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❆ ❜❛s✐❝ ✐♥tr♦❞✉❝t✐♦♥ t♦ r❡✈❡rs❡ ♠❛t❤❡♠❛t✐❝s ❍✉❣♦ ❍❡r❜❡❧✐♥ ✻ ▼❛r❝❤ ✷✵✶✷ ❏♦✉r♥é❡s ❈❛❧❝✉❧❛❜✐❧✐té ✷✵✶✷ P❛r✐s ✶

❖✉t❧✐♥❡ ✲ ❘❡✈❡rs❡ ♠❛t❤❡♠❛t✐❝s✿ ❜❛s✐❝s✱ t❤❡ ❜✐❣ ✜✈❡ ✭ RCA 0 ⊂ WKL 0 ⊂ ACA 0 ⊂ ATR 0 ⊂ Π 1 1 ✲ CA 0 ✮ ✲ ❈♦♠♣r❡❤❡♥s✐♦♥ ✈s ✐♥t❡r♣♦❧❛t✐♦♥✿ t♦✇❛r❞s ❛ ✉♥✐❢♦r♠ ♣r❡s❡♥t❛t✐♦♥ ♦❢ RCA 0 ✱ WKL 0 ✱ ACA 0 ✱ ATR 0 ✱ Π 1 1 ✲ CA 0 ✲ ❚♦✇❛r❞s ❛ ❝♦♠♣✉t❛t✐♦♥❛❧ ❝♦♥t❡♥t t♦ RCA 0 ✱ WKL 0 ✱ ACA 0 ✱ ATR 0 ✱ Π 1 1 ✲ CA 0 ✷

❚❤❡ ❜❛s✐s ♦❢ r❡✈❡rs❡ ♠❛t❤❡♠❛t✐❝s ❙t❛rt❡❞ ❜② ❍❛r✈❡② ❋r✐❡❞♠❛♥ ✐♥ t❤❡ ✼✵✬s✿ ❉❡t❡r♠✐♥❡ t❤❡ ❧♦❣✐❝❛❧ str❡♥❣t❤ ✉♥❞❡r❧②✐♥❣ st❛♥❞❛r❞ ♠❛t❤❡♠❛t✐❝❛❧ t❤❡♦r❡♠s ✱ ❡✳❣✳✿ ✲ RCA 0 s✉✣❝❡s t♦ ♣r♦✈❡✿ ❇❛✐r❡ ❝❛t❡❣♦r② t❤❡♦r❡♠✱ ■♥t❡r♠❡❞✐❛t❡ ❱❛❧✉❡ ❚❤❡♦r❡♠✱ ❙♦✉♥❞♥❡ss ♦❢ ♣r❡❞✐❝❛t❡ ❧♦❣✐❝✱ ✳✳✳ ✲ WKL 0 ❂ ❍❡✐♥❡✴❇♦r❡❧ ❝♦✈❡r✐♥❣ ❧❡♠♠❛ ❂ ●ö❞❡❧✬s ❈♦♠♣❧❡t❡♥❡ss ❚❤❡♦r❡♠ ❂ ❇r♦✉✇❡r✬s ❋✐①❡❞ P♦✐♥t ❚❤❡♦r❡♠ ❂ ❙❡♣❛r❛❜❧❡ ❍❛❤♥✴❇❛♥❛❝❤ ❚❤❡♦r❡♠ ❂ ❈♦✉♥t❛❜❧❡ ❝♦♠♠✉t❛t✐✈❡ r✐♥❣s ❤❛✈❡ ❛ ♣r✐♠❡ ✐❞❡❛❧ ❂ ✳✳✳ ✲ ACA 0 ❂ ❇♦❧③❛♥♦✴❲❡✐❡rstr❛ÿ ❚❤❡♦r❡♠ ❂ ❈♦✉♥t❛❜❧❡ ❝♦♠♠✉t❛t✐✈❡ r✐♥❣s ❤❛✈❡ ❛ ♠❛①✐♠❛❧ ✐❞❡❛❧ ❂ ❘❛♠s❡②✬s ❚❤❡♦r❡♠ ❢♦r ❝♦❧♦✉r✐♥❣✬s ♦❢ [ N ] n ❂ ✳✳✳ ✲ ATR 0 ❂ ❝♦✉♥t❛❜❧❡ ✇❡❧❧✲♦r❞❡r✐♥❣s ❛r❡ ❝♦♠♣❛r❛❜❧❡ ❂ P❡r❢❡❝t ❙❡t ❚❤❡♦r❡♠ ❂ ▲✉s✐♥✬s s❡♣❛r❛t✐♦♥ t❤❡♦r❡♠ ❂ ✳✳✳ ✲ Π 1 1 ✲ CA 0 ❂ tr❡❡s ❤❛✈❡ ❛ ❧❛r❣❡st ♣❡r❢❡❝t s✉❜tr❡❡ ❂ ❈❛♥t♦r✴❇❡♥❞✐①s♦♥ ❚❤❡♦r❡♠ ❂ ❙✐❧✈❡r✬s ❚❤❡♦r❡♠ ❂ ✳✳✳ ❆ r❡❢❡r❡♥❝❡ ❜♦♦❦ ❜② ❙t❡♣❤❡♥ ❙✐♠♣s♦♥✿ ❙✉❜s②st❡♠s ♦❢ s❡❝♦♥❞✲♦r❞❡r ❛r✐t❤♠❡t✐❝ ❬✶✾✾✾✱ ✷✵✵✻❪ ✸

❘❡✈❡rs❡ ♠❛t❤❡♠❛t✐❝s✿ s✉❜s②st❡♠s ♦❢ s❡❝♦♥❞✲♦r❞❡r ❛r✐t❤♠❡t✐❝ ✲ ❚❤❡ ❧❛♥❣✉❛❣❡ ♦❢ P❡❛♥♦ ❆r✐t❤♠❡t✐❝ ✭P❆✮ ✇✐t❤ ✐ts q✉❛♥t✐✜❝❛t✐♦♥ ♦✈❡r ♥❛t✉r❛❧ ♥✉♠❜❡rs ✭ n ✱ m ✱ ✳✳✳✮ ✲ ❊♥♦✉❣❤ t♦ t❛❧❦ ❛❜♦✉t ❧✐sts✱ ✐♥t❡❣❡rs✱ r❛t✐♦♥❛❧ ♥✉♠❜❡rs✱ ✜♥✐t❡ s❡q✉❡♥❝❡s✱ ❢♦r♠✉❧❛❡✱ ✜♥✐t❡ tr❡❡s✱ ✳✳✳ ✲ ◗✉❛♥t✐✜❝❛t✐♦♥ ♦✈❡r s❡ts ♦❢ ♥❛t✉r❛❧ ♥✉♠❜❡rs ✭ X ✱ Y ✱ ✳✳✳✮ ❛♥❞ ❛t♦♠✐❝ ❢♦r♠✉❧❛❡ n ∈ X ✲ ❊♥♦✉❣❤ t♦ t❛❧❦ ❛❜♦✉t ❝♦✉♥t❛❜❧❡ s❡q✉❡♥❝❡s✱ r❡❛❧ ♥✉♠❜❡rs✱ ❣r♦✉♣s✱ ✜❡❧❞s✱ ✈❡❝t♦r s♣❛❝❡s✱ ❢✉♥❝t✐♦♥s ♦✈❡r ♥❛t✉r❛❧ ♥✉♠❜❡rs✱ ✳✳✳ ✲ ■♥❞✉❝t✐♦♥ ❢♦r t❤♦s❡ ♣r♦♣❡rt✐❡s ❞❡✜♥❛❜❧❡ ❛s ❛ s❡t ✲ ❚✉♥✐♥❣ t❤❡ str❡♥❣t❤ ✈✐❛ ❝♦♠♣r❡❤❡♥s✐♦♥ ❛①✐♦♠s✱ ♣♦ss✐❜❧② ✐♠♣❛❝t✐♥❣ t❤❡ str❡♥❣t❤ ♦❢ ✐♥❞✉❝t✐♦♥ ✲ RCA 0 ❂ ❘❡❝✉rs✐✈❡ ❈♦♠♣r❡❤❡♥s✐♦♥ ❆①✐♦♠ ✭❈❆ ♦♥ ∆ 0 1 ✲❢♦r♠✉❧❛❡✮ ✲ WKL 0 ❂ RCA 0 ✰ ❲❡❛❦ ❑ö♥✐❣✬s ▲❡♠♠❛ ✲ ACA 0 ❂ RCA 0 ✰ ❆r✐t❤♠❡t✐❝ ❈♦♠♣r❡❤❡♥s✐♦♥ ❆①✐♦♠ ✭❈❆ ♦♥ Σ 0 1 ✲❢♦r♠✉❧❛❡✮ ✲ ATR 0 ❂ ACA 0 ✰ ❆r✐t❤♠❡t✐❝ ❚r❛♥s✜♥✐t❡ ❘❡❝✉rs✐♦♥ ✲ Π 1 1 ✲ CA 0 ❂ ACA 0 ✰ ❈♦♠♣r❡❤❡♥s✐♦♥ ❆①✐♦♠ ♦♥ Π 1 1 ✲❢♦r♠✉❧❛❡ ✹

P♦s✐t✐♦♥ ✐♥ ●ö❞❡❧ ❤✐❡r❛r❝❤②  ZF , ZFC , ...  str♦♥❣ ❩❡r♠❡❧♦ s❡t t❤❡♦r② ❙②st❡♠ ❋ ω. 2 + ●✐r❛r❞✬s ❙②st❡♠ ❋ ω  HOL ✭❈❤✉r❝❤✬s ❙✐♠♣❧❡ ❚②♣❡ ❚❤❡♦r②✮  ●✐r❛r❞✲❘❡②♥♦❧❞s✬ ❙②st❡♠ ❋ Z 2 ✭❢✉❧❧ ✷♥❞ ♦r❞❡r ❛r✐t❤♠❡t✐❝✮   Π 1  1 ✲ CA 0 Ψ 0 (Ω ω )     Ψ 0 ( ǫ Ω+1 ) ✭❇❛❝❤♠❛♥♥✲❍♦✇❛r❞✮ CZF ✭❆❝③❡❧✬s ❈♦♥str✉❝t✐✈❡ ❙❡t ❚❤❡♦r②✮    ♠❡❞✐✉♠ Γ 0 ✭❋❡❢❡r♠❛♥✲❙❝❤ütt❡✮ ATR 0 ●ö❞❡❧✬s ❙②st❡♠ ❚ ǫ 0 ACA 0     HA ω ✭✐♥t✉✐t✳ ❛r✐t❤♠✳ ✐♥ ✜♥✐t❡ t②♣❡s✮  ●ö❞❡❧✬s ❙②st❡♠ ❚ ǫ 0     ●ö❞❡❧✬s ❙②st❡♠ ❚ ǫ 0 PA   ω ω WKL 0 ♣r✐♠✳ r❡❝✳ ❢✉♥❝t✳   ω ω  RCA 0 ♣r✐♠✳ r❡❝✳ ❢✉♥❝t✳    ω ω ✇❡❛❦ ♣r✐♠✳ r❡❝✳ ❢✉♥❝t✳ IΣ 1 ω ω ♣r✐♠✳ r❡❝✳ ❢✉♥❝t✳ PRA ✭♣r✐♠✳ r❡❝✳ ❛r✐t❤♠❡t✐❝✮     ♣r✐♠✳ r❡❝✳ ❢✉♥❝t✳ ✉♣ t♦ p n ω 3  EFA ✭❡❧❡♠❡♥t❛r② ❢✉♥❝t✳ ❛r✐t❤♠❡t✐❝✮  ✺

❚❤❡ ❛r✐t❤♠❡t✐❝❛❧ ❤✐❡r❛r❝❤② A ∈ Σ 0 � A ✐s ❡q✉✐✈❛❧❡♥t t♦ s♦♠❡ ❢♦r♠✉❧❛ B ( n, m, Y ) ✇✐t❤ B ♣r✐♠✐t✐✈❡ r❡❝✉rs✐✈❡ 0 A ∈ Π 0 � A ✐s ❡q✉✐✈❛❧❡♥t t♦ s♦♠❡ ❢♦r♠✉❧❛ B ( n, m, Y ) ✇✐t❤ B ♣r✐♠✐t✐✈❡ r❡❝✉rs✐✈❡ 0 A ∈ Σ 0 n +1 � A ✐s ❡q✉✐✈❛❧❡♥t t♦ s♦♠❡ ❢♦r♠✉❧❛ ∃ n B ( n, m, Y ) ✇✐t❤ B ( n, m, Y ) ∈ Π 0 n A ∈ Π 0 n +1 � A ✐s ❡q✉✐✈❛❧❡♥t t♦ s♦♠❡ ❢♦r♠✉❧❛ ∀ n B ( n, m, Y ) ✇✐t❤ B ( n, m, Y ) ∈ Σ 0 n A ∈ ∆ 0 � A ∈ Σ 0 n ❛♥❞ A ∈ Π 0 n n ❋♦r♠✉❧❛❡ ✐♥ Σ 0 k ✭♦r Π 0 k ✮ ❛r❡ s❛✐❞ ❛r✐t❤♠❡t✐❝❛❧ ✻

❚❤❡ ❛♥❛❧②t✐❝❛❧ ❤✐❡r❛r❝❤② A ∈ Σ 1 � A ✐s ❡q✉✐✈❛❧❡♥t t♦ s♦♠❡ ❢♦r♠✉❧❛ B ( m, Y ) ∈ ∆ 0 n ❢♦r s♦♠❡ n 0 A ∈ Π 1 � A ✐s ❡q✉✐✈❛❧❡♥t t♦ s♦♠❡ ❢♦r♠✉❧❛ B ( m, Y ) ∈ ∆ 0 n ❢♦r s♦♠❡ n 0 A ∈ Σ 1 n +1 � A ✐s ❡q✉✐✈❛❧❡♥t t♦ s♦♠❡ ❢♦r♠✉❧❛ ∃ X B ( X, m, Y ) ✇✐t❤ B ( X, m, Y ) ∈ Π 1 n A ∈ Π 1 n +1 � A ✐s ❡q✉✐✈❛❧❡♥t t♦ s♦♠❡ ❢♦r♠✉❧❛ ∀ X B ( X, , m, Y ) ✇✐t❤ B ( X, m, Y ) ∈ Σ 1 n A ∈ ∆ 1 � A ∈ Σ 1 n ❛♥❞ A ∈ Π 1 n n ✼

■♥❞✉❝t✐♦♥ ♦♥ ❛❧❧ s❡ts ✈s Σ 0 1 ✲✐♥❞✉❝t✐♦♥ ❚❡❝❤♥✐❝❛❧❧②✱ ❛❧❧ s②st❡♠s ✐♥❝❧✉❞❡ ✐♥❞✉❝t✐♦♥ ♦✈❡r ❛❧❧ s❡ts✿ IND : 0 ∈ X ⇒ ∀ n ( n ∈ X ⇒ n + 1 ∈ X ) ⇒ ∀ n ( n ∈ X ) ❜✉t RCA 0 ❛♥❞ WKL 0 ✇❤✐❝❤ ❤❛✈❡ ✐♥❞✉❝t✐♦♥ ♦✈❡r Σ 0 1 ✲❢♦r♠✉❧❛❡✿ Σ 0 1 ✲ IND : A (0) ⇒ ∀ n ( A ( n ) ⇒ A ( n + 1)) ⇒ ∀ n A ( n ) ❢♦r A ( n ) ∈ Σ 0 1 ✭♣♦ss✐❜❧② ✇✐t❤ ♣❛r❛♠❡t❡rs✮ ❍♦✇❡✈❡r✱ ❛❧❧ ✺ s②st❡♠s ❝♦✉❧❞ ❜❡ ✉♥✐❢♦r♠❧② ❜❛s❡❞ ♦♥ t❤❡ Σ 0 1 ✲✐♥❞✉❝t✐♦♥ s❝❤❡♠❡✿ ❜❡❝❛✉s❡ A ❝❛♥ ❝♦♥t❛✐♥ ❢r❡❡ ✈❛r✐❛❜❧❡s n ✱ m ✱ ✳✳✳ ♦❢ ♥❛t✉r❛❧ ♥✉♠❜❡rs ❛♥❞ X ✱ Y ✱ ✳✳✳ ♦❢ s❡ts ♦❢ ♥❛t✉r❛❧ ♥✉♠❜❡rs✱ t❤❡ ❡✛❡❝t✐✈❡ str❡♥❣t❤ ♦❢ Σ 0 1 ✲✐♥❞✉❝t✐♦♥ ✐s ✐♥ ❢❛❝t ❣♦✈❡r♥❡❞ ❜② ✇❤✐❝❤ s❡ts ❛r❡ ❞❡✜♥❛❜❧❡ ✭✐✳❡✳ ❜② t❤❡ ❝♦♠♣r❡❤❡♥s✐♦♥ ❛①✐♦♠s✮✳ ❆❧t❡r♥❛t✐✈❡❧② ✭❆❋❆■❯✮✱ ❛❧❧ ✺ s②st❡♠s ❝♦✉❧❞ ❜❡ ✉♥✐❢♦r♠❧② ❜❛s❡❞ ♦♥ ✐♥❞✉❝t✐♦♥ ♦✈❡r ❛❧❧ s❡ts t♦♦✱ ✐❢ ❛❧❧ ♣r✐♠✳ r❡❝✳ ❢✉♥❝t✐♦♥s ❛r❡ t❛❦❡♥ ♣r✐♠✐t✐✈❡ ✐♥ RCA 0 ❛♥❞ WKL 0 ✳ ✽