ss r s t
play

ss r st - PowerPoint PPT Presentation

ss r st rr r ts ts trst rs


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

  2. ❆ ♣❛rt ♦❢ ❛ ❧❛r❣❡r ♣r♦❥❡❝t ❋✶✹✷✵ ❜② ❇❧❛ss✱ ▼✐❧❞❡♥❜❡r❣❡r✱ ❙❤❡❧❛❤ ❆ ❙✐♠♣❧❡ P ℵ 1 ✲P♦✐♥t ❛♥❞ ❛ ❙✐♠♣❧❡ P ℵ 2 ✲P♦✐♥t ✷ ✴ ✸✵

  3. ❘❡♠❛r❦ Pr❡s❡r✈❛t✐♦♥ ♦❢ ✲♣♦✐♥ts✳ ❙♦♠❡ ❞❡✈❡❧♦♣♠❡♥t ✳ ✳ ✳ ❚❤❡♦r❡♠✱ ❙❤❡❧❛❤ ✶✾✾✹ ❆♥② ❢♦r❝✐♥❣ ❛❞❞✐♥❣ ❛ r❡❛❧ ❞❡str♦②s ❛♥ ✉❧tr❛✜❧t❡r ♦✈❡r ✳ 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 ) ✑✳ ✸ ✴ ✸✵

  4. ❚❤❡♦r❡♠✱ ❙❤❡❧❛❤ ✶✾✾✹ ❆♥② ❢♦r❝✐♥❣ ❛❞❞✐♥❣ ❛ r❡❛❧ ❞❡str♦②s ❛♥ ✉❧tr❛✜❧t❡r ♦✈❡r ✳ 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 ✳ ✳ ✳ ✸ ✴ ✸✵

  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 ✳ ✸ ✴ ✸✵

  6. ❈♦✉♥t❛❜❧❡ s✉♣♣♦rt ✐t❡r❛t✐♦♥ ❚❤❡♦r❡♠✱ ❇❧❛ss✱ ❙❤❡❧❛❤ ▲❡t E ❜❡ ❛ P ✲♣♦✐♥t✳ ▲❡t � P α , Q β : β < γ, α ≤ γ � ❜❡ ❛ ❝♦✉♥t❛❜❧❡ s✉♣♣♦rt ✐t❡r❛t✐♦♥ s✉❝❤ t❤❛t ❡❛❝❤ P α ✐s ♣r♦♣❡r✳ ■❢ ❡❛❝❤ P α ✱ α < γ ✱ ♣r❡s❡r✈❡s E ✱ t❤❡♥ ❛❧s♦ P γ ♣r❡s❡r✈❡s E ✳ ✹ ✴ ✸✵

  7. ❚❤❡♥ 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❡❞✳ ❆ r❡q✉❡st t♦ ❞❡str♦② ❛ P ✲♣♦✐♥t ❙✉♣♣♦s❡✿ ✭✶✮ � P α , Q β : β < ω 2 , α ≤ ω 2 � ✐s ❛ ❝♦✉♥t❛❜❧❡ s✉♣♣♦rt ✐t❡r❛t✐♦♥ ♦❢ ♣r♦♣❡r ✐t❡r❛♥❞s✱ ❛♥❞ ✭✷✮ ✐♥ V P ω 2 t❤❡r❡ ✐s ❛ s✐♠♣❧❡ P ℵ 2 ✲♣♦✐♥t U ✳ ✺ ✴ ✸✵

  8. ❲❡ 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❡❞✳ ❆ r❡q✉❡st t♦ ❞❡str♦② ❛ P ✲♣♦✐♥t ❙✉♣♣♦s❡✿ ✭✶✮ � P α , Q β : β < ω 2 , α ≤ ω 2 � ✐s ❛ ❝♦✉♥t❛❜❧❡ s✉♣♣♦rt ✐t❡r❛t✐♦♥ ♦❢ ♣r♦♣❡r ✐t❡r❛♥❞s✱ ❛♥❞ ✭✷✮ ✐♥ V P ω 2 t❤❡r❡ ✐s ❛ s✐♠♣❧❡ P ℵ 2 ✲♣♦✐♥t U ✳ ❚❤❡♥ t❤❡r❡ ✐s ❛♥ ω 1 ✲❝❧✉❜ ♦❢ st❛❣❡s ❛t ✇❤✐❝❤ U ∩ V P α ✐s ❛ P ✲♣♦✐♥t✳ ✺ ✴ ✸✵

  9. ❙♦ ❛ ❢♦r❝✐♥❣ ❞❡str♦②✐♥❣ s♦♠❡ ✲♣♦✐♥ts ❛♥❞ ❦❡❡♣✐♥❣ ♦t❤❡rs ✐s r❡q✉❡st❡❞✳ ❆ r❡q✉❡st t♦ ❞❡str♦② ❛ P ✲♣♦✐♥t ❙✉♣♣♦s❡✿ ✭✶✮ � P α , Q β : β < ω 2 , α ≤ ω 2 � ✐s ❛ ❝♦✉♥t❛❜❧❡ s✉♣♣♦rt ✐t❡r❛t✐♦♥ ♦❢ ♣r♦♣❡r ✐t❡r❛♥❞s✱ ❛♥❞ ✭✷✮ ✐♥ V P ω 2 t❤❡r❡ ✐s ❛ s✐♠♣❧❡ P ℵ 2 ✲♣♦✐♥t U ✳ ❚❤❡♥ t❤❡r❡ ✐s ❛♥ ω 1 ✲❝❧✉❜ ♦❢ st❛❣❡s ❛t ✇❤✐❝❤ U ∩ V P α ✐s ❛ P ✲♣♦✐♥t✳ ❲❡ t❛❦❡ s✉❝❤ ❛ st❛❣❡ α ✱ ❛♥❞ ❝♦♥s✐❞❡r t❤❡ ❧❡❛st β > α s✉❝❤ t❤❛t t❤❡r❡ ✐s X ∈ U \ V P α ✳ U ∩ V P <β ✐s ❞❡str♦②❡❞ ❜② P β ✭❛♥❞ ❝♦♠♣❧❡♠❡♥t❡❞✮ ❧❛t❡r ✐♥ t❤❡ ✐t❡r❛t✐♦♥✳ ✺ ✴ ✸✵

  10. ❆ r❡q✉❡st t♦ ❞❡str♦② ❛ P ✲♣♦✐♥t ❙✉♣♣♦s❡✿ ✭✶✮ � P α , Q β : β < ω 2 , α ≤ ω 2 � ✐s ❛ ❝♦✉♥t❛❜❧❡ s✉♣♣♦rt ✐t❡r❛t✐♦♥ ♦❢ ♣r♦♣❡r ✐t❡r❛♥❞s✱ ❛♥❞ ✭✷✮ ✐♥ V P ω 2 t❤❡r❡ ✐s ❛ s✐♠♣❧❡ P ℵ 2 ✲♣♦✐♥t U ✳ ❚❤❡♥ t❤❡r❡ ✐s ❛♥ ω 1 ✲❝❧✉❜ ♦❢ st❛❣❡s ❛t ✇❤✐❝❤ U ∩ V P α ✐s ❛ P ✲♣♦✐♥t✳ ❲❡ t❛❦❡ s✉❝❤ ❛ st❛❣❡ α ✱ ❛♥❞ ❝♦♥s✐❞❡r t❤❡ ❧❡❛st β > α s✉❝❤ t❤❛t t❤❡r❡ ✐s X ∈ U \ V P α ✳ U ∩ V P <β ✐s ❞❡str♦②❡❞ ❜② P β ✭❛♥❞ ❝♦♠♣❧❡♠❡♥t❡❞✮ ❧❛t❡r ✐♥ t❤❡ ✐t❡r❛t✐♦♥✳ ❙♦ ❛ ❢♦r❝✐♥❣ ❞❡str♦②✐♥❣ s♦♠❡ P ✲♣♦✐♥ts ❛♥❞ ❦❡❡♣✐♥❣ ♦t❤❡rs ✐s r❡q✉❡st❡❞✳ ✺ ✴ ✸✵

  11. ❚❤❡♦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 ✇❡ ❤❛✈❡ ❙✉❜❢♦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} ✳ ✻ ✴ ✸✵

  12. ❍♦✇❡✈❡r✱ ▼❛t❡t ❢♦r❝✐♥❣ ♥♦t ❛❞❞ ❛♥ ✉♥s♣❧✐t r❡❛❧✳ ❉❡✜♥✐t✐♦♥ ❆ r❡❛❧ ✐s ❝❛❧❧❡❞ ❛♥ ✉♥s♣❧✐t r❡❛❧ ✐❢ ❡✈❡r② ✇❡ ❤❛✈❡ ❙✉❜❢♦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 ✳ ✻ ✴ ✸✵

Download Presentation
Download Policy: The content available on the website is offered to you 'AS IS' for your personal information and use only. It cannot be commercialized, licensed, or distributed on other websites without prior consent from the author. To download a presentation, simply click this link. If you encounter any difficulties during the download process, it's possible that the publisher has removed the file from their server.

Recommend


More recommend