ a)(∀m ∈ b)(n < m)✳ ✭✷✮ ❆ s❡q✉❡♥❝❡ ¯ a = an : n ∈ ω ♦❢ ♠❡♠❜❡rs ♦❢ P ✐s ❝❛❧❧❡❞ ✉♥♠❡s❤❡❞ ✐❢ ❢♦r ❛❧❧ n✱ an < an+1✳ ✭✸✮ ❇② (P)ω ✇❡ ❞❡♥♦t❡ t❤❡ s❡t ♦❢ ✉♥♠❡s❤❡❞ s❡q✉❡♥❝❡s ¯ a s✉❝❤ t❤❛t (∀n)(nor(an) ≥ n + 1)✳ ✭✹✮ ▲❡t ✳ ❲❡ ✇r✐t❡ ❢♦r ✳

✾ ✴ ✸✵

SLIDE 22

❈♦♥❞✐t✐♦♥s ✐♥ t❤❡ ♣✉r❡ ♣❛rt ♦❢ ❛ ❢♦r❝✐♥❣ ♦r❞❡r

❉❡✜♥✐t✐♦♥

✭✶✮ ❋♦r a, b ∈ P ✇✐t❤ a, b = ∅ ✇❡ ✇r✐t❡ a < b ✐❢ (∀n ∈ a)(∀m ∈ b)(n < m)✳ ✭✷✮ ❆ s❡q✉❡♥❝❡ ¯ a = an : n ∈ ω ♦❢ ♠❡♠❜❡rs ♦❢ P ✐s ❝❛❧❧❡❞ ✉♥♠❡s❤❡❞ ✐❢ ❢♦r ❛❧❧ n✱ an < an+1✳ ✭✸✮ ❇② (P)ω ✇❡ ❞❡♥♦t❡ t❤❡ s❡t ♦❢ ✉♥♠❡s❤❡❞ s❡q✉❡♥❝❡s ¯ a s✉❝❤ t❤❛t (∀n)(nor(an) ≥ n + 1)✳ ✭✹✮ ▲❡t ¯ a ∈ (P)ω✳ ❲❡ ✇r✐t❡ a ∈ ¯ a ❢♦r a ∈ {an : n ∈ ω}✳

✾ ✴ ✸✵

SLIDE 23

❚❤❡ ≤✲r❡❧❛t✐♦♥ ♦♥ t❤❡ ♣✉r❡ ♣❛rt

❉❡✜♥✐t✐♦♥

❋♦r s❡q✉❡♥❝❡s ¯ a✱ ¯ b ∈ (P)ω ✇❡ ✇r✐t❡ ¯ b ≤ ¯ a ♦r ✏¯ b str♦♥❣❡r t❤❛♥ ¯ a✑ ✐✛ t❤❡r❡ ✐s ❛ str✐❝t❧② ✐♥❝r❡❛s✐♥❣ ❢✉♥❝t✐♦♥ g: ω → ω s✉❝❤ t❤❛t ❢♦r ❛♥② n✱ bn ⊆ ag(n) ◦ · · · ◦ ag(n+1)−1, ❛♥❞ a ◦ b = {s ∪ t : s ∈ a, t ∈ b}✳

✶✵ ✴ ✸✵

SLIDE 24

Pr♦❥❡❝t✐♦♥ t♦ s✉❜s❡ts ♦❢ ω

❚❤❡ ♥❡①t t✇♦ ♥♦t✐♦♥s ❝♦♥♥❡❝t ❣❡♥❡r❛t❡❞ s❡ts ✐♥ (P)ω ✇✐t❤ s❡♠✐✜❧t❡rs ♦✈❡r ω✳

❉❡✜♥✐t✐♦♥

✭✶✮ ❋♦r ¯ a ∈ (P)ω ✇❡ ❧❡t set2(¯ a) = { an : n ∈ ω}✳ ❲❡ ✇r✐t❡ 2 t♦ ❞✐st✐♥❣✉✐s❤ ✐t ❢r♦♠ ♥♦t✐♦♥s t❤❛t ❛r❡ ✉s❡❞ ✐♥ ▼❛t❡t ❢♦r❝✐♥❣✳ ✭✷✮ ▲❡t ✳ ❚❤❡ ♣r♦❥❡❝t✐♦♥ ♦❢ ✐♥t♦ ✐s ✳

✶✶ ✴ ✸✵

SLIDE 25

Pr♦❥❡❝t✐♦♥ t♦ s✉❜s❡ts ♦❢ ω

❚❤❡ ♥❡①t t✇♦ ♥♦t✐♦♥s ❝♦♥♥❡❝t ❣❡♥❡r❛t❡❞ s❡ts ✐♥ (P)ω ✇✐t❤ s❡♠✐✜❧t❡rs ♦✈❡r ω✳

❉❡✜♥✐t✐♦♥

✭✶✮ ❋♦r ¯ a ∈ (P)ω ✇❡ ❧❡t set2(¯ a) = { an : n ∈ ω}✳ ❲❡ ✇r✐t❡ 2 t♦ ❞✐st✐♥❣✉✐s❤ ✐t ❢r♦♠ ♥♦t✐♦♥s t❤❛t ❛r❡ ✉s❡❞ ✐♥ ▼❛t❡t ❢♦r❝✐♥❣✳ ✭✷✮ ▲❡t H ⊆ (P)ω✳ ❚❤❡ ♣r♦❥❡❝t✐♦♥ ♦❢ H ✐♥t♦ [ω]ω ✐s Φ2(H) = {set2(¯ a) : ¯ a ∈ H}✳

✶✶ ✴ ✸✵

SLIDE 26

❆♥ ✉♥s♣❧✐t r❡❛❧

❚❤❡r❡ ✐s ❛ ❝♦♥♥❡❝t✐♦♥ t♦ ❛❞❞✐♥❣ ❛ r❡❛❧ t❤❛t ✐s ♥♦t s♣❧✐t ❜② ❛♥② r❡❛❧ ✐♥ t❤❡ ❣r♦✉♥❞ ♠♦❞❡❧ ❛♥❞ t♦ ✉❧tr❛✜❧t❡rs ♦✈❡r ω✿

▲❡♠♠❛

■❢ ¯ a ∈ (P)ω ❛♥❞ X ⊆ ω t❤❡♥ t❤❡r❡ ✐s ¯ b ≤ ¯ a s✉❝❤ t❤❛t set2(¯ b) ⊆ X ♦r set2(¯ b) ⊆ (ω \ X)✳

✶✷ ✴ ✸✵

SLIDE 27

❉✐❛❣♦♥❛❧ ❧♦✇❡r ❜♦✉♥❞s

❉❡✜♥✐t✐♦♥

▲❡t ¯ a ∈ (P)ω✱ n ∈ ω✳ ❲❡ ✇r✐t❡ (¯ a past n) ❢♦r ai : i ∈ [k, ω)✱ ✇❤❡r❡ k ✐s t❤❡ ♠✐♥✐♠❛❧ ♥✉♠❜❡r s✉❝❤ t❤❛t n ≤ min ak✳

❉❡✜♥✐t✐♦♥

▲❡t ❜❡ ❛ ✲❞❡s❝❡♥❞✐♥❣ s❡q✉❡♥❝❡ ✐♥ ✳ ❆ s❡q✉❡♥❝❡ ✐s ❛ ❞✐❛❣♦♥❛❧ ❧♦✇❡r ❜♦✉♥❞ ♦❢ ✐✛

✶✸ ✴ ✸✵

SLIDE 28

❉✐❛❣♦♥❛❧ ❧♦✇❡r ❜♦✉♥❞s

❉❡✜♥✐t✐♦♥

▲❡t ¯ a ∈ (P)ω✱ n ∈ ω✳ ❲❡ ✇r✐t❡ (¯ a past n) ❢♦r ai : i ∈ [k, ω)✱ ✇❤❡r❡ k ✐s t❤❡ ♠✐♥✐♠❛❧ ♥✉♠❜❡r s✉❝❤ t❤❛t n ≤ min ak✳

❉❡✜♥✐t✐♦♥

▲❡t ¯ an : n ∈ ω ❜❡ ❛ ≤✲❞❡s❝❡♥❞✐♥❣ s❡q✉❡♥❝❡ ✐♥ (P)ω✳ ❆ s❡q✉❡♥❝❡ ¯ b ∈ (P)ω ✐s ❛ ❞✐❛❣♦♥❛❧ ❧♦✇❡r ❜♦✉♥❞ ♦❢ (¯ an)n ✐✛ (∀b ∈ ¯ b)((¯ b past max(b)) ≤ ¯ amax(b)).

✶✸ ✴ ✸✵

SLIDE 29

❚r❡❡s ♦❢ ✜♥✐t❡ s❡ts

❉❡✜♥✐t✐♦♥

▲❡t ¯ a ∈ (P)ω✱ s ❛ ✜♥✐t❡ s❡t✱ s < min(a0)✳ Lev≤k(s, ¯ a) = {{s} ◦ ai0 ◦ · · · ◦ ain−1 : i0 < · · · < in−1 ≤ k}✳ T(s, ¯ a) = {Lev≤k(¯ a) : k ∈ ω.}✳

✶✹ ✴ ✸✵

SLIDE 30

▲❛r❣❡ ♠♦♥♦❝❤r♦♠❛t✐❝ tr❡❡s

❚❤❡♦r❡♠✱ ❇❧❛ss✱ ❙❤❡❧❛❤

❋♦r ❛♥② C : [ω]<ω → 2 ❛♥❞ ❛♥② ¯ a ∈ (P)ω t❤❡r❡ ✐s ¯ b ≤ ¯ a s✉❝❤ t❤❛t C ↾ T(¯ b) ✐s ❝♦♥st❛♥t✳ ◆♦✇ ✇❡ ✜① ❛ ✲♣♦✐♥t ❛♥❞ ❛ss✉♠❡ ❈❍✳

✶✺ ✴ ✸✵

SLIDE 31

▲❛r❣❡ ♠♦♥♦❝❤r♦♠❛t✐❝ tr❡❡s

❚❤❡♦r❡♠✱ ❇❧❛ss✱ ❙❤❡❧❛❤

❋♦r ❛♥② C : [ω]<ω → 2 ❛♥❞ ❛♥② ¯ a ∈ (P)ω t❤❡r❡ ✐s ¯ b ≤ ¯ a s✉❝❤ t❤❛t C ↾ T(¯ b) ✐s ❝♦♥st❛♥t✳ ◆♦✇ ✇❡ ✜① ❛ P✲♣♦✐♥t E ❛♥❞ ❛ss✉♠❡ ❈❍✳

✶✺ ✴ ✸✵

SLIDE 32

❙✉✐t❛❜❧❡ s❡ts ♦❢ ♣✉r❡ ♣❛rts ♦❢ ❝♦♥❞✐t✐♦♥s

❉❡✜♥✐t✐♦♥

❆ s❡t H ⊆ (P)ω ✐s ❝❛❧❧❡❞ ❛ s✉✐t❛❜❧❡ s❡t ✐❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞✿ ✭✶✮ ✭❯♣✇❛r❞s ❈❧♦s✉r❡✮ H ⊆ (P)ω✱ ❛♥❞ ¯ a ∈ H ❛♥❞ ¯ b ≥∗ ¯ a ✐♠♣❧✐❡s ¯ b ∈ H✱ ✭✷✮ ✭❊①✐st❡♥❝❡ ♦❢ ❉✐❛❣♦♥❛❧ ▲♦✇❡r ❇♦✉♥❞s✮ ■❢ ¯ an : n ∈ [ω]<ω ✐s ❛ ≤✲❞❡s❝❡♥❞✐♥❣ s❡q✉❡♥❝❡ ♦❢ ❡❧❡♠❡♥ts ♦❢ H t❤❡♥ t❤❡r❡ ✐s ¯ b ∈ H s✉❝❤ t❤❛t (∀b ∈ ¯ b)((¯ b past max(b)) ≤ ¯ amax(b))✳ ✭✸✮ ✭❋✉❧❧♥❡ss✮ ❋♦r ❛♥② Y ⊆ ω✱ t❤❡r❡ ✐s ¯ a ∈ H s✉❝❤ t❤❛t set2(¯ a) ⊆ Y ♦r set2(¯ a) ⊆ Y c✳ ✭✹✮ ✭❘❛♠s❡② ♣r♦♣❡rt②✿ ▼♦♥♦❝❤r♦♠❛t✐❝ tr❡❡s ♦❢ ♣♦ss✐❜✐❧✐t✐❡s✮ ❋♦r ❛♥② C : [ω]<ω → 2 ❛♥❞ ❛♥② ¯ a ∈ H t❤❡r❡ ✐s ¯ b ≤ ¯ a ∈ H s✉❝❤ t❤❛t C ↾ T(¯ b) ✐s ❝♦♥st❛♥t✳ ✭✺✮ ✭❆✈♦✐❞✐♥❣ E✮ ❲❡ r❡q✉✐r❡ Φ2(H) ≤RB E✳

✶✻ ✴ ✸✵

SLIDE 33

❆ ❢✉❧❧ ❇❧❛ss✕❙❤❡❧❛❤ ❢♦r❝✐♥❣

❉❡✜♥✐t✐♦♥

■♥ t❤❡ ❢♦r❝✐♥❣ ♦r❞❡r BS✱ ❝♦♥❞✐t✐♦♥s ❛r❡ ♣❛✐rs (s, ¯ a) s✉❝❤ t❤❛t s ∈ P<ω(ω) ❛♥❞ ¯ a ∈ (P)ω ❛♥❞ s < a0✳ ❚❤❡ ❢♦r❝✐♥❣ ♦r❞❡r ✐s (t,¯ b) ≤ (s, ¯ a) ✭r❡❝❛❧❧ t❤❡ str♦♥❣❡r ❝♦♥❞✐t✐♦♥ ✐s t❤❡ s♠❛❧❧❡r ♦♥❡✮ ✐✛ ✭✶✮ s ⊆ t ❛♥❞ ✭✷✮ t❤❡r❡ ✐s s✉❝❤ t❤❛t ❛♥❞ ✭✸✮ ✳

✶✼ ✴ ✸✵

SLIDE 34

❆ ❢✉❧❧ ❇❧❛ss✕❙❤❡❧❛❤ ❢♦r❝✐♥❣

❉❡✜♥✐t✐♦♥

■♥ t❤❡ ❢♦r❝✐♥❣ ♦r❞❡r BS✱ ❝♦♥❞✐t✐♦♥s ❛r❡ ♣❛✐rs (s, ¯ a) s✉❝❤ t❤❛t s ∈ P<ω(ω) ❛♥❞ ¯ a ∈ (P)ω ❛♥❞ s < a0✳ ❚❤❡ ❢♦r❝✐♥❣ ♦r❞❡r ✐s (t,¯ b) ≤ (s, ¯ a) ✭r❡❝❛❧❧ t❤❡ str♦♥❣❡r ❝♦♥❞✐t✐♦♥ ✐s t❤❡ s♠❛❧❧❡r ♦♥❡✮ ✐✛ ✭✶✮ s ⊆ t ❛♥❞ ✭✷✮ t❤❡r❡ ✐s k ∈ ω s✉❝❤ t❤❛t t s ∈ a0 ◦ · · · ◦ ak−1 ❛♥❞ ✭✸✮ ✳

✶✼ ✴ ✸✵

SLIDE 35

❆ ❢✉❧❧ ❇❧❛ss✕❙❤❡❧❛❤ ❢♦r❝✐♥❣

❉❡✜♥✐t✐♦♥

■♥ t❤❡ ❢♦r❝✐♥❣ ♦r❞❡r BS✱ ❝♦♥❞✐t✐♦♥s ❛r❡ ♣❛✐rs (s, ¯ a) s✉❝❤ t❤❛t s ∈ P<ω(ω) ❛♥❞ ¯ a ∈ (P)ω ❛♥❞ s < a0✳ ❚❤❡ ❢♦r❝✐♥❣ ♦r❞❡r ✐s (t,¯ b) ≤ (s, ¯ a) ✭r❡❝❛❧❧ t❤❡ str♦♥❣❡r ❝♦♥❞✐t✐♦♥ ✐s t❤❡ s♠❛❧❧❡r ♦♥❡✮ ✐✛ ✭✶✮ s ⊆ t ❛♥❞ ✭✷✮ t❤❡r❡ ✐s k ∈ ω s✉❝❤ t❤❛t t s ∈ a0 ◦ · · · ◦ ak−1 ❛♥❞ ✭✸✮ ¯ b ≤ an : n ≥ k✳

✶✼ ✴ ✸✵

SLIDE 36

❙✉❜❢♦r❝✐♥❣s ♦❢ ❇❧❛ss✕❙❤❡❧❛❤ ❢♦r❝✐♥❣

❉❡✜♥✐t✐♦♥

✐✈❡♥ ❛ s✉✐t❛❜❧❡ s❡t H ✐♥ (P)ω✱ t❤❡ ♥♦t✐♦♥ ♦❢ ❢♦r❝✐♥❣ BS(H)

❝♦♥s✐sts ♦❢ ❛❧❧ ♣❛✐rs (s, ¯ a) s✉❝❤ t❤❛t ¯ a ∈ H ❛♥❞ s < min(a0)✳ ❚❤❡ ♦r❞❡r r❡❧❛t✐♦♥ ✐s ❛s ✐♥ BS✳ ❲❡ ♥❛♠❡ t❤❡ ❣❡♥❡r✐❝ r❡❛❧s✿

❉❡✜♥✐t✐♦♥

▲❡t G ❜❡ BS(H)✲❣❡♥❡r✐❝ ♦✈❡r V ✳ ❲❡ ❝❛❧❧ W =

{s : ∃¯

a(s, ¯ a) ∈ G} t❤❡ BS(H)✲❣❡♥❡r✐❝ r❡❛❧✳

✶✽ ✴ ✸✵

SLIDE 37

Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❢♦r❝✐♥❣

❘❡♠❛r❦

■❢ Φ2(H) ✐s ❛♥ ✉❧tr❛✜❧t❡r✱ t❤❡♥ BS(H) ❞❡str♦②s t❤❡ ✉❧tr❛✜❧t❡r Φ2(H)✱ s✐♥❝❡ W ❞✐❛❣♦♥❛❧✐s❡s Φ2(H)✳ ▼♦r❡♦✈❡r✱ BS(H) ❞❡str♦②s ❛♥② ✉❧tr❛✜❧t❡r U s✉❝❤ t❤❛t Φ2(H) ≤RB U✳ ❚❤❡ ❣❡♥❡r✐❝ r❡❛❧ ✐s ♥♦t s♣❧✐t ❜② ❛♥② s❡t ✐♥ t❤❡ ❣r♦✉♥❞ ♠♦❞❡❧✿

▲❡♠♠❛

▲❡t H ❜❡ ❛ s✉✐t❛❜❧❡ s❡t ✐♥ (P)ω✳ ■❢ X ⊆ ω✱ X ∈ V t❤❡♥ ❛❢t❡r ❢♦r❝✐♥❣ ✇✐t❤ BS(H) ✇❡ ❤❛✈❡ W ⊆∗ X ♦r W ⊆∗ (ω \ X)✳

✶✾ ✴ ✸✵

SLIDE 38

Pr❡s❡r✈❛t✐♦♥

Pr♦♣♦s✐t✐♦♥

✭Pr♦♣✳ ✷✳✾ ✐♥ ❇s❙❤✿✷✹✷✮ ▲❡t A ˜ ❜❡ ❛ BS(H)✲♥❛♠❡ ❢♦r ❛ s✉❜s❡t ♦❢ ω✳ ❚❤❡♥ ❡✈❡r② ❝♦♥❞✐t✐♦♥ (s, ¯ a) ❤❛s ❛ ✵✲❡①t❡♥s✐♦♥ (s,¯ b) ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt②✳ ■❢ ℓ ∈ ω✱ ✐❢ n = n(ℓ) ✐s t❤❡ ♥✉♠❜❡r s✉❝❤ t❤❛t bℓ ⊆ n✱ ✐❢ t ∈ b0 ◦ · · · ◦ bℓ−1✱ ❛♥❞ ✐❢ i < n(ℓ − 1)✱ t❤❡♥ (t,¯ b past n(ℓ − 1)) ❞❡❝✐❞❡s ✇❤❡t❤❡r i ∈ A ˜ ✳

❚❤❡♦r❡♠

▲❡t ❜❡ ❛♥ s✉✐t❛❜❧❡ s❡t✱ ❛♥❞ ❧❡t ❜❡ ❛ ✲♣♦✐♥t s✉❝❤ t❤❛t ✳ ❚❤❡♥ ❝♦♥t✐♥✉❡s t♦ ❣❡♥❡r❛t❡ ❛♥ ✉❧tr❛✜❧t❡r ❛❢t❡r ✇❡ ❢♦r❝❡ ✇✐t❤ ✳

✷✵ ✴ ✸✵

SLIDE 39

Pr❡s❡r✈❛t✐♦♥

Pr♦♣♦s✐t✐♦♥

✭Pr♦♣✳ ✷✳✾ ✐♥ ❇s❙❤✿✷✹✷✮ ▲❡t A ˜ ❜❡ ❛ BS(H)✲♥❛♠❡ ❢♦r ❛ s✉❜s❡t ♦❢ ω✳ ❚❤❡♥ ❡✈❡r② ❝♦♥❞✐t✐♦♥ (s, ¯ a) ❤❛s ❛ ✵✲❡①t❡♥s✐♦♥ (s,¯ b) ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt②✳ ■❢ ℓ ∈ ω✱ ✐❢ n = n(ℓ) ✐s t❤❡ ♥✉♠❜❡r s✉❝❤ t❤❛t bℓ ⊆ n✱ ✐❢ t ∈ b0 ◦ · · · ◦ bℓ−1✱ ❛♥❞ ✐❢ i < n(ℓ − 1)✱ t❤❡♥ (t,¯ b past n(ℓ − 1)) ❞❡❝✐❞❡s ✇❤❡t❤❡r i ∈ A ˜ ✳

❚❤❡♦r❡♠

▲❡t H ❜❡ ❛♥ s✉✐t❛❜❧❡ s❡t✱ ❛♥❞ ❧❡t E ❜❡ ❛ P✲♣♦✐♥t s✉❝❤ t❤❛t Φ2(H) ≤RB E✳ ❚❤❡♥ E ❝♦♥t✐♥✉❡s t♦ ❣❡♥❡r❛t❡ ❛♥ ✉❧tr❛✜❧t❡r ❛❢t❡r ✇❡ ❢♦r❝❡ ✇✐t❤ BS(H)✳

✷✵ ✴ ✸✵

SLIDE 40

Pr♦♦❢ s❦❡t❝❤

T(s,¯ b) ♠♦♥♦❝❤r♦♠❛t✐❝ ❛s ✐♥ Pr♦♣♦s✐t✐♦♥✳ ▲❡t ❢♦r ℓ ∈ ω✱ n(ℓ) = max( bℓ)✳ A(t) = {i : (∃ℓ)(i < n(ℓ) ∧ (t, (¯ b past n(ℓ))) i ∈ A ˜ )}✳ ❆ss✉♠❡ ❛❧❧ A(t) ∈ E✱ ❛♥❞ B ⊆∗ A(t)✱ B ∈ E✳ ■♥❞✉❝t✐✈❡❧② ❞❡✜♥❡ ❛ s❡q✉❡♥❝❡ ζ(n) : n ∈ ω ♦❢ ♥❛t✉r❛❧ ♥✉♠❜❡rs✱ st❛rt✐♥❣ ✇✐t❤ ζ(0) = 0✱ ❛♥❞ ✐♥❝r❡❛s✐♥❣ s♦ r❛♣✐❞❧② t❤❛t✱ ✐❢ t ∈ Lev≤ζ(k)(s,¯ b)✱ t❤❡♥ ✭✐✮ B A(t) ⊆ ζ(k + 1)✱ ❛♥❞ ✭✐✐✮ ✐❢ i ∈ A(t) ❛♥❞ i < n(ζ(k))✱ t❤❡♥ (t, (¯ b past max(bζ(k)))) i ∈ A✳ ❋✐♥❞ ¯ c ∈ H✱ s✉❝❤ t❤❛t set2(¯ c) ❛✈♦✐❞s B ✐♥ ❛ str♦♥❣ s❡♥s❡✳ ❚❤❡♥ (s, ¯ c) B ∩ X2 ⊂ A ˜ ✳

✷✶ ✴ ✸✵

SLIDE 41

❆ r❡q✉❡st ❢♦r ♠♦r❡ ♣r❡s❡r✈❛t✐♦♥ t❤❡♦r❡♠s ❢♦r ❝s✐

❈♦♥❥❡❝t✉r❡ ♦❢ ❛♥ ■♥❞✉❝t✐♦♥ ▲❡♠♠❛

❚❤❡r❡ ✐s ❛ ❝♦✉♥t❛❜❧❡ s✉♣♣♦rt ✐t❡r❛t✐♦♥ ♦❢ ♣r♦♣❡r ❢♦r❝✐♥❣s Pα, Qβ : β < ω2, α ≤ ω2 s✉❝❤ t❤❛t ❢♦r ❡❛❝❤ β < ω t❤❡r❡ ✐s ❛ Pβ✲♥❛♠❡ Hβ ❢♦r s✉✐t❛❜❧❡ s❡ts s✉❝❤ t❤❛t ❢♦r ❛♥② α ≤ ω2✱ t❤❡ ✐♥✐t✐❛❧ s❡❣♠❡♥t Pγ, Qβ, Hβ : β < α, γ ≤ α ❢✉❧✜❧s✿ ✭P✶✮ ❋♦r ❛❧❧ γ < α✱ Pγ✏Q ˜

γ = BS(Hγ) ❢♦r ❛ s✉✐t❛❜❧❡ s❡t Hγ

❛❞❞✐♥❣ Wγ ∧ Φ2(Hγ) ≤RB E✑✳ ✭P✷✮ Pα ✐s ♣r♦♣❡r ❛♥❞ Pα ✏E ❣❡♥❡r❛t❡s ❛♥ ✉❧tr❛✜❧t❡r✑✳ ✭P✸✮ Pα Hα = {¯ a ∈ (P)ω : (∀γ < α)(∃¯ b ∈ (P)ω)(¯ b ≤ ¯ a ∧ set2(¯ b) ⊆∗ Wγ)}.

✷✷ ✴ ✸✵

SLIDE 42

❘❡✈✐s✐♥❣ t❤❡ ❝❤♦✐❝❡ ♦❢ Hα

❉❡✜♥✐t✐♦♥

✭✶✮ ❆ s✉✐t❛❜❧❡ s❡t C ✐s ❝❛❧❧❡❞ ❝❡♥tr❡❞ ✐❢ ❛♥② ✜♥✐t❡ s✉❜s❡t ♦❢ C ❤❛s ❛ ❧♦✇❡r ❜♦✉♥❞ ✐♥ C✳ ✭✷✮ ❆ ❝❡♥tr❡❞ s✉✐t❛❜❧❡ s❡t ✐s ❝❛❧❧❡❞ ❛ ♠❛①✐♠❛❧ ❝❡♥tr❡❞ s❡t ✐❢ ❢♦r ❛♥② t❤❡r❡ ✐s t❤❛t ✐s ✐♥❝♦♠♣❛t✐❜❧❡ ✇✐t❤ ✳ ❲❡ ♥❛♠❡ t❤❡ ❣❡♥❡r✐❝ ♠❛①✐♠❛❧ ❝❡♥tr❡❞ s❡t✿

❉❡✜♥✐t✐♦♥

❲❡ ❞❡♥♦t❡ ❜② t❤❡ ❣❡♥❡r✐❝ ✜❧t❡r✳

❋❛❝t♦r✐s❛t✐♦♥

✷✸ ✴ ✸✵

SLIDE 43

❘❡✈✐s✐♥❣ t❤❡ ❝❤♦✐❝❡ ♦❢ Hα

❉❡✜♥✐t✐♦♥

✭✶✮ ❆ s✉✐t❛❜❧❡ s❡t C ✐s ❝❛❧❧❡❞ ❝❡♥tr❡❞ ✐❢ ❛♥② ✜♥✐t❡ s✉❜s❡t ♦❢ C ❤❛s ❛ ❧♦✇❡r ❜♦✉♥❞ ✐♥ C✳ ✭✷✮ ❆ ❝❡♥tr❡❞ s✉✐t❛❜❧❡ s❡t ✐s ❝❛❧❧❡❞ ❛ ♠❛①✐♠❛❧ ❝❡♥tr❡❞ s❡t ✐❢ ❢♦r ❛♥② ¯ a ∈ C t❤❡r❡ ✐s ¯ b ∈ C t❤❛t ✐s ✐♥❝♦♠♣❛t✐❜❧❡ ✇✐t❤ ¯ a✳ ❲❡ ♥❛♠❡ t❤❡ ❣❡♥❡r✐❝ ♠❛①✐♠❛❧ ❝❡♥tr❡❞ s❡t✿

❉❡✜♥✐t✐♦♥

❲❡ ❞❡♥♦t❡ ❜② t❤❡ ❣❡♥❡r✐❝ ✜❧t❡r✳

❋❛❝t♦r✐s❛t✐♦♥

✷✸ ✴ ✸✵

SLIDE 44

❘❡✈✐s✐♥❣ t❤❡ ❝❤♦✐❝❡ ♦❢ Hα

❉❡✜♥✐t✐♦♥

✭✶✮ ❆ s✉✐t❛❜❧❡ s❡t C ✐s ❝❛❧❧❡❞ ❝❡♥tr❡❞ ✐❢ ❛♥② ✜♥✐t❡ s✉❜s❡t ♦❢ C ❤❛s ❛ ❧♦✇❡r ❜♦✉♥❞ ✐♥ C✳ ✭✷✮ ❆ ❝❡♥tr❡❞ s✉✐t❛❜❧❡ s❡t ✐s ❝❛❧❧❡❞ ❛ ♠❛①✐♠❛❧ ❝❡♥tr❡❞ s❡t ✐❢ ❢♦r ❛♥② ¯ a ∈ C t❤❡r❡ ✐s ¯ b ∈ C t❤❛t ✐s ✐♥❝♦♠♣❛t✐❜❧❡ ✇✐t❤ ¯ a✳ ❲❡ ♥❛♠❡ t❤❡ ❣❡♥❡r✐❝ ♠❛①✐♠❛❧ ❝❡♥tr❡❞ s❡t✿

❉❡✜♥✐t✐♦♥

❲❡ ❞❡♥♦t❡ ❜② CH t❤❡ (H, ≤) ❣❡♥❡r✐❝ ✜❧t❡r✳

❋❛❝t♦r✐s❛t✐♦♥

BS(H) = (H, ≤) ∗ BS(CH)

✷✸ ✴ ✸✵

SLIDE 45

❆ ❢♦r❝✐♥❣ ✇✐t❤ ❝❡♥tr❡❞ s❡ts

■♥❞✉❝t✐♦♥ ❈♦♥❥❡❝t✉r❡

❚❤❡r❡ ✐s ❛ ❝♦✉♥t❛❜❧❡ s✉♣♣♦rt ✐t❡r❛t✐♦♥ ♦❢ ♣r♦♣❡r ❢♦r❝✐♥❣s Pα, Qβ : β < ω2, α ≤ ω2 s✉❝❤ t❤❛t ❢♦r ❡❛❝❤ β < ω t❤❡r❡ ✐s ❛ Pβ✲♥❛♠❡ Hβ ❢♦r s✉✐t❛❜❧❡ s❡ts s✉❝❤ t❤❛t ❢♦r ❛♥② α ≤ ω2✱ t❤❡ ✐♥✐t✐❛❧ s❡❣♠❡♥t Pγ, Qβ, Cβ : β < α, γ ≤ α ❢✉❧✜❧s✿ ✭P✶✬✮ ❋♦r ❛❧❧ γ < α✱ Pγ ✏Qγ = BS(Cγ) ❢♦r ❛ ♠❛①✐♠❛❧ ❝❡♥tr❡❞ s✉✐t❛❜❧❡ s❡t Cγ✑✳✳ ▲❡t Wγ ❞❡♥♦t❡ t❤❡ BS(Cγ)✲❣❡♥❡r✐❝ r❡❛❧✳ ✭P✷✮ Pα ✐s ♣r♦♣❡r ❛♥❞ Pα ✏E ❣❡♥❡r❛t❡s ❛♥ ✉❧tr❛✜❧t❡r✑✳ ✭P✸✬✮ Pα (∀γ < α)

Cγ ⊆ Cα ∧ (∀¯

a ∈ Cα)(∃¯ b ∈ Cα)(¯ b ≤ ¯ a ∧ set2(¯ b) ⊆∗ Wγ)

✷✹ ✴ ✸✵

SLIDE 46

♦♦❞ ♥❡✇s ❛❜♦✉t t❤❡ ❧✐♠✐t st❡♣s ♦❢ ✉♥❝♦✉♥t❛❜❧❡ ❝♦✜♥❛❧✐t②

❋♦r α < ω2 ♦❢ ✉♥❝♦✉♥t❛❜❧❡ ❝♦✜♥❛❧✐t②✱ Cα = {Cβ : β < α} ✐s ❛ ♠❛①✐♠❛❧ ❝❡♥tr❡❞ s✉✐t❛❜❧❡ s❡t✳

✷✺ ✴ ✸✵

SLIDE 47

❖♥ t❤❡ ❝♦♥❞✐t✐♦♥ set2(¯ b) ⊆∗ Wα

❉❡✜♥✐t✐♦♥

❋♦r ❛ s❡q✉❡♥❝❡ ¯ a ∈ (P)ω ❛♥❞ X ⊆ ω ✐s s✉❝❤ t❤❛t C = {n : nor(¯ an ↾ X) ≥ n − 1} ✐s ✐♥✜♥✐t❡✱ t❤❡♥ ✇❡ ✇❡ ✇r✐t❡ ¯ a ↾ X ❢♦r ¯ an ↾ X : n ∈ C \ min(C)✳

▲❡♠♠❛

▲❡t (s, ¯ a) ∈ BS(C) ❛♥❞ ❧❡t G ❜❡ BS(C)✲❣❡♥❡r✐❝ ❛♥❞ ❧❡t W ❞❡♥♦t❡ t❤❡ BS(C)✲❣❡♥❡r✐❝ r❡❛❧✳ ❚❤❡♥ V [G] | = ¯ a ↾ W ∈ (P)ω✳

✷✻ ✴ ✸✵

SLIDE 48

❚♦✇❛r❞s ❝♦✈❡r✐♥❣ ♠♦❞❡❧s

❲❡ ❞❡✜♥❡ t✉♣❧❡ Rn,α : n ∈ ω ♦❢ r❡❧❛t✐♦♥s s✉❝❤ t❤❛t t❤❡r❡ ✐s ❛ s✉✐t❛❜❧❡ s❡t ❛s ✐♥ ✭P✸✮ ✐s ❡q✉✐✈❛❧❡♥t t♦ V Pα | = (∀f ∈ dom(R0,α))(∃¯ g ∈ range(R0,α))(

n∈ω

fRn,αg). ❲❡ ❧❡t ¯ R = Rn,α : α ≤ ω2, n ∈ ω✱ α ❜❡✐♥❣ t❤❡ st❛❣❡✱ n t❤❡ s✐③❡ ♦❢ t❤❡ ✑♠✐st❛❦❡✑ ✐♥ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❦✐♥❞ ✑❢♦r ❛❧❧ ❜✉t ✜♥✐t❡❧② ♠❛♥②✑✳ ✇✐❧❧ ❜❡ ❛ ❝❧♦s❡❞ r❡❧❛t✐♦♥ ✐♥ ✐♥ t❤❡ ❇❛✐r❡ s♣❛❝❡ t♦♣♦❧♦❣②✳ ❚❤❡ ♦r❞✐♥❛❧ ✐♥❞✐❝❛t❡s t❤❡ st❛❣❡✳ ❚❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ r❡❧❛t✐♦♥✱ ❛♥❞ t❤❡ ♣r♦♦❢ t❤❛t ✐t ✐s ♣r❡s❡r✈❡❞✱ ❛r❡ ❝❛rr✐❡❞ ♦♥ s✐♠✉❧t❛♥❡♦✉s❧② ❜② ✐♥❞✉❝t✐♦♥ ♦♥ ✳ ◆♦✇ s✉♣♣♦s❡ ✱ ✱ ❛r❡ ❞❡✜♥❡❞ ❛♥❞ ✇❡ ✇♦r❦ ✐♥ ✿

✷✼ ✴ ✸✵

SLIDE 49

❚♦✇❛r❞s ❝♦✈❡r✐♥❣ ♠♦❞❡❧s

❲❡ ❞❡✜♥❡ t✉♣❧❡ Rn,α : n ∈ ω ♦❢ r❡❧❛t✐♦♥s s✉❝❤ t❤❛t t❤❡r❡ ✐s ❛ s✉✐t❛❜❧❡ s❡t ❛s ✐♥ ✭P✸✮ ✐s ❡q✉✐✈❛❧❡♥t t♦ V Pα | = (∀f ∈ dom(R0,α))(∃¯ g ∈ range(R0,α))(

n∈ω

fRn,αg). ❲❡ ❧❡t ¯ R = Rn,α : α ≤ ω2, n ∈ ω✱ α ❜❡✐♥❣ t❤❡ st❛❣❡✱ n t❤❡ s✐③❡ ♦❢ t❤❡ ✑♠✐st❛❦❡✑ ✐♥ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❦✐♥❞ ✑❢♦r ❛❧❧ ❜✉t ✜♥✐t❡❧② ♠❛♥②✑✳ Rn,α ✇✐❧❧ ❜❡ ❛ ❝❧♦s❡❞ r❡❧❛t✐♦♥ ✐♥ dom(Rn,α) × range(Rn,α) ✐♥ t❤❡ ❇❛✐r❡ s♣❛❝❡ t♦♣♦❧♦❣②✳ ❚❤❡ ♦r❞✐♥❛❧ α ✐♥❞✐❝❛t❡s t❤❡ st❛❣❡✳ ❚❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ r❡❧❛t✐♦♥✱ ❛♥❞ t❤❡ ♣r♦♦❢ t❤❛t ✐t ✐s ♣r❡s❡r✈❡❞✱ ❛r❡ ❝❛rr✐❡❞ ♦♥ s✐♠✉❧t❛♥❡♦✉s❧② ❜② ✐♥❞✉❝t✐♦♥ ♦♥ ✳ ◆♦✇ s✉♣♣♦s❡ ✱ ✱ ❛r❡ ❞❡✜♥❡❞ ❛♥❞ ✇❡ ✇♦r❦ ✐♥ ✿

✷✼ ✴ ✸✵

SLIDE 50

❚♦✇❛r❞s ❝♦✈❡r✐♥❣ ♠♦❞❡❧s

❲❡ ❞❡✜♥❡ t✉♣❧❡ Rn,α : n ∈ ω ♦❢ r❡❧❛t✐♦♥s s✉❝❤ t❤❛t t❤❡r❡ ✐s ❛ s✉✐t❛❜❧❡ s❡t ❛s ✐♥ ✭P✸✮ ✐s ❡q✉✐✈❛❧❡♥t t♦ V Pα | = (∀f ∈ dom(R0,α))(∃¯ g ∈ range(R0,α))(

n∈ω

fRn,αg). ❲❡ ❧❡t ¯ R = Rn,α : α ≤ ω2, n ∈ ω✱ α ❜❡✐♥❣ t❤❡ st❛❣❡✱ n t❤❡ s✐③❡ ♦❢ t❤❡ ✑♠✐st❛❦❡✑ ✐♥ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❦✐♥❞ ✑❢♦r ❛❧❧ ❜✉t ✜♥✐t❡❧② ♠❛♥②✑✳ Rn,α ✇✐❧❧ ❜❡ ❛ ❝❧♦s❡❞ r❡❧❛t✐♦♥ ✐♥ dom(Rn,α) × range(Rn,α) ✐♥ t❤❡ ❇❛✐r❡ s♣❛❝❡ t♦♣♦❧♦❣②✳ ❚❤❡ ♦r❞✐♥❛❧ α ✐♥❞✐❝❛t❡s t❤❡ st❛❣❡✳ ❚❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ r❡❧❛t✐♦♥✱ ❛♥❞ t❤❡ ♣r♦♦❢ t❤❛t ✐t ✐s ♣r❡s❡r✈❡❞✱ ❛r❡ ❝❛rr✐❡❞ ♦♥ s✐♠✉❧t❛♥❡♦✉s❧② ❜② ✐♥❞✉❝t✐♦♥ ♦♥ α < ω2✳ ◆♦✇ s✉♣♣♦s❡ Pα✱ Cβ✱ β < α ❛r❡ ❞❡✜♥❡❞ ❛♥❞ ✇❡ ✇♦r❦ ✐♥ V Pα✿

✷✼ ✴ ✸✵

SLIDE 51

❆ r❡❧❛t✐♦♥

❉❡✜♥✐t✐♦♥

f = (¯ aℓ : ℓ ∈ ω, X, C, h) ✐s ❝❛❧❧❡❞ ❛ t❛s❦ ✐✛ ✭✶✮ (¯ aℓ)ℓ ✐s ❛ ≤✲❞❡s❝❡♥❞✐♥❣ s❡q✉❡♥❝❡ ✐♥ (P)ω✱ ✭✷✮ C : [ω]ω → 2✱ ✭✸✮ h: ω → ω ❛ ✜♥✐t❡✲t♦✲♦♥❡ ❢✉♥❝t✐♦♥✳

❉❡✜♥✐t✐♦♥

▲❡t f ❜❡ ❛ t❛s❦✳ ❲❡ s❛② ¯ g ❛♥s✇❡rs t❤❡ t❛s❦✱ ✐✛ ✭✶✮ ¯ g ∈ (P)ω✳ ✭✷✮ ¯ g ✐s ❛ ❞✐❛❣♦♥❛❧ ❧♦✇❡r ❜♦✉♥❞ ♦❢ (¯ aℓ)ℓ✳ ✭✸✮ C ↾ T(¯ g) ✐s ❝♦♥st❛♥t✮✳ ✭✹✮ ∃E ∈ Eh[E] ∩ h[set2(¯ g)] = ∅✳

✷✽ ✴ ✸✵

SLIDE 52

❆ r❡❧❛t✐♦♥✱ ❝♦♥t✐♥✉❡❞

❉❡✜♥✐t✐♦♥

❆ss✉♠❡ t❤❛t Hγ : γ < α ✐s ❛♥ s❡q✉❡♥❝❡ ♦❢ s✉✐t❛❜❧❡ s❡ts Hγ ∈ V Pγ ❛♥❞ ✐♥ Hγ : γ < α ∈ V Pα✳ ✭❛✮ ❚❤❡ ❞♦♠❛✐♥ ♦❢ Rn,α ✐s t❤❡ s❡t ♦❢ f = ((¯ aℓ)ℓ, C, h) s✉❝❤ t❤❛t f ✐s ❛ t❛s❦ ✐♥ V Pα ❛♥❞ s✉❝❤ t❤❛t ❛❧❧ ¯ aℓ ❛r❡ ❝♦♠♣❛t✐❜❧❡ ✇✐t❤ Cγ✱ γ < α✱ (∀ℓ)(∀γ < α)(∃¯ bℓ ≤ ¯ aℓ)(set2(¯ bℓ) ⊆∗ Wγ). ✭❜✮ ❚❤❡ r❛♥❣❡ ♦❢ Rn,α ✐s t❤❡ s❡t ♦❢ ¯ g t❤❛t ❛r❡ ❝♦♠♣❛t✐❜❧❡ s✉❝❤ t❤❛t (∀γ < α)((∃¯ b ≤ ¯ g)(set2(¯ b) ⊆∗ Wγ)✳ ✭❝✮ ❲❡ ✇r✐t❡ fRn,αg ✐✛ ✭✶✮ (∀ℓ ∈ ω)

((¯

g past gℓ), past n + 1) ≤ ¯ amax(gℓ)+1

✭✸✮ C ↾ T(¯

g past n) ✐s ❝♦♥st❛♥t✮✳ ✭✹✮ ∃E ∈ Eh[set2(¯ g)] ∩ h[E] ⊆ n✳

✷✾ ✴ ✸✵

SLIDE 53

❚❤❡ ✐♥t❡♥❞❡❞ ✉s❡

▲❡♠♠❛

❚❤❡r❡ ✐s ❛ s✉✐t❛❜❧❡ s❡t Cα ❛s ✐♥ ✭P✸✬✮ ✐✛ V Pα | = (∀f ∈ dom(R0,α))(∃¯ g ∈ range(R0,α))(

n∈ω fRn,α¯

g))✳

✸✵ ✴ ✸✵