Counterexamples in Computable Continuum Theory Takayuki Kihara - PowerPoint PPT Presentation

co-c.e. closed Dendroids Example A Cantor fan and a harmonic comb are dendroids, but not dendrites. ✓ ✏ Cantor set Cantor fan Harmonic comb ✒ ✑ Here a harmonic comb is defined by: ( ) ( ) [ 0 , 1 ] × { 0 } ∪ ( { 0 } ∪ { 1 / n : n ∈ N } ) × [ 0 , 1 ] . Takayuki Kihara Counterexamples in Computable Continuum Theory

Effectiveness for Tree-Like Continua Remark D endroids is approximated by T rees . Takayuki Kihara Counterexamples in Computable Continuum Theory

Effectiveness for Tree-Like Continua Remark D endroids is approximated by T rees . Main Theorem C omputable D endrites is not approximated by C o - c . e . T rees . 1 Takayuki Kihara Counterexamples in Computable Continuum Theory

Effectiveness for Tree-Like Continua Remark D endroids is approximated by T rees . Main Theorem C omputable D endrites is not approximated by C o - c . e . T rees . 1 C o - c . e . D endrites is not approximated by C omputable D endrites . 2 Takayuki Kihara Counterexamples in Computable Continuum Theory

Effectiveness for Tree-Like Continua Remark D endroids is approximated by T rees . Main Theorem C omputable D endrites is not approximated by C o - c . e . T rees . 1 C o - c . e . D endrites is not approximated by C omputable D endrites . 2 C omputable D endroids is not approximated by C o - c . e . D endrites . 3 Takayuki Kihara Counterexamples in Computable Continuum Theory

Effectiveness for Tree-Like Continua Remark D endroids is approximated by T rees . Main Theorem C omputable D endrites is not approximated by C o - c . e . T rees . 1 C o - c . e . D endrites is not approximated by C omputable D endrites . 2 C omputable D endroids is not approximated by C o - c . e . D endrites . 3 C o - c . e . D endroids is not approximated by C omputable D endroids . 4 Takayuki Kihara Counterexamples in Computable Continuum Theory

Effectiveness for Tree-Like Continua Remark D endroids is approximated by T rees . Main Theorem C omputable D endrites is not approximated by C o - c . e . T rees . 1 C o - c . e . D endrites is not approximated by C omputable D endrites . 2 C omputable D endroids is not approximated by C o - c . e . D endrites . 3 C o - c . e . D endroids is not approximated by C omputable D endroids . 4 Not every contractible planar co-c.e. dendroid contains a 5 computable point. Takayuki Kihara Counterexamples in Computable Continuum Theory

Effectiveness for Tree-Like Continua Remark D endroids is approximated by T rees . Main Theorem C omputable D endrites is not approximated by C o - c . e . T rees . 1 C o - c . e . D endrites is not approximated by C omputable D endrites . 2 C omputable D endroids is not approximated by C o - c . e . D endrites . 3 C o - c . e . D endroids is not approximated by C omputable D endroids . 4 Not every contractible planar co-c.e. dendroid contains a 5 computable point. This is the solution to Question of Le Roux, and Ziegler . Takayuki Kihara Counterexamples in Computable Continuum Theory

Theorem C o - c . e . D endrites is not approximated by C omputable D endrites . ✓ ✏ 0 -rising 1 -rising 1 -rising Approximation of Basic Dendrite Basic Dendrite ✒ ✑ Basic Dendrite has 2 n many n -risings of height 2 − n . Takayuki Kihara Counterexamples in Computable Continuum Theory

Fix a non-computable c.e. set A ⊆ N . The Basic construction around an n -rising is following: ✓ ✏ n ∈ A n -rising n � A ✒ ✑ n ∈ A = ⇒ an n -rising will be a cut point . n � A = ⇒ an n -rising will be a ramification point . Takayuki Kihara Counterexamples in Computable Continuum Theory

To prove the theorem, we need to prepare some tools. Takayuki Kihara Counterexamples in Computable Continuum Theory

To prove the theorem, we need to prepare some tools. Lemma Every subdendrite of Ψ( 2 <ω ) is homeomorphic to Ψ( T ) 1 for a subtree T ⊆ 2 <ω . T ⊆ 2 <ω is co-c.e. closed (c.e., computable, resp.) tree 2 iff Ψ( T ) ⊆ R 2 is co-c.e. closed (c.e., computable, resp.) dendrite. Every computable subdendrite D ⊆ Ψ( 2 <ω ) 3 there exists a computable subtree T ⊆ 2 <ω such that D ⊆ Ψ( T ) holds, and D and Ψ( T ) has same paths. Takayuki Kihara Counterexamples in Computable Continuum Theory

Definition (Cenzer-K.-Weber-Wu 2009) A co-c.e. closed subset P of Cantor space is tree-immune if a co-c.e. tree T P ⊆ 2 <ω has no infinite computable subtree. Here T P = { σ ∈ 2 <ω : ( ∃ f ⊃ σ ) f ∈ P } Example The set of all consistent complete extensions of Peano Arithmetic is tree-immune. Lemma Let P be a tree-immune co-c.e. closed subset of Cantor space, and D ⊆ Ψ( T P ) be any computable subdendrite. Then D contains no path. Takayuki Kihara Counterexamples in Computable Continuum Theory

Now we start True Construction . ✓ ✏ Tree-immune Π 0 1 set Approximating basic n -rising Basic n -rising ✒ ✑ An n -rising has a copy of a tree-immune co-c.e. closed set of scale 2 − n . Takayuki Kihara Counterexamples in Computable Continuum Theory

Fix a non-computable Σ 0 1 set A ⊆ N . The True Construction around an n -rising is following: ✓ ✏ n ∈ A Tree-immune Π 0 1 set n � A ✒ ✑ n ∈ A = ⇒ any top of an n -rising will be a cut point . n � A = ⇒ any top of an n -rising will be inaccessible by computable dendrites. Takayuki Kihara Counterexamples in Computable Continuum Theory

Fix a non-computable Σ 0 1 set A ⊆ N . The True Construction around an n -rising is following: ✓ ✏ n ∈ A Tree-immune Π 0 1 set n � A ✒ ✑ n ∈ A = ⇒ If a dendrite D passes this n -rising, then D contains a top of this n -rising. n � A = ⇒ Any computable dendrite contains no top of n -rising. Takayuki Kihara Counterexamples in Computable Continuum Theory

Theorem (Restated) C o - c . e . D endrites is not approximated by C omputable D endrites . The construction of the co-c.e. closed dendrite H is completed. Takayuki Kihara Counterexamples in Computable Continuum Theory

Theorem (Restated) C o - c . e . D endrites is not approximated by C omputable D endrites . The construction of the co-c.e. closed dendrite H is completed. Let D ⊆ H be any computable dendrite. Takayuki Kihara Counterexamples in Computable Continuum Theory

Theorem (Restated) C o - c . e . D endrites is not approximated by C omputable D endrites . The construction of the co-c.e. closed dendrite H is completed. Let D ⊆ H be any computable dendrite. It suffices to show that D cannot pass 2 distinct risings. Takayuki Kihara Counterexamples in Computable Continuum Theory

Theorem (Restated) C o - c . e . D endrites is not approximated by C omputable D endrites . The construction of the co-c.e. closed dendrite H is completed. Let D ⊆ H be any computable dendrite. It suffices to show that D cannot pass 2 distinct risings. If D passes m , n -risings, then D passes a k -rising for all k ≥ min { m , n } . Takayuki Kihara Counterexamples in Computable Continuum Theory

Theorem (Restated) C o - c . e . D endrites is not approximated by C omputable D endrites . The construction of the co-c.e. closed dendrite H is completed. Let D ⊆ H be any computable dendrite. It suffices to show that D cannot pass 2 distinct risings. If D passes m , n -risings, then D passes a k -rising for all k ≥ min { m , n } . Since D is co-c.e. closed, we can enumerate all k such that D contains no top of any k -rising. Takayuki Kihara Counterexamples in Computable Continuum Theory

Theorem (Restated) C o - c . e . D endrites is not approximated by C omputable D endrites . The construction of the co-c.e. closed dendrite H is completed. Let D ⊆ H be any computable dendrite. It suffices to show that D cannot pass 2 distinct risings. If D passes m , n -risings, then D passes a k -rising for all k ≥ min { m , n } . Since D is co-c.e. closed, we can enumerate all k such that D contains no top of any k -rising. This enumeration yields the complement of a c.e. set A . Takayuki Kihara Counterexamples in Computable Continuum Theory

Theorem (Restated) C o - c . e . D endrites is not approximated by C omputable D endrites . The construction of the co-c.e. closed dendrite H is completed. Let D ⊆ H be any computable dendrite. It suffices to show that D cannot pass 2 distinct risings. If D passes m , n -risings, then D passes a k -rising for all k ≥ min { m , n } . Since D is co-c.e. closed, we can enumerate all k such that D contains no top of any k -rising. This enumeration yields the complement of a c.e. set A . This contradicts non-computability of A . Takayuki Kihara Counterexamples in Computable Continuum Theory

Theorem C omputable D endroids is not approximated by C o - c . e . D endrites . Takayuki Kihara Counterexamples in Computable Continuum Theory

Theorem C omputable D endroids is not approximated by C o - c . e . D endrites . We will use harmonic combs in place of the Basic Dendrite . Takayuki Kihara Counterexamples in Computable Continuum Theory

Theorem C omputable D endroids is not approximated by C o - c . e . D endrites . We will use harmonic combs in place of the Basic Dendrite . Before starting the construction, we take account of the fact that topologist’s sine curve is not path-connected. Takayuki Kihara Counterexamples in Computable Continuum Theory

Theorem C omputable D endroids is not approximated by C o - c . e . D endrites . We will use harmonic combs in place of the Basic Dendrite . Before starting the construction, we take account of the fact that topologist’s sine curve is not path-connected. It means that we can not cut-pointize infinite many risings, on one harmonic comb. Takayuki Kihara Counterexamples in Computable Continuum Theory

Theorem C omputable D endroids is not approximated by C o - c . e . D endrites . We will use harmonic combs in place of the Basic Dendrite . Before starting the construction, we take account of the fact that topologist’s sine curve is not path-connected. It means that we can not cut-pointize infinite many risings, on one harmonic comb. Our idea is using a computable approximation of a certain limit computable function . Takayuki Kihara Counterexamples in Computable Continuum Theory

Theorem C omputable D endroids is not approximated by C o - c . e . D endrites . We will use harmonic combs in place of the Basic Dendrite . Before starting the construction, we take account of the fact that topologist’s sine curve is not path-connected. It means that we can not cut-pointize infinite many risings, on one harmonic comb. Our idea is using a computable approximation of a certain limit computable function . One harmonic comb replaces one rising. Takayuki Kihara Counterexamples in Computable Continuum Theory

Theorem C omputable D endroids is not approximated by C o - c . e . D endrites . We will use harmonic combs in place of the Basic Dendrite . Before starting the construction, we take account of the fact that topologist’s sine curve is not path-connected. It means that we can not cut-pointize infinite many risings, on one harmonic comb. Our idea is using a computable approximation of a certain limit computable function . One harmonic comb replaces one rising. The Basic Dendroid will be constructed by connecting infinitely many harmonic combs. Takayuki Kihara Counterexamples in Computable Continuum Theory

✓ ✏ 0 -harmonic comb 1 -h.c. 1 -h.c. Basic Dendroid ✒ ✑ Basic Dendroid has 2 n many n -harmonic combs of height 2 − n . Each n -harmonic comb has infinitely many risings . Takayuki Kihara Counterexamples in Computable Continuum Theory

✓ ✏ ( n , 2 ) -rising ( n , 1 ) -rising ( n , 0 ) -rising ( n , ω ) -rising n -harmonic comb ✒ ✑ Basic Dendroid has 2 n many n -harmonic combs of height 2 − n . Each n -harmonic comb has ( ω + 1 ) -many risings ; They are ( n , α ) -risings for α < ω + 1 . Takayuki Kihara Counterexamples in Computable Continuum Theory

To prove the theorem, we need the following lemma. Takayuki Kihara Counterexamples in Computable Continuum Theory

To prove the theorem, we need the following lemma. Lemma There exists a limit computable function p such that, for every uniformly c.e. sequence { U n } of cofinite c.e. sets, it holds that p ( n ) ∈ U n for almost all n . Takayuki Kihara Counterexamples in Computable Continuum Theory

To prove the theorem, we need the following lemma. Lemma There exists a limit computable function p such that, for every uniformly c.e. sequence { U n } of cofinite c.e. sets, it holds that p ( n ) ∈ U n for almost all n . Proof { V e } : an effective enumeration of uniformly c.e. decreasing sequence of c.e. sets. σ ( e , x ) = { i ≤ e : x ∈ ( V i ) e } : The e -state of x . p ( e ) chooses x to maximize the e -state. Takayuki Kihara Counterexamples in Computable Continuum Theory

p = lim s p s : a limit computable function in the previous lemma. The construction on an n -harmonic comb is following: ✓ ✏ p s ( n ) = m ( n , m ) -rising p s ( n ) � m ✒ ✑ ( ∃ s ) p s ( n ) = m = ⇒ an ( n , m ) -rising will be a cut point . ( ∀ s ) p s ( n ) � m = ⇒ an ( n , m ) -rising will be a ramification point . Takayuki Kihara Counterexamples in Computable Continuum Theory

p = lim s p s : a limit computable function in the previous lemma. The construction on an n -harmonic comb is following: ✓ ✏ p s ( n ) = m ( n , m ) -rising p s ( n ) � m ✒ ✑ Since p ( n ) = lim s p s ( n ) changes his mind at most finitely often, he cut-pointizes only finitely many risings on an n -harmonic comb. Takayuki Kihara Counterexamples in Computable Continuum Theory

p = lim s p s : a limit computable function in the previous lemma. The construction on an n -harmonic comb is following: ✓ ✏ p s ( n ) = m ( n , m ) -rising p s ( n ) � m ✒ ✑ Thus each n -harmonic comb, actually, will be homeomorphic to a harmonic comb. The construction yields computable dendroid K . Takayuki Kihara Counterexamples in Computable Continuum Theory

Recall that a dendrite is a locally connected dendroid. On a harmonic comb, any top of almost all rising must be inaccessible by a dendrite. ✓ ✏ Locally connected D n -harmonic comb ⊇ D contains tops of only three risings; ( n , 1 ) -rising; ( n , 4 ) -rising; ( n , ω ) -rising ✒ ✑ Takayuki Kihara Counterexamples in Computable Continuum Theory

✓ ✏ p s ( n ) = m ( n , m ) -rising p s ( n ) � m ✒ ✑ ( ∃ s ) p s ( n ) = m = ⇒ any top of an ( n , m ) -rising will be a cut point . Meanwhile, any top of almost all risings will be inaccessible by a given dendrite. Takayuki Kihara Counterexamples in Computable Continuum Theory

✓ ✏ p s ( n ) = m ( n , m ) -rising p s ( n ) � m ✒ ✑ ( ∃ s ) p s ( n ) = m = ⇒ If a dendrite D passes an ( n , m ) -rising, then D contains a top of an ( n , m ) -rising. Meanwhile, any dendrite contains no top of almost all risings. Takayuki Kihara Counterexamples in Computable Continuum Theory

✓ ✏ p s ( n ) = m ( n , m ) -rising p s ( n ) � m ✒ ✑ U D n : the set of all ( n , m ) -risings whose top is not accessed by a dendrite D . Then U D n is cofinite for all n . If D passes n -harmonic comb then p ( n ) � U D n . Takayuki Kihara Counterexamples in Computable Continuum Theory

Theorem (Restated) C omputable D endroids is not approximated by C o - c . e . D endrites . K : the computable dendroid in the construction. Takayuki Kihara Counterexamples in Computable Continuum Theory

Theorem (Restated) C omputable D endroids is not approximated by C o - c . e . D endrites . K : the computable dendroid in the construction. D : a co-c.e. closed subdendrite of K . Takayuki Kihara Counterexamples in Computable Continuum Theory

Theorem (Restated) C omputable D endroids is not approximated by C o - c . e . D endrites . K : the computable dendroid in the construction. D : a co-c.e. closed subdendrite of K . U n : the set of all ( n , m ) -risings whose top is not accessed by a dendrite D . Takayuki Kihara Counterexamples in Computable Continuum Theory

Theorem (Restated) C omputable D endroids is not approximated by C o - c . e . D endrites . K : the computable dendroid in the construction. D : a co-c.e. closed subdendrite of K . U n : the set of all ( n , m ) -risings whose top is not accessed by a dendrite D . U n is cofinite by previous observation. Takayuki Kihara Counterexamples in Computable Continuum Theory

Theorem (Restated) C omputable D endroids is not approximated by C o - c . e . D endrites . K : the computable dendroid in the construction. D : a co-c.e. closed subdendrite of K . U n : the set of all ( n , m ) -risings whose top is not accessed by a dendrite D . U n is cofinite by previous observation. { U n } is uniformly c.e., since D is co-c.e. closed. Takayuki Kihara Counterexamples in Computable Continuum Theory

Theorem (Restated) C omputable D endroids is not approximated by C o - c . e . D endrites . K : the computable dendroid in the construction. D : a co-c.e. closed subdendrite of K . U n : the set of all ( n , m ) -risings whose top is not accessed by a dendrite D . U n is cofinite by previous observation. { U n } is uniformly c.e., since D is co-c.e. closed. It suffices to show that D cannot pass 2 distinct combs. Takayuki Kihara Counterexamples in Computable Continuum Theory

Theorem (Restated) C omputable D endroids is not approximated by C o - c . e . D endrites . K : the computable dendroid in the construction. D : a co-c.e. closed subdendrite of K . U n : the set of all ( n , m ) -risings whose top is not accessed by a dendrite D . U n is cofinite by previous observation. { U n } is uniformly c.e., since D is co-c.e. closed. It suffices to show that D cannot pass 2 distinct combs. Takayuki Kihara Counterexamples in Computable Continuum Theory

Theorem (Restated) C omputable D endroids is not approximated by C o - c . e . D endrites . K : the computable dendroid in the construction. D : a co-c.e. closed subdendrite of K . U n : the set of all ( n , m ) -risings whose top is not accessed by a dendrite D . U n is cofinite by previous observation. { U n } is uniformly c.e., since D is co-c.e. closed. It suffices to show that D cannot pass 2 distinct combs. If D passes an n -comb, it must hold that p ( n ) � U n . Takayuki Kihara Counterexamples in Computable Continuum Theory

Theorem (Restated) C omputable D endroids is not approximated by C o - c . e . D endrites . K : the computable dendroid in the construction. D : a co-c.e. closed subdendrite of K . U n : the set of all ( n , m ) -risings whose top is not accessed by a dendrite D . U n is cofinite by previous observation. { U n } is uniformly c.e., since D is co-c.e. closed. It suffices to show that D cannot pass 2 distinct combs. If D passes an n -comb, it must hold that p ( n ) � U n . It contradicts our choice of p which satisfies p ( n ) ∈ U n for almost all n . Takayuki Kihara Counterexamples in Computable Continuum Theory

Observation (Restated) Not every nonempty n -connected co-c.e. closed set in R n + 2 contains a computable point, for any n ∈ N . Takayuki Kihara Counterexamples in Computable Continuum Theory

Observation (Restated) Not every nonempty n -connected co-c.e. closed set in R n + 2 contains a computable point, for any n ∈ N . Every nonempty n -connected co-c.e. closed set in R n + 1 contains a computable point, for n = 0 . Takayuki Kihara Counterexamples in Computable Continuum Theory

Observation (Restated) Not every nonempty n -connected co-c.e. closed set in R n + 2 contains a computable point, for any n ∈ N . Every nonempty n -connected co-c.e. closed set in R n + 1 contains a computable point, for n = 0 . Question (Le Roux-Ziegler) Does every simply connected planar co-c.e. 1 closed set contain a computable point? Does every contractible Euclidean co-c.e. closed set contain a 2 computable point? Takayuki Kihara Counterexamples in Computable Continuum Theory

Observation (Restated) Not every nonempty n -connected co-c.e. closed set in R n + 2 contains a computable point, for any n ∈ N . Every nonempty n -connected co-c.e. closed set in R n + 1 contains a computable point, for n = 0 . Question (Le Roux-Ziegler) Does every simply connected planar co-c.e. 1 closed set contain a computable point? Does every contractible Euclidean co-c.e. closed set contain a 2 computable point? Main Theorem Not every nonempty contractible planar co-c.e. closed set contains a computable point. Takayuki Kihara Counterexamples in Computable Continuum Theory

A fat approximation of Cantor set: ✓ ✏ Fat approx. of Cantor set A construction of Cantor set ✒ ✑ P : a co-c.e. closed subset of Cantor set. P s : a fat approximation of P at stage s . l s , r s : the leftmost and rightmost of P s . Takayuki Kihara Counterexamples in Computable Continuum Theory

A fat approximation of Cantor set: ✓ ✏ Fat approx. of Cantor set A construction of Cantor set ✒ ✑ P : a co-c.e. closed subset of Cantor set. P s : a fat approximation of P at stage s . l s , r s : the leftmost and rightmost of P s . [ l s , l s + 1 ] ∩ P s , [ r s + 1 , r s ] ∩ P s contains intervals I l s , I r s . Takayuki Kihara Counterexamples in Computable Continuum Theory

A fat approximation of Cantor set: ✓ ✏ Fat approx. of Cantor set A construction of Cantor set ✒ ✑ P : a co-c.e. closed subset of Cantor set. P s : a fat approximation of P at stage s . l s , r s : the leftmost and rightmost of P s . [ l s , l s + 1 ] ∩ P s , [ r s + 1 , r s ] ∩ P s contains intervals I l s , I r s . We call these intervals I l s , I r s ⊆ P s \ P s + 1 free blocks. Takayuki Kihara Counterexamples in Computable Continuum Theory

Prepare a stretched co-c.e. closed class D − 0 = P × [ 0 , 1 ] . ✓ ✏ Free block Body Free block Stretched ✒ ✑ P ⊆ R 1 : a co-c.e. closed set without computable points. P s : a fat approximation of P (Note that P = ∩ s P s ). D − 0 = [ 0 , 1 ] × P 0 . Takayuki Kihara Counterexamples in Computable Continuum Theory

D 0 is the following connected closed set. ✓ ✏ Free block Body Free block Stretched ✒ ✑ The desired co-c.e. closed set D will be obtained by carving D 0 . Takayuki Kihara Counterexamples in Computable Continuum Theory

Destination α ∈ R : an incomputable left-c.e. real. There is a computable sequence { J s } of rational open intervals s.t. min J s → α as s → ∞ . diam ( J s ) → 0 as s → ∞ . Either J s + 1 ⊂ J s or max J s < min J s + 1 , for each s . Takayuki Kihara Counterexamples in Computable Continuum Theory

Our construction starts with D 0 . ✓ ✏ Free block Body Free block Stretched ✒ ✑ Takayuki Kihara Counterexamples in Computable Continuum Theory

By carving free blocks, stretch P 0 toward max J 0 . ✓ ✏ max J 0 ✒ ✑ Takayuki Kihara Counterexamples in Computable Continuum Theory

By carving free blocks, stretch P 0 toward min J 0 . ✓ ✏ min J 0 ✒ ✑ Takayuki Kihara Counterexamples in Computable Continuum Theory

Proceed one step with a fat approximation of P . ✓ ✏ min J 0 max J 0 ✒ ✑ Takayuki Kihara Counterexamples in Computable Continuum Theory

D 1 is defined by this, ✓ ✏ Zoom ✒ ✑ Takayuki Kihara Counterexamples in Computable Continuum Theory

D 1 is defined by this, ✓ ✏ Zoom ✒ ✑ If J 1 ⊂ J 0 , then the construction of D 2 is similar as that of D 1 . i.e., on the top block, stretch toward max J 1 and back to min J 1 , by caving free blocks. Takayuki Kihara Counterexamples in Computable Continuum Theory

D 1 is defined by this, ✓ ✏ Zoom ✒ ✑ If J 1 ⊂ J 0 , then the construction of D 2 is similar as that of D 1 . i.e., on the top block, stretch toward max J 1 and back to min J 1 , by caving free blocks. Takayuki Kihara Counterexamples in Computable Continuum Theory

D 1 is defined by this, ✓ ✏ Zoom ✒ ✑ If J 1 ⊂ J 0 , then the construction of D 2 is similar as that of D 1 . i.e., on the top block, stretch toward max J 1 and back to min J 1 , by caving free blocks. In general, similar for J s + 1 ⊂ J s . Takayuki Kihara Counterexamples in Computable Continuum Theory

D 1 is defined by this, ✓ ✏ Zoom ✒ ✑ If J 1 ⊂ J 0 , then the construction of D 2 is similar as that of D 1 . i.e., on the top block, stretch toward max J 1 and back to min J 1 , by caving free blocks. In general, similar for J s + 1 ⊂ J s . Only the problem is the case of J s + 1 � J s ! Takayuki Kihara Counterexamples in Computable Continuum Theory

In the case of J s + 1 � J s : ✓ ✏ Overview of D s (above D p ) J s + 1 J s J p ✒ ✑ Pick the greatest p ≤ s such that J s + 1 ⊂ J p . Takayuki Kihara Counterexamples in Computable Continuum Theory

In the case of J s + 1 � J s : ✓ ✏ Overview of D s (above D p ) J s + 1 J p ✒ ✑ Go back to D p by caving free blocks into the shape of P . Takayuki Kihara Counterexamples in Computable Continuum Theory

✓ ✏ Overview of D s (above D p ) J s + 1 J p ✒ ✑ By caving free blocks on D p into the shape of P , stretch toward max J s + 1 and back to min J s + 1 . Takayuki Kihara Counterexamples in Computable Continuum Theory

Main Theorem (Restated) Not every nonempty contractible planar co-c.e. closed set contains a computable point. D = ∩ s D s is co-c.e. closed. Takayuki Kihara Counterexamples in Computable Continuum Theory

Main Theorem (Restated) Not every nonempty contractible planar co-c.e. closed set contains a computable point. D = ∩ s D s is co-c.e. closed. D is obtained by bundling [ 0 , 1 ] × P at ( α, y ) ∈ R 2 for some y . Takayuki Kihara Counterexamples in Computable Continuum Theory

Main Theorem (Restated) Not every nonempty contractible planar co-c.e. closed set contains a computable point. D = ∩ s D s is co-c.e. closed. D is obtained by bundling [ 0 , 1 ] × P at ( α, y ) ∈ R 2 for some y . D is path-connected by the property of an approximation { J s } of the incomputable left-c.e. real α . Takayuki Kihara Counterexamples in Computable Continuum Theory

Main Theorem (Restated) Not every nonempty contractible planar co-c.e. closed set contains a computable point. D = ∩ s D s is co-c.e. closed. D is obtained by bundling [ 0 , 1 ] × P at ( α, y ) ∈ R 2 for some y . D is path-connected by the property of an approximation { J s } of the incomputable left-c.e. real α . Therefore, D is homeomorphic to Cantor fan, and contractible. Takayuki Kihara Counterexamples in Computable Continuum Theory

Main Theorem (Restated) Not every nonempty contractible planar co-c.e. closed set contains a computable point. D = ∩ s D s is co-c.e. closed. D is obtained by bundling [ 0 , 1 ] × P at ( α, y ) ∈ R 2 for some y . D is path-connected by the property of an approximation { J s } of the incomputable left-c.e. real α . Therefore, D is homeomorphic to Cantor fan, and contractible. Stretching [ 0 , 1 ] × P cannot introduce new computable points. Takayuki Kihara Counterexamples in Computable Continuum Theory

Main Theorem (Restated) Not every nonempty contractible planar co-c.e. closed set contains a computable point. D = ∩ s D s is co-c.e. closed. D is obtained by bundling [ 0 , 1 ] × P at ( α, y ) ∈ R 2 for some y . D is path-connected by the property of an approximation { J s } of the incomputable left-c.e. real α . Therefore, D is homeomorphic to Cantor fan, and contractible. Stretching [ 0 , 1 ] × P cannot introduce new computable points. Of course, ( α, y ) is also incomputable. Takayuki Kihara Counterexamples in Computable Continuum Theory

Main Theorem (Restated) Not every nonempty contractible planar co-c.e. closed set contains a computable point. D = ∩ s D s is co-c.e. closed. D is obtained by bundling [ 0 , 1 ] × P at ( α, y ) ∈ R 2 for some y . D is path-connected by the property of an approximation { J s } of the incomputable left-c.e. real α . Therefore, D is homeomorphic to Cantor fan, and contractible. Stretching [ 0 , 1 ] × P cannot introduce new computable points. Of course, ( α, y ) is also incomputable. Hence, D has no computable points. Takayuki Kihara Counterexamples in Computable Continuum Theory