Caristi’s fixed point theorem and strong systems of arithmetic David Fern´ andez-Duque Mathematics Department, Ghent University David.FernandezDuque@UGent.be Joint with Paul Shafer, Henry Towsner, and Keita Yokoyama. Wormshop 2017 Moscow, Russia David Fern´ andez-Duque Caristi’s theorem and strong arithmetics Wormshop ’17 1 / 16
Fixed point theorems in analysis Theorem (Banach, 1922) Let X be a complete metric space and f : X → X be a contraction; that is, there is ρ < 1 such that d ( f ( x ) , f ( y )) < ρ · d ( x, y ) for all x, y ∈ X . Then, there is x ∗ ∈ X such that f ( x ∗ ) = x ∗ . David Fern´ andez-Duque Caristi’s theorem and strong arithmetics Wormshop ’17 2 / 16
Fixed point theorems in analysis Theorem (Banach, 1922) Let X be a complete metric space and f : X → X be a contraction; that is, there is ρ < 1 such that d ( f ( x ) , f ( y )) < ρ · d ( x, y ) for all x, y ∈ X . Then, there is x ∗ ∈ X such that f ( x ∗ ) = x ∗ . Theorem (Brouwer, 1910) Let D be a disk in R n and f : D → D be continuous. Then, there is x ∗ ∈ X such that f ( x ∗ ) = x ∗ . David Fern´ andez-Duque Caristi’s theorem and strong arithmetics Wormshop ’17 2 / 16
Caristi systems and Caristi’s fixed point theorem Definition A Caristi system is a triple ( X , V, f ) , where • X is a complete separable metric space, • V : X → R ≥ 0 is a lower semi-continuous function, and • f : X → X is an arbitrary function , such that � � ∀ x ∈X d ( x, f ( x )) ≤ V ( x ) − V ( f ( x )) . Theorem (Caristi, 1976) If ( X , V, f ) is a Caristi system, then f has a fixed point. • Henceforth, a metric space is a complete separable metric space. • We call the V in a Caristi system a potential. David Fern´ andez-Duque Caristi’s theorem and strong arithmetics Wormshop ’17 3 / 16
Proofs of Caristi’s theorem 1 Caristi’s proof (simplified by Chi Song Wong) David Fern´ andez-Duque Caristi’s theorem and strong arithmetics Wormshop ’17 4 / 16
Proofs of Caristi’s theorem 1 Caristi’s proof (simplified by Chi Song Wong) 2 Proof by EVP. Theorem (Ekeland, 1974) Every lower semi-continuous function V : X → R ≥ 0 has a critical point, i.e. a point x ∗ ∈ X such that � � ∀ y ∈ X d ( x ∗ , y ) ≤ V ( x ∗ ) − V ( y ) → y = x ∗ David Fern´ andez-Duque Caristi’s theorem and strong arithmetics Wormshop ’17 4 / 16
The Big Five subsystems of second-order arithmetic • Two sorts: natural numbers, sets of naturals. • Real numbers, infinite trees, etc. can all be coded in SOA. • All systems have Σ 0 1 -induction, elementary arithmetical axioms. RCA 0 Computable sets exist ( ∆ 0 1 comprehension) WKL 0 RCA 0 + “every infinite binary tree has an infinite path” ACA 0 { n ∈ N : ϕ ( n ) } exists, ϕ arithmetical ATR 0 Transfinite recursion for arithmetical formulas. Π 1 1 -CA 0 { n ∈ N : ∀ X ⊆ N ϕ ( n, X ) } exists, ϕ arithmetical David Fern´ andez-Duque Caristi’s theorem and strong arithmetics Wormshop ’17 5 / 16
The strength of continuous Caristi’s theorem Theorem (F-D S T Y) The following are equivalent over RCA 0 . 1 ACA 0 . 2 Caristi’s theorem for continuous functions. 3 Caristi’s theorem for continuous potentials and continuous functions. Proof. (1 → 2). By EVP. David Fern´ andez-Duque Caristi’s theorem and strong arithmetics Wormshop ’17 6 / 16
The strength of continuous Caristi’s theorem Theorem (F-D S T Y) The following are equivalent over RCA 0 . 1 ACA 0 . 2 Caristi’s theorem for continuous functions. 3 Caristi’s theorem for continuous potentials and continuous functions. Proof. (1 → 2). By EVP. (3 → 1). Use the fact that ACA 0 is equivalent to the statement that every decreasing sequence of positive reals has an infimium. David Fern´ andez-Duque Caristi’s theorem and strong arithmetics Wormshop ’17 6 / 16
Compactness and Caristi’s theorem Theorem (F-D S T Y) For compact metric spaces X : • Caristi for l.s.c. V and continuous f is equivalent to WKL 0 . • Caristi for continuous V and continuous f is equivalent to WKL 0 . David Fern´ andez-Duque Caristi’s theorem and strong arithmetics Wormshop ’17 7 / 16
Compactness and Caristi’s theorem Theorem (F-D S T Y) For compact metric spaces X : • Caristi for l.s.c. V and continuous f is equivalent to WKL 0 . • Caristi for continuous V and continuous f is equivalent to WKL 0 . Baire functions: • Continuous functions are Baire class 0 • ω -limits of Baire class < ξ functions are Baire class ξ . David Fern´ andez-Duque Caristi’s theorem and strong arithmetics Wormshop ’17 7 / 16
Compactness and Caristi’s theorem Theorem (F-D S T Y) For compact metric spaces X : • Caristi for l.s.c. V and continuous f is equivalent to WKL 0 . • Caristi for continuous V and continuous f is equivalent to WKL 0 . Baire functions: • Continuous functions are Baire class 0 • ω -limits of Baire class < ξ functions are Baire class ξ . Theorem (F-D S T Y) For compact metric spaces X : • Caristi for l.s.c. V and Baire class 1 f is equivalent to ACA 0 . David Fern´ andez-Duque Caristi’s theorem and strong arithmetics Wormshop ’17 7 / 16
Caristi vs. ATR 0 Theorem (F-D S T Y) Caristi for Baire class 1 functions f : X → X (with arbitrary X and l.s.c. V ) implies ATR 0 . David Fern´ andez-Duque Caristi’s theorem and strong arithmetics Wormshop ’17 8 / 16
Caristi vs. ATR 0 Theorem (F-D S T Y) Caristi for Baire class 1 functions f : X → X (with arbitrary X and l.s.c. V ) implies ATR 0 . Facts: � �� � 1 ϕ - TR := ∀≺ WO ( ≺ ) → ∃ Z ∀ ξ ∀ n n ∈ Z ξ ↔ ϕ ( n, Z ≺ ξ ) • Z ξ = { m ∈ N : � m, ξ � ∈ Z } , • Z ≺ ξ = { m ∈ N : ∃ ζ ≺ ξ � m, ζ � ∈ Z } . David Fern´ andez-Duque Caristi’s theorem and strong arithmetics Wormshop ’17 8 / 16
Caristi vs. ATR 0 Theorem (F-D S T Y) Caristi for Baire class 1 functions f : X → X (with arbitrary X and l.s.c. V ) implies ATR 0 . Facts: � �� � 1 ϕ - TR := ∀≺ WO ( ≺ ) → ∃ Z ∀ ξ ∀ n n ∈ Z ξ ↔ ϕ ( n, Z ≺ ξ ) • Z ξ = { m ∈ N : � m, ξ � ∈ Z } , • Z ≺ ξ = { m ∈ N : ∃ ζ ≺ ξ � m, ζ � ∈ Z } . 2 ATR 0 ≡ RCA 0 + Σ 0 1 - TR . David Fern´ andez-Duque Caristi’s theorem and strong arithmetics Wormshop ’17 8 / 16
Caristi vs. ATR 0 Theorem (F-D S T Y) Caristi for Baire class 1 functions f : X → X (with arbitrary X and l.s.c. V ) implies ATR 0 . Facts: � �� � 1 ϕ - TR := ∀≺ WO ( ≺ ) → ∃ Z ∀ ξ ∀ n n ∈ Z ξ ↔ ϕ ( n, Z ≺ ξ ) • Z ξ = { m ∈ N : � m, ξ � ∈ Z } , • Z ≺ ξ = { m ∈ N : ∃ ζ ≺ ξ � m, ζ � ∈ Z } . 2 ATR 0 ≡ RCA 0 + Σ 0 1 - TR . 3 If ϕ is Σ 0 1 , there is a sequence of trees ( T ξ ) ξ ∈ N such that any path g through T ξ codes Z ξ . David Fern´ andez-Duque Caristi’s theorem and strong arithmetics Wormshop ’17 8 / 16
Caristi for Baire class 1 F implies ATR 0 . Proof idea. Fix ( T i ) i<ω . We define a Caristi system ( X , F, V ) as follows. • X be the set of sequences of paths ( g i ) i<ω . • F ( � g ) = lim n → ω F n , where F n ( � g ) replaces g ξ by an ‘ n -approximation’ of the path through T ξ , whenever: 1 g ξ ( n ) �∈ T ξ , and 2 ξ < n is the ≺ -minimum satisfying 1. g ) = � { 2 − n : g n is not a path through T i } . • V ( � This defines a Caristi system, whose fixed point ( g ∗ i ) i<ω such that each g ∗ i is a path through T i . � David Fern´ andez-Duque Caristi’s theorem and strong arithmetics Wormshop ’17 9 / 16
Leftmost paths Theorem (Marcone) Π 1 1 - CA 0 is equivalent to the statement “every ill-founded tree T ⊆ N < N has a leftmost path.” Definition (Towsner) • The transfinite leftmost path principle states that if T ⊆ N < N is ill-founded and α is a well-order, then there is a path f ∗ through T such that no path through T is both Σ T ⊕ f ∗ and to the left of f ∗ . α • TLPP 0 is RCA 0 plus the transfinite leftmost path principle. TLPP 0 is strictly between ATR 0 and Π 1 1 -CA 0 . David Fern´ andez-Duque Caristi’s theorem and strong arithmetics Wormshop ’17 10 / 16
Caristi’s fixed point theorem for Baire functions Theorem (F-D S T Y) Caristi for Baire functions f : X → X (with arbitrary X and l.s.c. V ) is equivalent to TLPP 0 . Thus in the general case: • Caristi is equivalent to TLPP 0 • Ekeland is equivalent to Π 1 1 -CA 0 David Fern´ andez-Duque Caristi’s theorem and strong arithmetics Wormshop ’17 11 / 16
Strength of Caristi’s proof Recall: Caristi’s proof relies on uncountable Caristi sequences. Definition Fix a Caristi system ( X , V, f ) . A Caristi sequence (from x 0 ) is a well-order ( L, ≺ ) and a sequence ( x ℓ : ℓ ∈ L ) ⊆ X such that x min L = x 0 x S ( ℓ ) = f ( x ℓ ) x ℓ = lim k<ℓ x k ( ℓ ∈ Lim ) David Fern´ andez-Duque Caristi’s theorem and strong arithmetics Wormshop ’17 12 / 16
Maximal sequences Call a Caristi sequence proper if the x ℓ ’s are all distinct. Lemma (Maximal sequence principle) Given a Caristi system ( X , V, f ) with f arithmetical and x 0 ∈ X , there is a proper Caristi sequence with no strict, proper extensions. David Fern´ andez-Duque Caristi’s theorem and strong arithmetics Wormshop ’17 13 / 16
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