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▼✐❧❧❡r ❋♦r❝✐♥❣ ✇✐t❤ ❈❛♥❥❛r ❯❧tr❛✜❧t❡rs

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❥♦✐♥t ✇♦r❦ ✇✐t❤ ❈❤r✐st✐❛♥ ❇rä✉♥✐♥❣❡r ❍❛♠❜✉r❣ ❙❡t ❚❤❡♦r② ❲♦r❦s❤♦♣ ♦♥❧✐♥❡ ❈♦♥❢❡r❡♥❝❡ ❤♦st❡❞ ❜② ❍❛♠❜✉r❣ ❏✉♥❡ ✷✵ ✕ ✷✶✱ ✷✵✷✵

✶ ✴ ✸✵

❋✐♥✐t❡ ❙tr✐❝t❧② ■♥❝r❡❛s✐♥❣ ❙❡q✉❡♥❝❡s ✕ ❇❧♦❝❦s

❉❡✜♥✐t✐♦♥

❚❤❡ s❡t ♦❢ ✜♥✐t❡ str✐❝t❧② ✐♥❝r❡❛s✐♥❣ s❡q✉❡♥❝❡s ♦❢ ♥❛t✉r❛❧ ♥✉♠❜❡rs ✐s ❝❛❧❧❡❞ ω↑<ω✳ ❚❤❡ ❧❡♥❣t❤ ♦❢ s ∈ ω↑<ω✱ |s|✱ ✐s ✐ts ❞♦♠❛✐♥✳ ❋♦r s, t ∈ ω<ω✱ ✇❡ s❛② ✏t ❡①t❡♥❞s s✑ ♦r ✏s ✐s ❛♥ ✐♥✐t✐❛❧ s❡❣♠❡♥t ♦❢ t✑ ❛♥❞ ✇r✐t❡ s t ✐❢ dom(s) ⊆ dom(t) ❛♥❞ s = t ↾ dom(s)✳ ❋♦r ✇❡ ❤❛✈❡ ❛♥❞ ✈✐❝❡ ✈❡rs❛✱ ❣♦✐♥❣ ❢r♦♠ ✜♥✐t❡ s✉❜s❡ts ♦❢ t♦ t❤❡✐r ✐♥❝r❡❛s✐♥❣ ❡♥✉♠❡r❛t✐♦♥s✳

✷ ✴ ✸✵

❋✐♥✐t❡ ❙tr✐❝t❧② ■♥❝r❡❛s✐♥❣ ❙❡q✉❡♥❝❡s ✕ ❇❧♦❝❦s

❉❡✜♥✐t✐♦♥

❚❤❡ s❡t ♦❢ ✜♥✐t❡ str✐❝t❧② ✐♥❝r❡❛s✐♥❣ s❡q✉❡♥❝❡s ♦❢ ♥❛t✉r❛❧ ♥✉♠❜❡rs ✐s ❝❛❧❧❡❞ ω↑<ω✳ ❚❤❡ ❧❡♥❣t❤ ♦❢ s ∈ ω↑<ω✱ |s|✱ ✐s ✐ts ❞♦♠❛✐♥✳ ❋♦r s, t ∈ ω<ω✱ ✇❡ s❛② ✏t ❡①t❡♥❞s s✑ ♦r ✏s ✐s ❛♥ ✐♥✐t✐❛❧ s❡❣♠❡♥t ♦❢ t✑ ❛♥❞ ✇r✐t❡ s t ✐❢ dom(s) ⊆ dom(t) ❛♥❞ s = t ↾ dom(s)✳ ❋♦r s, t ∈ ω↑<ω ✇❡ ❤❛✈❡ s t ⇔ range(s) ⊑ range(t) ❛♥❞ ✈✐❝❡ ✈❡rs❛✱ ❣♦✐♥❣ ❢r♦♠ ✜♥✐t❡ s✉❜s❡ts ♦❢ ω t♦ t❤❡✐r ✐♥❝r❡❛s✐♥❣ ❡♥✉♠❡r❛t✐♦♥s✳

✷ ✴ ✸✵

❚r❡❡s

❉❡✜♥✐t✐♦♥

❆ s✉❜s❡t p ⊆ ω↑<ω t❤❛t ✐s ❝❧♦s❡❞ ✉♥❞❡r ✐♥✐t✐❛❧ s❡❣♠❡♥ts ✐s ❝❛❧❧❡❞ ❛ tr❡❡✳ ❚❤❡ ❡❧❡♠❡♥ts ♦❢ ❛ tr❡❡ ❛r❡ ❝❛❧❧❡❞ ♥♦❞❡s✳ ❆ ♥♦❞❡ s ∈ p ✐s ❝❛❧❧❡❞ ❛ s♣❧✐tt✐♥❣ ♥♦❞❡ ♦❢ p ✐❢ s ❤❛s ♠♦r❡ t❤❛♥ ♦♥❡ ❞✐r❡❝t ⊳✲s✉❝❝❡ss♦r ✐♥ p ❛♥❞ ω✲s♣❧✐tt✐♥❣ ♥♦❞❡ ♦❢ p ✐❢ s ❤❛s ✐♥✜♥✐t❡❧② ♠❛♥② ❞✐r❡❝t ⊳✲s✉❝❝❡ss♦rs ✐♥ p✳ ❚❤❡ s❡t ♦❢ s♣❧✐tt✐♥❣ ♥♦❞❡s ♦❢ p ✐s ❞❡♥♦t❡❞ ❜② sp(p) ✇❤✐❧❡ ω✲ sp(p) ❞❡♥♦t❡s t❤❡ s❡t ♦❢ ω✲s♣❧✐tt✐♥❣ ♥♦❞❡s ♦❢ p✳

✸ ✴ ✸✵

❙♣❡❝✐❛❧ ❙❡ts ♦❢ ❇❧♦❝❦s

❉❡✜♥✐t✐♦♥

✭✶✮ ❋♦r ❛♥② s❡t A ✇❡ ✇r✐t❡ [A]<ω = {t : t ⊆ A, |t| < ω}✳ ❚❤❡ ❡❧❡♠❡♥ts ♦❢ Fin = [ω]<ω \ {∅} ❛r❡ ❝❛❧❧❡❞ ❜❧♦❝❦s✳ ✭✷✮ ▲❡t F ❜❡ ❛ ✜❧t❡r ♦✈❡r ω✳ ❲❡ ❧❡t F<ω ={[A]<ω \ {∅} : A ∈ F} (F<ω)+ ={B ⊆ Fin : ∀A ∈ F([A]<ω ∩ B = ∅)}

✹ ✴ ✸✵

❋❛♠✐❧✐❡s ♦❢ ❙✉♣❡r♣❡r❢❡❝t ❚r❡❡s

• ✉③♠á♥ ❛♥❞ ❑❛❧❛❥❞③✐❡✈s❦✐ ✐♥tr♦❞✉❝❡❞ ❛ ❢❛♠✐❧② ♦❢ ▼✐❧❧❡r ❢♦r❝✐♥❣s

PT(F)✱ F ❛ ✜❧t❡r ♦✈❡r ω✱ ❡①t❡♥❞✐♥❣ t❤❡ ❋ré❝❤❡t ✜❧t❡r✳

❉❡✜♥✐t✐♦♥

▲❡t F ❜❡ ❛ ✜❧t❡r ♦✈❡r ω✳ ❚❤❡ ❢♦r❝✐♥❣ PT(F) ❝♦♥s✐sts ♦❢ ❛❧❧ p ⊆ ω↑<ω s✉❝❤ t❤❛t ❢♦r ❡❛❝❤ s ∈ p t❤❡r❡ ✐s t s✱ s✉❝❤ t❤❛t t ∈ ω✲ sp(p) ❛♥❞ sucsplp(t) := {range(r) \ range(t) : r ❛ ⊳✲♠✐♥✐♠❛❧ ✐♥✜♥✐t❡❧② s♣❧✐tt✐♥❣ ♥♦❞❡ ♦❢ p ❛❜♦✈❡ t} ∈ (F<ω)+. ❙✉❝❤ ❛ ✐s ❝❛❧❧❡❞ ❛♥ ✲s♣❧✐tt✐♥❣ ♥♦❞❡✳ ❲❡ ❢✉rt❤❡r♠♦r❡ r❡q✉✐r❡ ♦❢ t❤❛t ❡❛❝❤ ✲s♣❧✐tt✐♥❣ ♥♦❞❡ ✐s ❛♥ ✲s♣❧✐tt✐♥❣ ♥♦❞❡ ❛♥❞ t❤❡r❡ ✐s ❛ ✉♥✐q✉❡ ✲♠✐♥✐♠❛❧ ✲s♣❧✐tt✐♥❣ ♥♦❞❡ ❝❛❧❧❡❞ t❤❡ tr✉♥❦ ♦❢ ✱ ✳ ❚❤❡ s❡t ♦❢ ✲s♣❧✐tt✐♥❣ ♥♦❞❡s ♦❢ ✐s ❞❡♥♦t❡❞ ❜② ✲ ✳

✺ ✴ ✸✵

❋❛♠✐❧✐❡s ♦❢ ❙✉♣❡r♣❡r❢❡❝t ❚r❡❡s

• ✉③♠á♥ ❛♥❞ ❑❛❧❛❥❞③✐❡✈s❦✐ ✐♥tr♦❞✉❝❡❞ ❛ ❢❛♠✐❧② ♦❢ ▼✐❧❧❡r ❢♦r❝✐♥❣s

PT(F)✱ F ❛ ✜❧t❡r ♦✈❡r ω✱ ❡①t❡♥❞✐♥❣ t❤❡ ❋ré❝❤❡t ✜❧t❡r✳

❉❡✜♥✐t✐♦♥

▲❡t F ❜❡ ❛ ✜❧t❡r ♦✈❡r ω✳ ❚❤❡ ❢♦r❝✐♥❣ PT(F) ❝♦♥s✐sts ♦❢ ❛❧❧ p ⊆ ω↑<ω s✉❝❤ t❤❛t ❢♦r ❡❛❝❤ s ∈ p t❤❡r❡ ✐s t s✱ s✉❝❤ t❤❛t t ∈ ω✲ sp(p) ❛♥❞ sucsplp(t) := {range(r) \ range(t) : r ❛ ⊳✲♠✐♥✐♠❛❧ ✐♥✜♥✐t❡❧② s♣❧✐tt✐♥❣ ♥♦❞❡ ♦❢ p ❛❜♦✈❡ t} ∈ (F<ω)+. ❙✉❝❤ ❛ t ✐s ❝❛❧❧❡❞ ❛♥ F✲s♣❧✐tt✐♥❣ ♥♦❞❡✳ ❲❡ ❢✉rt❤❡r♠♦r❡ r❡q✉✐r❡ ♦❢ p t❤❛t ❡❛❝❤ ω✲s♣❧✐tt✐♥❣ ♥♦❞❡ ✐s ❛♥ F✲s♣❧✐tt✐♥❣ ♥♦❞❡ ❛♥❞ t❤❡r❡ ✐s ❛ ✉♥✐q✉❡ ⊳✲♠✐♥✐♠❛❧ ω✲s♣❧✐tt✐♥❣ ♥♦❞❡ ❝❛❧❧❡❞ t❤❡ tr✉♥❦ ♦❢ p✱ tr(p)✳ ❚❤❡ s❡t ♦❢ F✲s♣❧✐tt✐♥❣ ♥♦❞❡s ♦❢ p ✐s ❞❡♥♦t❡❞ ❜② F✲ sp(p)✳

✺ ✴ ✸✵

P❧❛✐♥ s❧✐❞❡ ❢♦r ❛ s❦❡t❝❤

✻ ✴ ✸✵

• ✉③♠á♥✬s ❛♥❞ ❑❛❧❛❥❞③✐❡✈s❦✐✬s ❘❡s✉❧ts

▲❡♠♠❛

❚❤❡ ❢♦r❝✐♥❣ PT(F) ❤❛s t❤❡ ♣✉r❡ ❞❡❝✐s✐♦♥ ♣r♦♣❡rt②✳ p ∈ PT(F)✱ s ∈ F✲ sp(p)✱ D ⊆ PT(F) ♦♣❡♥ ❞❡♥s❡✳ ❚❤❡♥ E(p, s, D) =⊑ − min{range(t)\range(s) : ∃q ≤ p(tr(q) = t∧q ∈ D)} ✐s ✐♥ (F<ω)+✳ ▲❡t f ∈ F✳ p ↾↾ F ❝♦♥t❛✐♥s ♦♥❧② t❤♦s❡ ♥♦❞❡s t ∈ p ❢♦r ✇❤✐❝❤ range(t) \ tr(p) ⊆ F✳ ❲❡ ❤❛✈❡ p ↾↾ F ≤0 p✳

✼ ✴ ✸✵

• ✉③♠á♥✬s ❛♥❞ ❑❛❧❛❥❞③✐❡✈s❦✐✬s ❘❡s✉❧ts

▲❡♠♠❛

❚❤❡ ❢♦r❝✐♥❣ PT(F) ❤❛s t❤❡ ♣✉r❡ ❞❡❝✐s✐♦♥ ♣r♦♣❡rt②✳ p ∈ PT(F)✱ s ∈ F✲ sp(p)✱ D ⊆ PT(F) ♦♣❡♥ ❞❡♥s❡✳ ❚❤❡♥ E(p, s, D) =⊑ − min{range(t)\range(s) : ∃q ≤ p(tr(q) = t∧q ∈ D)} ✐s ✐♥ (F<ω)+✳ ▲❡t f ∈ F✳ p ↾↾ F ❝♦♥t❛✐♥s ♦♥❧② t❤♦s❡ ♥♦❞❡s t ∈ p ❢♦r ✇❤✐❝❤ range(t) \ tr(p) ⊆ F✳ ❲❡ ❤❛✈❡ p ↾↾ F ≤0 p✳

✼ ✴ ✸✵

❋♦r❝✐♥❣ ✇✐t❤ Fσ✲❋✐❧t❡rs

❉❡✜♥✐t✐♦♥

✭✶✮ ❚❤❡ ♣❛rt✐❛❧ ♦r❞❡r Fσ ✐s t❤❡ ❢♦r❝✐♥❣ ✇✐t❤ Fσ✲✜❧t❡rs ♦✈❡r ω✳ ❙tr♦♥❣❡r ✜❧t❡rs ❛r❡ s✉♣❡r✜❧t❡rs✳ ✭✷✮ ■❢ F ✐s ❛ ✜❧t❡r✱ t❤❡♥ Fσ(F) ✐s t❤❡ ❢♦r❝✐♥❣ ✇✐t❤ Fσ✲✜❧t❡rs t❤❛t ❛r❡ ❝♦♠♣❛t✐❜❧❡ ✇✐t❤ F✱ ✐✳❡✳ G ∈ Fσ(F) ✐✛ G ✐s ❛♥ Fσ✲✜❧t❡r ❛♥❞ G ⊆ F+ = {X ⊆ ω : ∀(F ∈ F)(X ∩ F = ∅)}✳

❉❡✜♥✐t✐♦♥

▲❡t G ❜❡ ❛♥ Fσ(F)✲❣❡♥❡r✐❝ ✜❧t❡r✳ ❲❡ ❧❡t U ❜❡ ❛ Fσ(F)✲♥❛♠❡ ❢♦r t❤❡ ✉♥✐♦♥ ♦❢ G✳ ❇② ❛ ❞❡♥s✐t② ❛r❣✉♠❡♥t✱ t❤❡ ♣♦s❡t Fσ(F) ❢♦r❝❡s t❤❛t U ✐s ❛♥ ✉❧tr❛✜❧t❡r t❤❛t ❝♦♥t❛✐♥s F ❛s ❛ s✉❜s❡t✳

✽ ✴ ✸✵

❈❛♥❥❛r ❋✐❧t❡rs

❉❡✜♥✐t✐♦♥

F ✐s ❝❛❧❧❡❞ ❛ ❈❛♥❥❛r ✜❧t❡r ✐❢ ❢♦r ❛♥② s❡q✉❡♥❝❡ Xn : n < ω ♦❢ ❡❧❡♠❡♥ts ♦❢ (F<ω)+ t❤❡r❡ ✐s ❛ s❡q✉❡♥❝❡ Yn ∈ [Xn]<ω s✉❝❤ t❤❛t {Yn : n < ω} ∈ (F<ω)+✳ ❍r✉➨á❦ ❛♥❞ ▼✐♥❛♠✐ s❤♦✇❡❞✿ ❆ ✜❧t❡r ✐s ❈❛♥❥❛r ✐✛ ▼❛t❤✐❛s ❢♦r❝✐♥❣ ✇✐t❤ s❡❝♦♥❞ ❝♦♠♣♦♥❡♥ts ✐♥ t❤❡ ✜❧t❡r ❞♦❡s ♥♦t ❛❞❞ ❛ ❞♦♠✐♥❛t✐♥❣ r❡❛❧✳ ▼♦r❡ ❡q✉✐✈❛❧❡♥t ❢♦r♠✉❧❛t✐♦♥s ❛r❡ ❣✐✈❡♥ ❜② ❇❧❛ss ❍r✉s❛❦ ❱❡r♥❡r✱ ❈❤♦❞♦✉♥s❦② ❘❡♣♦✈s ❩❞♦♠s❦②②✱ ●✉③♠á♥ ❍r✉➨á❦ ▼❛rt✐♥❡③✳

✾ ✴ ✸✵

• ❡♥❡r✐❝ ❈❛♥❥❛r ❯❧tr❛✜❧t❡rs

▲❡♠♠❛

❈❛♥❥❛r✳ ❚❤❡ ❣❡♥❡r✐❝ ✜❧t❡r U ♦❢ t❤❡ ❢♦r❝✐♥❣ Fσ ✐s s✉❝❤ t❤❛t ▼❛t❤✐❛s ❢♦r❝✐♥❣ ✇✐t❤ ✐t ❞♦❡s ♥♦t ❛❞❞ ❛ ❞♦♠✐♥❛t✐♥❣ r❡❛❧✳

✶✵ ✴ ✸✵

PT(U) ❛♥❞ ❖t❤❡r ❯❧tr❛✜❧t❡rs

❆♥♦t❤❡r ✐♠♣♦rt❛♥t ❝♦♥❝❡♣t ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✳

❉❡✜♥✐t✐♦♥

✭✶✮ ❆ ❢✉♥❝t✐♦♥ h: ω → ω ✐s ❝❛❧❧❡❞ ✜♥✐t❡✲t♦✲♦♥❡ ✐❢ ❢♦r ❛♥② n✱ t❤❡ ♣r❡✐♠❛❣❡ ♦❢ {n}✱ ✐✳❡✳ h−1[{n}]✱ ✐s ✜♥✐t❡ ✭t❤✐s ✐♥❝❧✉❞❡s t❤❡ ♣♦ss✐❜✐❧✐t② ♦❢ ❜❡✐♥❣ ❡♠♣t②✮✳ ✭✷✮ ▲❡t F ❛♥❞ U ❜❡ ✉❧tr❛✜❧t❡rs ♦✈❡r ω✳ F ❛♥❞ U ❛r❡ ❝❛❧❧❡❞ ♥❡❛r❧② ❝♦❤❡r❡♥t ✐❢ t❤❡r❡ ✐s ❛ ✜♥✐t❡✲t♦✲♦♥❡ ❢✉♥❝t✐♦♥ h s✉❝❤ t❤❛t h(F) = h(U) ✇❤❡r❡ h(U) = {X ⊆ ω : h−1[X] ∈ U}✳ ✭✸✮ ❆ ✜❧t❡r F ✐s ❝❛❧❧❡❞ ❛❧♠♦st ✉❧tr❛ ✐❢ t❤❡r❡ ✐s ❛ ✜♥✐t❡✲t♦✲♦♥❡ ♠❛♣✲ ♣✐♥❣ h s✉❝❤ t❤❛t h(F) ✐s ❛♥ ✉❧tr❛✜❧t❡r✳

✶✶ ✴ ✸✵

PT(U) ❛♥❞ ❖t❤❡r ❯❧tr❛✜❧t❡rs✱ ■■

▲❡♠♠❛

▲❡t W ❜❡ ❛ P✲♣♦✐♥t✳ ✭❛✮ ■❢ U ✐s ❛ ❈❛♥❥❛r ✉❧tr❛✜❧t❡r t❤❛t ✐s ♥♦t ♥❡❛r❧② ❝♦❤❡r❡♥t t♦ W✱ t❤❡♥ ❢♦r❝✐♥❣ ✇✐t❤ PT(U) ♣r❡s❡r✈❡s W✳ ✭❜✮ ■❢ ❛ ❈❛♥❥❛r ✜❧t❡r F ✐s ♥♦t ❛❧♠♦st ✉❧tr❛ ✭s❡❡ ❉❡❢✳ ✷✳✸✭✸✮✮✱ t❤❡♥ Fσ(F) ∗ PT(U ˜ ) ♣r❡s❡r✈❡s W✳

✶✷ ✴ ✸✵

❇♦rr♦✇✐♥❣ ❢r♦♠ ▲✐♥❡❛r ❈r❡❛t✉r❡s

❉❡✜♥✐t✐♦♥

▲❡t f : ω → ω ❜❡ ❛ str✐❝t❧② ✐♥❝r❡❛s✐♥❣ ❢✉♥❝t✐♦♥ ✇✐t❤ f(0) = 0✳ ❆ ❝♦♥❞✐t✐♦♥ p ∈ PT(F) ✐s s❛✐❞ t♦ ❤❛✈❡ f✲❜❧♦❝❦str✉❝t✉r❡ ✐❢ (∀t ∈ F✲ sp(p))(∀r ∈ sucsplp(t)) (∃k ∈ ω)(range(r) \ range(t) ⊆ [f(k), f(k + 1))). ✭✵✳✶✮

✶✸ ✴ ✸✵

❇❛❝❦ t♦ ▼❛t❤✐❛s ❋♦r❝✐♥❣ ♦r t❤❡ ❈❛♥❥❛r ●❛♠❡

▲❡♠♠❛

▲❡t F ❜❡ ❈❛♥❥❛r ❛♥❞ p ∈ PT(F)✳ ❚❤❡r❡ ✐s ❛♥ f ∈ ω↑ω ✇✐t❤ f(0) = 0 ❛♥❞ t❤❡r❡ ✐s ❛ q ≤0 p ✇✐t❤ f✲❜❧♦❝❦str✉❝t✉r❡✳ Pr❡❞❡❝❡ss♦r ❧❡♠♠❛ ❜② ●✉③♠á♥ ❛♥❞ ❑❛❧❛❥❞③✐❡✈s❦✐

▲❡♠♠❛

▲❡t F ❜❡ ❈❛♥❥❛r ❛♥❞ Xn : n < ω✱ Xn ∈ (F<ω)+✳ ❚❤❡r❡ ✐s ❛♥ f ∈ ω↑ω ✇✐t❤ f(0) = 0 s✉❝❤ t❤❛t {Xn ∩ P(f(n)) : n < ω} ∈ (F<ω)+✳

✶✹ ✴ ✸✵

Pr❡♠✐s❡

❚❤❡♦r❡♠

▲❡t α ≤ ω1 ❛♥❞ ❧❡t P = Pγ, Qβ : γ ≤ α, β < α ❜❡ ❞❡✜♥❡❞ ❜② ✐♥❞✉❝t✐♦♥ ♦♥ α ≤ ω1 ❛s ❢♦❧❧♦✇s✿ ✭✶✮ P0 = {0}✱ ❛♥❞ ✭✷✮ ❋♦r β < α ✇❡ ❤❛✈❡✿ ■❢ ✲ ❢♦r γ < β✱ rγ ✐s t❤❡ PT(Uγ)✲❣❡♥❡r✐❝ r❡❛❧ ♦✈❡r VPγ∗Fσ(Fγ)✱ ✲ Pβ Fβ = filter({range(rγ) : γ < β})✱ ❛♥❞ ✲ Uβ t❤❡ Fσ(Fβ)✲❣❡♥❡r✐❝ ✜❧t❡r ♦✈❡r VPβ✱ t❤❡♥ Pβ Qβ = Fσ(Fβ) ∗ PT(Uβ)✳ ✭✸✮ Pα Fα = filter({range(rγ) : γ < α})✳

✶✺ ✴ ✸✵

❈♦♥❝❧✉s✐♦♥

❚❤❡♥

t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞s ✭❆✮ Pα ✐s ♣r♦♣❡r ❛♥❞ ❢♦r❝✐♥❣ ✇✐t❤ Pα ♣r❡s❡r✈❡s ❛♥② P✲♣♦✐♥t ✐♥ {VPβ : β < α}✳ ❋♦r α < ω2✱ ✇❡ ❤❛✈❡ |Pα| ≤ ℵ1✳ ✭❇✮ ❋♦r ❛♥② β < α ✐❢ cf(β) ≤ ω t❤❡♥ Pβ Fβ ✐s ♥♦t ♥❡❛r❧② ✉❧tr❛✳ ❛♥❞ Pβ ∗ Fσ(Fβ) Uβ ✐s ❛ ❈❛♥❥❛r ✉❧tr❛✜❧t❡r ❛♥❞ ♥♦t ♥❡❛r❧② ❝♦❤❡r❡♥t W ❢♦r ❛♥② W ∈ VPβ.

✶✻ ✴ ✸✵

▼♦r❡ ❈♦♥❝❧✉s✐♦♥s

✭❈✮ ▲❡t α = ω1✳ Pα Fα = Uα ✐s ❛ ❈❛♥❥❛r ✉❧tr❛✜❧t❡r ❛♥❞ ♥♦t ♥❡❛r❧② ❝♦❤❡r❡♥t ❛♥② P✲♣♦✐♥t W ∈

• {VPγ : γ < α}.

✭❉✮ ∀γ < β < α✱ Pβ+1 rβ ⊆∗ rγ✳

✶✼ ✴ ✸✵

❙♦♠❡ ❙t❡♣s ♦❢ t❤❡ Pr♦♦❢

❲❡ ♣r♦✈❡ t❤❡ ❧❡♠♠❛ ❜② ✐♥❞✉❝t✐♦♥ ♦♥ α✳ ❋✐rst s✉♣♣♦s❡ t❤❛t α ≤ ω2 ✐s ❛ ❧✐♠✐t ♦r❞✐♥❛❧ ❛♥❞ t❤❡ ❧❡♠♠❛ ✐s ♣r♦✈❡❞ ❢♦r γ < α✳ ❋♦r ❝♦♥❝❧✉s✐♦♥ ✭❆✮ ✇❡ ❝✐t❡✿

❚❤❡♦r❡♠

✭❇❧❛ss✱ ❙❤❡❧❛❤✮ ■❢ W ✐s ❛ P✲♣♦✐♥t✱ α ✐s ❛ ❧✐♠✐t ♦r❞✐♥❛❧ ❛♥❞ Pα = Pγ : γ < α ✐s t❤❡ ❝♦✉♥t❛❜❧❡ s✉♣♣♦rt ❧✐♠✐t ❛♥❞ ❢♦r γ < α✱ t❤❡ ❢♦r❝✐♥❣ Pγ ✐s ♣r♦♣❡r ❛♥❞ ♣r❡s❡r✈❡s W✱ t❤❡♥ Pα ✐s ♣r♦♣❡r ❛♥❞ ♣r❡s❡r✈❡s W✳ ❆❧s♦ t❤❡ st❛t❡♠❡♥t ♦♥ t❤❡ s✐③❡ ♦❢ t❤❡ ❢♦r❝✐♥❣ ♦r❞❡r ✐s ✐♥ t❤❡ ♣r♦♣❡r ❢♦r❝✐♥❣ ❜♦♦❦✳

✶✽ ✴ ✸✵

❆❜♦✉t t❤❡ ❘❛♥❣❡s ♦❢ t❤❡ ❚r❡❡ ◆♦❞❡s

▲❡♠♠❛

▲❡t F ❜❡ ❈❛♥❥❛r ❛♥❞ p ∈ PT(F)✳ ❚❤❡r❡ ✐s ❛♥ f ∈ ω↑ω ✇✐t❤ f(0) = 0 ❛♥❞ t❤❡r❡ ✐s ❛ q ≤0 p ✇✐t❤ f✲❜❧♦❝❦str✉❝t✉r❡✳

✶✾ ✴ ✸✵

F ✐s ❈❛♥❥❛r ✐✛ ■ ❞♦❡s ♥♦t ❤❛✈❡ ❛ ❲✐♥♥✐♥❣ ❙tr❛t❡❣② ✐♥ t❤❡ ❈❛♥❥❛r ●❛♠❡ ❢♦r F

❉❡✜♥✐t✐♦♥

❋♦r ❛ ✜❧t❡r F ✇❡ ❝♦♥s✐❞❡r t❤❡ ❈❛♥❥❛r ●❛♠❡ G (F)✳ P❧❛②❡r ■ ❛♥❞ ♣❧❛②❡r ■■ ❛❧t❡r♥❛t❡❧② ♣❧❛② s❡ts X0, Y0, X1, Y1, . . . ✳ ❚❤❡ r✉❧❡s ❛r❡ Xi ∈ (F<ω)+✱ Yi ∈ [Xi]<ω \ {∅} ❢♦r ❡✈❡r② i ∈ ω✳ ■ X0 X1 X2 . . . ■■ Y0 Y1 Y2 ❆❢t❡r ω r♦✉♥❞s✱ ♣❧❛②❡r ■■ ✇✐♥s ✐❢

n∈ω Yn ∈ (F<ω)+✳

✷✵ ✴ ✸✵

❆ ▲❡♠♠❛ ❛❜♦✉t ❉✐r❡❝t✐♦♥s

▲❡♠♠❛

▲❡t W ❜❡ ❛ P✲♣♦✐♥t✳ ■❢ U ✐s ❛ ❈❛♥❥❛r ✉❧tr❛✜❧t❡r t❤❛t ✐s ♥♦t ♥❡❛r❧② ❝♦❤❡r❡♥t t♦ W✱ t❤❡♥ ❢♦r❝✐♥❣ ✇✐t❤ PT(U) ♣r❡s❡r✈❡s W✳

✷✶ ✴ ✸✵

❋r❡s❤ ❉✐r❡❝t✐♦♥s ❛❢t❡r ❍❛❧❢✲❙t❡♣s

▲❡♠♠❛

■❢ ❛ ✜❧t❡r F ✐s ♥♦t ❛❧♠♦st ✉❧tr❛ ✭s❡❡ ❉❡❢✳ ✷✳✸✭✸✮✮ ❛♥❞ Fσ(F) U ˜ ✐s ❈❛♥❥❛r✱ t❤❡♥ Fσ(F) ∗ PT(U ˜ ) ♣r❡s❡r✈❡s W✳

✷✷ ✴ ✸✵

❆❜♦✉t Fσ✲❋♦r❝✐♥❣

❉❡✜♥✐t✐♦♥

▲❡t X ⊆ Fin✳ ❲❡ ❧❡t C(X) = {A ⊆ ω : ∀s ∈ X(s ∩ A = ∅)}.

▲❡♠♠❛ ❜② ●✉③♠❛♥ ❛♥❞ ❑❛❧❛❥❞③✐❡s✈❦②

▲❡t G ❜❡ ❛ ✜❧t❡r✳ F Fσ(G ) X ∈ (U(G ) ˜

<ω)+ ✐✛

C(X) ⊆ filter(F ∪ G )

✏⇒✑ ▲❡t H ∈ filter(F ∪ G )✳ ❚❤❡♥ Hc ✐s filter(F ∪ G )✲♣♦s✐t✐✈❡ ❛♥❞ F ≥ F ∪ {Hc} ✐s ❛ ❝♦♥❞✐t✐♦♥ ✐♥ Fσ(G )✳ ❚❤❡♥ ∃s ∈ X, s ⊆ Hc✳ ❚❤✉s H ∈ C(X)✳ ✏⇐✑ ❙✉♣♣♦s❡ C(X) ⊆ filter(F ∪ G )✳ ❚❤❡♥ ∀A ∈ C(X)✱ Ac ∈ U(G )✳ ❍❡♥❝❡ ❢♦r ❛♥② D ∈ U(G )✱ Dc ∈ C(X) ❛♥❞ ❤❡♥❝❡ ∃s ∈ X(s ⊆ D)✳

✷✸ ✴ ✸✵

❆❜♦✉t t❤❡ ❈❛♥❥❛r✐t② ♦❢ Uω1

▲❡♠♠❛

▲❡t α = ω1✳ Pα ❢♦r❝❡s t❤❛t Fα ✐s ❛ ❈❛♥❥❛r ✉❧tr❛✜❧t❡r t❤❛t ✐s ♥♦t ♥❡❛r❧② ❝♦❤❡r❡♥t t♦ ❛♥② P✲♣♦✐♥t ✐♥

γ<α V Pγ✳

❆♥ ❡❛s② ❞❡♥s✐t② ❛r❣✉♠❡♥t s❤♦✇s t❤❛t ❢♦r cf(α) = ω1✱ Pα Fα = Uα ✐s ✉❧tr❛ s✐♥❝❡ ❛♥② ♥❛♠❡ ❢♦r ❛ s✉❜s❡t ♦❢ ω ❛♣♣❡❛rs ✐♥ s♦♠❡ V Pγ✱ γ < α✳

✷✹ ✴ ✸✵

❆♥② ♥❛♠❡ h ❢♦r ❛ ✜♥✐t❡✲t♦✲♦♥❡ ❢✉♥❝t✐♦♥ ❛♣♣❡❛rs ❛t s♦♠❡ Pβ✱ β < α✳ ▲❡t W ❜❡ ❛❧s♦ ✐♥ VPβ✳ ❚❤❡♥ Pβ+1 ∃X ∈ Uβ∃Y ∈ Wh[X] ∩ h[Y ] = ∅ ❛♥❞ ❜② ▲❡♠♠❛ ❱■■✱ ✼✳✶✸❜ ❑✉♥❡♥ t❤✐s ✐s ♣r❡s❡r✈❡❞ ✉♣✇❛r❞s✳ ❍❡♥❝❡ Pα ❢♦r❝❡s t❤❛t Uα ✐s ♥♦t ♥❡❛r❧② ❝♦❤❡r❡♥t t♦ ❛♥② P✲♣♦✐♥t ✐♥ V Pγ✱ γ < α✳

✷✺ ✴ ✸✵

❙❦❡t❝❤ ♦❢ ❛ Pr♦♦❢

❚❤❡ ♦♥❧② ♥♦t s♦ ❡❛s② st❛t❡♠❡♥t ✐s✿ Pα ❢♦r❝❡s t❤❛t Uα ✐s ❈❛♥❥❛r✳ ❇② ✐♥❞✉❝t✐♦♥ ❤②♣♦t❤❡s✐s ✇❡ ❦♥♦✇ t❤❛t ❢♦r β < α t❤❡ ♥❛♠❡ Uβ ✐s ❢♦r❝❡❞ ❜② Pβ ∗ Fσ(Fβ) t♦ ❜❡ ❛ ❈❛♥❥❛r ✉❧tr❛✜❧t❡r✳ ❊✈❡r② ♥❛♠❡ ❢♦r ❛♥ ω✲s❡q✉❡♥❝❡ ♦❢ s❡ts ♦❢ ❜❧♦❝❦s ❛♣♣❡❛rs ❢♦r t❤❡ ✜rst t✐♠❡ ❛t ❛♥ ✐t❡r❛t✐♦♥ st❛❣❡ ♦❢ ❝♦✉♥t❛❜❧❡ ❝♦✜♥❛❧✐t②✳ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ p ∈ Pα ❛♥❞ ❛ Pα✲♥❛♠❡ Xn : n < ω s✉❝❤ t❤❛t p (∀n)(Xn ∈ ((Uα)<ω)+). ❚❤✉s ❢♦r s♦♠❡ β0 < α✱ ✇❡ ❤❛✈❡ t❤❛t Xn : n < ω ✐s ❡q✉✐✈❛❧❡♥t t♦ ❛♥ Pβ0✲♥❛♠❡✳ ❲✳❧✳♦✳❣✳✱ ❧❡t Xn : n < ω ❜❡ ❛ Pβ0✲♥❛♠❡✳ (p ↾ β0, p1(β0)) Pβ0∗Fσ(Fβ0 ) (∀n)(Xn ∈ ((Uβ0)<ω)+). ❙✐♥❝❡ Pβ0 ∗ Fσ(Fβ0) ❢♦r❝❡s t❤❛t Uβ0 ✐s ❈❛♥❥❛r✱ t❤❡r❡ ✐s ❛ Pβ0✲♥❛♠❡ ❢♦r ❛ s❡q✉❡♥❝❡ Yn : n < ω s✉❝❤ t❤❛t

✷✻ ✴ ✸✵

❈s✐ ♦❢ Pr♦♣❡r ❋♦r❝✐♥❣s ✇✐t❤ ❘❡❛❧s ❛s ❈♦♥❞✐t✐♦♥s

(p ↾ β0, p1(β0)) Pβ0∗Fσ(Fβ0) (∀n)(Yn ∈ [Xn]<ω∧

• {Yn : n < ω} ∈ ((Uβ0)<ω)+.

❇✉t t❤❡♥ ❜② t❤❡ ❝❤❛r❛❝t❡r✐s❛t✐♦♥ ♦❢ Fσ✲❢♦r❝✐♥❣ (p ↾ β0, p1(β0)) Pα (∀n)(Yn ∈ [Xn]<ω∧

• {Yn : n < ω} ∈ ((Uα)<ω)+.

❚❤✐s ✐s s❡❡♥ ❛s ❢♦❧❧♦✇s✿ ❲❡ ❛ss✉♠❡ t❤❛t q ≥ p✱ q ∈ Pα ❛♥❞ q Y ∈ Uα✳ ❆❣❛✐♥ t❤❡r❡ ✐s β1 < α s✉❝❤ t❤❛t q ∈ Pβ1 ❛♥❞ Y ✐s ❛ Pβ1✲♥❛♠❡ ❛♥❞ (q ↾ β1, q1(β1)) Pβ1∗Fσ(Fβ1)) Y ∈ Uβ1. ❲❡ ❛ss✉♠❡ t❤❛t β1 ≥ β0✳

✷✼ ✴ ✸✵

▼♦st♦✇s❦✐✬s ❆❜s♦❧✉t❡♥❡ss ❚❤❡♦r❡♠

❲❡ ❧❡t Z = {Yn : n < ω}✳ ❇② t❤❡ ❝❤❛r❛❝t❡r✐s❛t✐♦♥ ♦❢ Fσ✲❢♦r❝✐♥❣ ✇❡ ❤❛✈❡ ✐♥ VPβ0↾(p↾β0)✱ C(Z) ⊆ filter(p1(β0) ∪ {rγ : γ ∈ β0}). ❚❤✐s ✐s ❛ Π1

1✲r❡❧❛t✐♦♥ ♦❢ Z ❛♥❞ p✱ ❛♥❞ ❤❡♥❝❡ ❤♦❧❞s✱ ❛❣❛✐♥ ❜② ▲❡♠♠❛

❱■■✱ ✼✳✶✸❜ ❑✉♥❡♥✱ ❛❧s♦ ✐♥ VPβ1↾(q↾β1)✳ ❙♦ ✐♥ VPβ1↾(p↾β0)✱ C(Z) ⊆ filter(p1(β0) ∪ {rγ : γ ∈ β0}). ■♥ t❤❡ s❛♠❡ ♠♦❞❡❧ ✇❡ ❝❛♥ ✐♥❝r❡❛s❡ t❤❡ ✜❧t❡r ❛s ❢♦❧❧♦✇s✿ C(Z) ⊆ filter(q1(β1) ∪ {rγ : γ ∈ β1}).

✷✽ ✴ ✸✵

❊♥❞ ♦❢ Pr♦♦❢

❍❡♥❝❡ ❜② t❤❡ ❝❤❛r❛❝t❡r✐s❛t✐♦♥ ♦❢ Fσ ❢♦r❝✐♥❣✱ q ↾ β1 Pβ1∗Fσ(Fβ1) Z ∈ (U<ω

β1 )+✱ ❛♥❞ s✐♥❝❡ q ❛♥❞ Y ❛♥❞ β1 ✇❡r❡

❛r❜✐tr❛r②✱ ❛♥❞ ✇❡ ❛r❡ ❞♦♥❡✳

✷✾ ✴ ✸✵

❚❤❛♥❦ ②♦✉✦

✸✵ ✴ ✸✵