aa r X i v : . [ m a t h . L O ] J a n Very weak fragments of weak K˝onig’s lemma
Stephen G. Simpson and Keita Yokoyama [email protected] [email protected] It is well-known that any finite Π -class of 2 N has a computable real. Then, how can we understandthis in the context of reverse mathematics? We need to formalize the statement carefully. Within RCA , the assertion “every infinite binary tree which has at most one path has a path” is alreadyequivalent to WKL since the negation of WKL implies the existence of an infinite binary tree withno path. In this note, we will consider the following very weak versions of K¨onig’s lemma. Definition 1.1. . WKL( pf - bd ): an infinite binary tree T ⊆ < N has a path if there exists c ∈ N such that for any prefix-free set P ⊆ T , | P | ≤ c .2 . WKL( w - bd ): an infinite binary tree T ⊆ < N has a path if there exists c ∈ N such that forany n ∈ N , | T = n | ≤ c , where T = n = { σ ∈ T | lh( σ ) = n } .3 . WKL( ext - bd ): an infinite binary tree T ⊆ < N has a path if there exists c ∈ N such that forany n ∈ N , | T = n ext | ≤ c , where T = n ext = { σ ∈ T | lh( σ ) = n ∧ σ is extendible } .4 . Σ -WKL( pf - bd ): an Σ infinite binary tree T ⊆ < N has a path if there exists c ∈ N suchthat for any prefix-free set P ⊆ T , | P | ≤ c . Here, Σ tree is described by an enumeration ofbinary strings { σ i } i ∈ N such that T = { σ i | i ∈ N } forms a tree.5 . Σ -WKL( w - bd ): an Σ infinite binary tree T ⊆ < N has a path if there exists c ∈ N suchthat for any n ∈ N , | T = n | ≤ c . This statement is known as Chaitin’s lemma.6 . Σ -WKL( ext - bd ): an Σ infinite binary tree T ⊆ < N has a path if there exists c ∈ N suchthat for any n ∈ N , | T = n ext | ≤ c .7 . KL( pf - bd ): an infinite binary tree T ⊆ N < N has a path if there exists c ∈ N such that forany prefix-free set P ⊆ T , | P | ≤ c .8 . KL( w - bd ): an infinite binary tree T ⊆ N < N has a path if there exists c ∈ N such that for any n ∈ N , | T = n | ≤ c .9 . KL( ext - bd ): an finitely-branching infinite binary tree T ⊆ N < N has a path if there exists c ∈ N such that for any n ∈ N , | T = n ext | ≤ c . The work of the second author is partially supported by JSPS KAKENHI grant number 16K17640, JSPS fellow-ship for research abroad, JSPS-NUS Bilateral Joint Research Projects J150000618 (PI’s: K. Tanaka, C. T. Chong),and JSPS Core-to-Core Program (A. Advanced Research Networks).The major part of this work was done when the second author visited Vanderbilt university in July 2016. → →
1, 6 → → → → tree iseasily interpreted as a tree in N < N . Indeed, for a given a Σ tree T = { σ i } i ∈ N , set ˆ T ⊆ N < N as τ ∈ ˆ T ↔ ∀ i ≤ j < lh( τ )(lh( σ τ ( i ) ) = i ∧ σ τ ( i ) ⊆ σ τ ( j ) ), then, T and ˆ T is isomorphic and a path ofˆ T computes a path of T . Thus, we have 7 → →
1, 8 → → → → -definable set can be described as a unique path of a Σ tree. Proposition 1.1 ( RCA , folklore) . For any Σ -definable set A ⊆ N , there exists a Σ tree T suchthat | T = n ext | = 1 for any n ∈ N and A is a path of T .Proof. Write n ∈ A ↔ ∃ mθ ( m, n ) for some Σ formula θ . Then, define a Σ tree T as σ ∈ T ↔∃ m ∈ N ∀ i < lh( σ )( σ ( i ) = 1 ↔ ∃ m ′ < m θ ( m ′ , i )). It is easy to check that this T is the desired. Corollary 1.2.
The following are equivalent over
RCA . . ACA . . Σ - WKL( ext - bd ) . . KL( ext - bd ) . Thus, they are not computably weak any more. On the other hand, all other statements inDefinition 1.1 is computably true, in other words, it is true in any ω -model of RCA . However,we will see that most of them are not provable within RCA , thus they require some non-trivialinduction.Throughout this note, we will use the following well-known fact. Lemma 1.3 ( RCA ) . If IΣ fails, then there exists a set X and a Π ,X -set A such that A isunbounded and | A | ≤ c for some c ∈ N . (Here, A is said to be unbounded if ∀ n ∈ N ∃ m ≥ n m ∈ A ,and | A | ≤ c means that for any finite set F ⊆ A (coded by a natural number), | F | ≤ c .) KL and induction Theorem 2.1.
The following are equivalent over
RCA . . BΣ . . Σ - WKL( pf - bd ) . . KL( pf - bd ) .Proof. We first show 1 →
3. Let b ∈ N , and let T ⊆ N < N be an infinite tree such that for any prefix-free P ⊆ T , | P | ≤ b . By Σ -induction, b = max { b ′ ≤ b | ∃ P ⊆ T ( P is prefix-free and | P | = b ′ ) } exists. Take prefix-free P ⊆ T so that | P | = b . Then, every element in T is compatible withsome member in P by the maximality. Put ˆ T = { σ ∈ T | ∃ τ ∈ P σ ⊇ τ } . Then, ˆ T is infinite. Forany σ ∈ ˆ T , there exists at most one immediate extension. (Presume that σ and σ are immediateextensions of σ and σ ⊃ τ ∈ P , then P \ { τ } ∪ { σ , σ } is a prefix-free set, which contradicts themaximality of P .) By BΣ , there exists τ ∈ P such that there exist infinitely many extensions of τ in ˆ T . Thus, any extension of τ has exactly one immediate extension. Hence one can computea path extending τ .3 → ¬ → ¬
2. Since BΣ is equivalent to RT , we assume that there exists afunction h : N → b for some b ∈ N such that h − ( x ) is finite for all x < b . Put T = { x y ∈ < N |∃ x < b ∃ y ′ ≥ y h ( y ′ ) = x } . Then, T is a Σ tree, and it is infinite since 1 h ( n ) n ∈ T for any n ∈ N .Moreover, any prefix-free subset of T is bounded by b . Since h − ( x ) is finite for all x < b , there isno infinite path of T , thus we have ¬ Lemma 2.2.
RCA + IΣ proves WKL( ext - bd ) .Proof. Let T ⊆ < N be an infinite tree and let b ∈ N be a bound for T = n ext . By IΣ there exists b ≤ b such that b = min { a ≤ b | ∀ n ( | T = n ext | ≤ a ) } . Take n ∈ N such that | T = n ext | = b , and let σ ∈ T = n ext . Then, any extendible extension of σ has exactly one immediate extendible extension.(More formally, for any σ ′ ⊇ σ , there exist n ∈ N and i < τ ∈ T = n , τ ⊇ σ ′ → τ ⊇ σ ′ ⌢ i .) Thus, one can compute a path extending σ . Theorem 2.3.
The following are equivalent over
RCA . . IΣ . . Σ - WKL( w - bd ) . . KL( w - bd ) .Proof. We first show 1 →
3. Let T ⊆ N < N be an infinite tree and let b ∈ N be a bound for T = n . By IΣ there exists b ≤ b such that b = max { a ≤ b | ∀ n ∃ m ( | T = m ext | ≥ a ) } . Then, the Σ set X = { m ∈ N | | T = m ext | ≥ b } is infinite, and hence there exists an infinite set X ⊆ X . Bymaximality, | T = m ext | = b for all but finite m ∈ X , so we may assume that | T = m ext | = b holds for all m ∈ X . Write X = { m < m < . . . } , and put σ i,j be the j -th left-most element of T = m i . Notethat one can compute the double sequence {{ σ i,j | j < b } | i ∈ N } from T and X . Now, definean infinite tree S ⊆ b < N as τ ∈ S ↔ ∀ i ≤ j < lh( τ )( σ i,τ ( i ) ⊆ σ j,τ ( j ) ). Then, S has a path byWKL( w - bd ), which is provable from IΣ by Lemma 2.2. One can easily retrieve a path of T froma path of S .3 → ¬ → ¬
2. By Lemma 1.3, there exists a set X and a Π ,X -set A such that A is unbounded and | A | ≤ c for some c ∈ N . Note that A cannot exist as a set since the cardinalityof an unbounded set won’t be bounded within RCA . Write n ∈ A ↔ ∀ mθ ( m, n, X ) where θ is aΣ -formula. Then, define a Σ tree T as σ ∈ T ↔ ∃ m > lh( σ )( ∀ i < lh( σ )( σ ( i ) = 1 ↔ ∀ m ′ < m θ ( m ′ , i, X )) ∧ |{ i < lh( σ ) | ∀ m ′ < m θ ( m ′ , i, X ) }| ≤ c ) . Then, T is infinite since A ↾ n := { x ∈ A | x < n } ∈ T for any n ∈ N . ( A ↾ n always exists by boundedΣ -comprehension which is available within RCA .) By the definition of T , for any σ, τ ∈ T suchthat lh( σ ) = lh( τ ), if σ ( i ) < τ ( i ) for some i < lh( σ ), then σ ( i ) ≤ τ ( i ) for all i < lh( σ ). Thus, thereare at most c -many elements in T = n for any n ∈ N . A path of T should be identical with A , so T cannot have a path. 3 Computably true fragments of
WKL and induction
The situation is more complicated when we consider weak fragments of WKL since they are allprovable within
WKL , which is Π -conservative over RCA . Thus, they never imply pure inductionaxioms. Still, they may require some induction when WKL fails badly.First, we see that a bound for prefix-free subsets is enough to find a recursive path even within RCA . Proposition 3.1.
WKL( pf - bd ) is provable within RCA .Proof. Let T ⊆ < N be an infinite tree and let c ∈ N such that for any prefix-free set P ⊆ T , | P | ≤ c . By IΣ there exists c ≤ c and a prefix-free set P ⊆ T such that | P | = c and there isno prefix-free set P ′ ⊆ T with | P ′ | > c . Then, there exists an extendible node σ ∈ T such that σ ∈ P . By the maximality of c and P , there is no incomparable τ, τ ′ ∈ T extending σ , henceone may easily computes a path of T extending σ .Next we focus on WKL( w - bd ) and WKL( ext - bd ). Indeed, they are equivalent and strictly inbetween RCA and WKL . Theorem 3.2.
WKL( w - bd ) and WKL( ext - bd ) are equivalent over RCA .Proof. Let T ⊆ < N be an infinite tree such that ∀ n ∈ N | T = n ext | ≤ c for some c ∈ N . We willconstruct a tree T ′ ⊆ < N with | T ′ = n | ≤ c such that any path of T ′ computes a path of T . Put T s = { σ ∈ T | ∃ τ ∈ T =lh( σ )+ s σ ⊆ τ } . We recursively define { s n } n ∈ N as s n = min { s > s n − || T = ns | ≤ c } . Such s n always exists since | T = n ext | ≤ c . Now T ′ ⊆ < N is defined as follows: τ ∈ T ′ ↔ ∃ σ ∈ T ≤ s ∀ s < lh( τ )( τ ( s ) = σ ( n ) if ∃ n ≤ s s = s n ∧ τ ( s ) = 2 if ∀ n ≤ s s = s n ) . Now, any string τ ∈ T ′ is of the form τ = j ⌢ i ⌢ j ⌢ i ⌢ . . . ⌢ j ⌢n − i n − such that h j , j , . . . , j n − i ∈ T = ns for some s ≥ s n . Thus, | T ′ = n | ≤ c . For a given a path of T ′ , one can easily compute a pathof T by removing all 2’s. Theorem 3.3 ( RCA ) . If IΣ fails, there exists a tree T ∈ S such that ∀ n ∈ N | T = n ext | ≤ c for some c ∈ N and [ T ] does not contain any T -recursive elements.Proof. We argue within (
M, S ) | = RCA + ¬ IΣ . Then, there exists a set X ∈ S and an X -r.e. set A such that N \ A is unbounded and | N \ A | ≤ c for some c ∈ N . Such A cannot be a memberof S since there is no unbounded set whose cardinality is bounded within RCA . Thus, A is not X -recursive in ( M, S ). By formalizing the construction of [2, Lemma 8.2], there exists a pair ofdisjoint X -r.e. sets B and B which splits A such that their separating class does not contain any X -recursive elements. Take an X -recursive tree T such that [ T ] is the separating class of B and B . If σ, τ ∈ T = n ext , σ ( i ) = τ ( i ) for all i ∈ A . Thus, | T = n ext | ≤ c . Corollary 3.4.
RCA does not imply WKL( ext - bd ) . Corollary 3.5.
The following are equivalent over
RCA . . IΣ . . recursive- WKL( w - bd ) : an infinite binary tree T ⊆ < N has a T -recursive path if there exists c ∈ N such that for any n ∈ N , | T = n | ≤ c . . recursive- WKL( ext - bd ) : an infinite binary tree T ⊆ < N has a T -recursive path if there exists c ∈ N such that for any n ∈ N , | T = n ext | ≤ c . Corollary 3.6.
Over
RCA , WKL( w - bd ) plus the assertion “there exists a set X such that for anyset Y , Y ≤ T X ” implies IΣ . Definition 3.1 (Very smallness, Binns/Kjos-Hanssen[1]) . VSMALL asserts the following: aninfinite binary tree T ⊆ < N has a path if for any function f : N → N , there exists n ∈ N such thatfor any m ≥ n , | T = f ( m )ext | < m . Proposition 3.7.
Over
RCA , the disjunction of IΣ and VSMALL implies
WKL( w - bd ) .Proof. By Lemma 2.2 and the definition of VSMALL.
Question 3.2.
Is WKL( w - bd ) strictly weaker than IΣ ∨ VSMALL over
RCA ? References [1] Stephen Binns and Bjørn Kjos-Hanssen. Finding paths through narrow and wide trees.
J.Symbolic Logic , 74(1):349–360, 2009.[2] Stephen G. Simpson. Π sets and models of WKL . In Reverse mathematics 2001 , volume 21of