A Decomposition of Brouwer's Fan Theorem

We introduce axioms L FAN and C FAN , where the former follows from the law of excluded middle and the latter follows from the axiom of countable choice. Then we show that Brouwer's fan theorem is constructively equivalent to L FAN + C FAN. This decomposition of the fan theorem into a logical axiom and a function existence axiom contributes to the programme of constructive reverse mathematics. The objective of constructive reverse mathematics is to classify theorems by means of logical axioms (that means, statements which follow from the law of excluded middle) and function existence axioms (that means, statements which follow from the axiom of choice). Given a theorem T, one aims at determining a logical axiom L T and a function existence axiom C T such that, in the framework of a suitable base system, T is equivalent to L T + C T. A decomposition of the weak König lemma can be found in Ishihara [3]. In this paper, we provide a classification of Brouwer's fan theorem. A suitable framework for carrying out constructive reverse mathematics is a function-based, intuitionistic formal system, like HA ω. In this system, if A(n) is equivalent to a quantifier-free formula, one may prove: (1) ∃α ∈ {0, 1} N ∀n (α(n) = 0 ↔ A(n)) , Let {0, 1} * denote the set of all finite binary sequences u, v, w. We write |u| for the length of u and u * v for the concatenation of u and v. If i ∈ {0, 1}, we write u * i for u * (i) and i * u for (i) * u. We use Greek letters for infinite binary sequences. We write αn for (α(0),. .. , α(n − 1)).

The objective of constructive reverse mathematics is to classify theorems by means of logical axioms (that means, statements which follow from the law of excluded middle) and function existence axioms (that means, statements which follow from the axiom of choice).Given a theorem T, one aims at determining a logical axiom L T and a function existence axiom C T such that, in the framework of a suitable base system, T is equivalent to L T + C T .A decomposition of the weak König lemma can be found in Ishihara [3].In this paper, we provide a classification of Brouwer's fan theorem.
A suitable framework for carrying out constructive reverse mathematics is a functionbased, intuitionistic formal system, like HA ω .In this system, if A(n) is equivalent to a quantifier-free formula, one may prove: If i ∈ {0, 1}, we write u * i for u * (i) and i * u for (i) * u.We use Greek letters for infinite binary sequences.We write αn for (α(0), . . ., α(n − 1)).A subset A of {0, 1} * is (For example, (1) implies that, for every quantifier-free formula C(n), the set A function α is an infinite path of A if ∀n (αn ∈ A).The weak König lemma reads as follows.
WKL Every infinite tree has an infinite path.
The following axiom is called the lesser limited principle of omniscience.
LLPO If a function α has the property Note that this axiom is a consequence of the law of excluded middle.The following characterisation of LLPO in terms of trees has also been mentioned in Berger et al. [2].

Lemma 1
The following axioms are equivalent.

3
(2) If T is an infinite tree, then either is an infinite tree.
Proof "1.⇒ 2." Assume LLPO and let T be an infinite tree.Define α by Furthermore, define β by By LLPO, there is an i ∈ {0, 1} such that ∀n (β(2n + i) = 0).We show that T i is an infinite tree.Suppose that there is an m such that i * w / ∈ T for all w with |w| = m.Then α(2m + i) = 1.Since T is an infinite tree, we have ∀k (α(2k + (1 − i)) = 0).Therefore, there exists an n ≤ m such that β(2n + i) = 1.This contradiction shows that T i is indeed an infinite tree."2.⇒ 1." Assume that the function γ has the property

Define an infinite tree T by
The following statement is a version of the axiom of countable choice.
C WKL Every Π 0 1 -spread has an infinite path.The following decomposition of WKL can be found in Ishihara [3].We recall it, because we want to compare it with the main result of this paper, the decomposition of FAN.We even give a proof of it, because there is a slight difference between C WKL and the choice axiom used in Ishihara [3].
Proof Assume WKL.First, we show LLPO by applying Lemma 1.Let T be an infinite tree.By WKL, there exists an infinite path α of T. Set i = α(0).Then Therefore, T i is an infinite tree.
Next, we show C WKL .To this end, let A be Π 0 1 -spread.There is a function G :

Define an infinite tree T by
Then there exists an infinite path α of T, which is also an infinite path of A. Now assume both LLPO and C WKL .Let T be an infinite tree.By LLPO and Lemma 1 we obtain that the set A, given by is a spread.Therefore, by C WKL , there is an infinite path α of A. This function α is also an infinite path of T.
A detachable subset B of {0, 1} * is Brouwer's fan theorem for detachable bars reads as follows.
FAN Every bar is a uniform bar.
The following Lemma can be found in Ishihara [4].

Lemma 3
The following statements are equivalent.

• FAN
• For every bar B which is closed under extension there exists an N such that ∀u |u| = N → u ∈ B .
• Every bar which is closed under extension is a uniform bar.
Berger and Ishihara [1] have shown that the statement 'every infinite tree with at most one infinite path has an infinite path' is constructively equivalent to FAN.See also Schwichtenberg [5] for a more formal proof of this result.This characterisation of FAN, together with Proposition 2, gives rise to the definition of the axioms L FAN and C FAN .
A subset A of {0, 1} * has at most one infinite path if L FAN If T is an infinite tree with at most one infinite path, then either C FAN If A is a Π 0 1 -spread and B is a bar, then there exists a u ∈ A ∩ B.
The axioms introduced so far are related as follows.
Lemma 4 LLPO implies L FAN and C WKL implies C FAN .
Proof The fact that LLPO implies L FAN follows from Lemma 1.In order to prove the second implication, assume C WKL and fix a bar B. Assume further that A is a Π 0 1 -spread.By C WKL , there is an infinite path α of A. Since B is a bar, there exists an n such that αn ∈ B. Thus αn ∈ A ∩ B. Now we are ready to prove the decomposition of FAN.
Again, since B is closed under extension, Lemma 3 implies the existence of an n such that Since T is an infinite tree, it contains an element u of length n implies that But T is an infinite tree, therefore, T i must be an infinite tree as well.
Next, we show that FAN implies C FAN .Let A be a Π 0 1 -spread.Then for every n there is a u with |u| = n such that every restriction of u belongs to A. Let B be a bar.By FAN, there is an N such that Suppose that u is of length N and that every restriction of u belongs to A. Then there is an n ≤ N such that un ∈ A ∩ B.
Finally, we show that the combination of L FAN and C FAN implies FAN.Let B be a bar which is closed under extension.Define If () ∈ B, then B is a uniform bar.Assume now that () / ∈ B and define a subset P of {0, 1} * by A sequence u belongs to P if and only if it is the largest element of {0, 1} * \ B, with respect to the ordering ≺.Since B is closed under extension, we can conclude that u ∈ P if and only if Therefore, P is detachable.Since ≺ is a total relation on {0, 1} * , we can conclude that P has at most one element, that means Furthermore, if there exists an element u of P, then every w with |w| = |u| + 1 belongs to B, which implies that B is a uniform bar.Define an infinite tree T by Note that here we use the assumption that () / ∈ B, because B = {0, 1} * would imply that T = ∅.If a sequence u belongs to T, then either it does not belong to B or else some proper restriction of u belongs to P. We show that T has at most one infinite path.Fix α and β and suppose that there is an m such that α(m) = β(m).Since B is both a bar and closed under extension, there is an n such that αn = βn, αn ∈ B, and βn ∈ B. We will derive a contradiction from the assumption that both αn and βn are in T. By the definition of T, there are k, l < n such that we can conclude that T is a Π 0 1 -set.Next, we show that T is a spread.Since T is an infinite tree, () ∈ T .Assume that u ∈ T .Then T u = {w | u * w ∈ T} is an infinite tree.By L FAN there exists i ∈ {0, 1} such that (T u ) i is an infinite tree, which implies that u * i is in T .This concludes the proof that T is a spread.Now C FAN yields the existence of a u such that u ∈ T ∩ B. Since we have T ⊆ T, we even obtain that u ∈ B ∩ T. By the definition of T, some restriction of u lies in P. Therefore, B is a uniform bar.
Overall, we obtain the following picture.

Proposition 5 FAN
↔ L FAN + C FAN Proof First we show that FAN implies L FAN .Let T be an infinite tree with at most one infinite path.This implies ∀α, β ∃n 0 * αn / ∈ T ∨ 1 * βn / ∈ T .Fix α and define a bar B by B = {u | 0 * α|u| / ∈ T ∨ 1 * u / ∈ T} .Since B is closed under extension, Lemma 3 implies the existence of an n such that ∀u |u| = n → u ∈ B .Thus we have ∀α ∃n ∀β 0 * αn / ∈ T ∨ 1 * βn / ∈ T .Define another bar B by If αk = βl, then αn = βn.If αk = βl, then αk and βl are two distinct elements of P. Either case leads to a contradiction.Therefore, T has at most one infinite path.Define a subset T of T by u ∈ T def ⇔ {w | u * w ∈ T} is an infinite tree.Since we have ∀u u ∈ T ↔ ∀m ∃w |w| = m ∧ u * w ∈ T ,