Unique Paths as Formal Points

A point-free formulation of the König Lemma for trees with uniformly at most one infinite path allows for a constructive proof without unique choice.


Introduction
The (Weak) König Lemma says that every infinite (binary) tree has an infinite path.Kleene [17] gave a recursive counterexample to this statement: a recursive infinite binary tree without recursive infinite path.Moreover, the Weak König Lemma for detachable trees (WKL) is equivalent [13,4] to Bishop's Lesser Limited Principle of Omniscience (LLPO)-a fragment of the Law of Excluded Middle that fails to hold recursively-in conjunction with an instance of dependent choice.One thus cannot expect to find a constructive proof of WKL as it stands, let alone of König's Lemma.
By a constructive proof we mean one that is carried out within Bishop-style constructive mathematics (BISH) [5], the overall framework of this note.Due to the choice of intuitionistic logic characteristic of BISH, one or the other assumption needs to be made explicit that would be automatic with classical logic: an assertion A is said to be decidable whenever A ∨ ¬A holds; 1 a subset S of a set T is called detachable if x ∈ S is decidable for each x ∈ T ; and a set S is named discrete if x = y is decidable for each pair x, y ∈ S. Likewise, one says that a set S is inhabited if it has an element, which is to be distinguished from the double negation that S be nonempty.
After this digression on the method of the present paper, we come back to its principal object: infinite paths of infinite trees.Kleene's example can be ruled out by requiring 1 For a brief discussion of this use of "decidable" in contrast to "recursive" we refer to [16].Note in this context that every theorem of BISH is valid in Russian recursive mathematics, which is BISH plus Markov's principle and (an appropriate form of) Church's thesis [10].
that each tree under consideration have at most one infinite path.When one restricts WKL to trees of this sort, the resulting principle WKL! is equivalent [3,15,30] to Brouwer's Fan Theorem for detachable bars (FAN).This parallels the equivalence [2] of FAN and the minimum principle MIN! analogous to WKL!An analysis [28] of the latter equivalence has shown, however, that FAN is only needed for the step from the given uniqueness hypothesis to the canonical uniform variant of it.In fact FAN is necessary and sufficient for this first step, whereas the second step, from the uniform uniqueness precondition to the desired existence statement, is a mere instance of completeness. 2  To analyse the equivalence of FAN and WKL! in a similar way, we recall [15,30] that FAN suffices for the assertion (*) "every detachable binary tree with at most one infinite path has uniformly at most one infinite path".Moreover, (*) implies WKL!: every detachable binary tree with uniformly at most one infinite path has a longest path [15], which is an infinite path of the tree whenever, in addition, the tree is infinite.(In fact WKL is equivalent [14] to the Longest Path Lemma (LPL): every detachable binary tree has a longest path.)In combination with the implication from WKL! to FAN, this yields that FAN is also necessary for the assertion (*).
In the present paper we give a direct proof of the statement that every infinite tree with uniformly at most one infinite path actually has an infinite path.It is noteworthy that the tree under consideration need neither be detachable nor binary; it suffices that its nodes are finite words over a discrete alphabet.As a by-product we observe that if-as in the binary case-the alphabet happens to be finite, then for every detachable binary tree with uniformly at most one infinite path there is a spread which is an infinite path of the given tree whenever the tree is infinite.The spread we construct need not be contained in the tree under consideration unless this is infinite; at least in this sense the spread is a generalisation of a longest path.
In the spirit of formal topology [26,27] we understand infinite paths as linear spreadsor, equivalently, as infinite chains-rather than as infinite sequences.This allows us 2 As a proof paradigm this instance of completeness can be traced back to Russian recursive mathematics [11,23] (see also [22]; applications include the construction of Haar measures); has proved productive in constructive analysis see [7,8] (see also [9] and [5,Chapter 7]; the use of uniform uniqueness is often somewhat implicit [12]); stood right at the beginnings of proof mining [18,19] (see also [20] and [21, ; here uniform uniqueness is put as a modulus of uniqueness); has repeatedly shown up in computable analysis [6,32,33]; and most recently has reoccurred in abstract Stone duality [1,31].We refer to [29] for a reminder of this time-honoured argument, for a treatment of it in a choice-free environment, and for a few more details about its impressive history.
to get by without any choice, whereas one might have to use unique countable choice to extract an infinite sequence from an infinite path.We also work with a first-order characterisation of "uniformly at most one infinite path" [15,30], which, in particular, is formulated without infinite sequences.Spreads have further been used in contexts which lack uniqueness, to avoid choosing sequences [25].

Preliminaries
By m, n, k, . . .we mean integers 0. Let A throughout be an inhabited discrete set.Examples are A = N and A = {0, 1}.We use u, v, w, . . .as variables for the elements of As usual the length of u ∈ A n is defined as |u| = n.The empty sequence () is the only one of length 0. The n-th finite initial segment un = (u (0) , . . ., u (n − 1)) of u with |u| n has length n.Note that u|u| = u and u0 = ().
We say that a subset T of A * is unbounded (in length) if and bounded (in length) if If A is finite, which we do not suppose in general, then T ⊆ A * is (un)bounded precisely when it is (in)finite.
We identify each sequence (i) of length 1 with its only element i ∈ A, and denote the concatenation of u and v by their juxtaposition uv.For example, if w = ui for some i ∈ A, then w is an immediate successor of u.We write u w if u is a restriction of w: that is, u w ↔ ∃v (uv = w) .
This partial order on A * is decidable, because A is discrete.Clearly, () is the least element of A * with respect to .
A subset T of A * is closed under restrictions if u w → w ∈ T → u ∈ T .

T Coquand and P Schuster
Any such T is inhabited if and only if () ∈ T ; moreover, it is unbounded precisely when ∀n ∃u |u| = n ∧ u ∈ T , and bounded precisely when A tree is an inhabited subset T of A * which is closed under restrictions.A tree T is a spread if every element of T has an (immediate) successor in T : that is, If T is a spread, then-by induction-for every n there is u ∈ T with |u| = n, which is to say that T is unbounded; in other words, a spread T has at least one element of any given length.Note that we do not generally suppose that a tree be detachable.
In particular, a chain T has at most one element u of any given length: this condition is even equivalent to "T is a chain" whenever T is closed under restrictions.A chain is (un)bounded precisely when it is (in)finite.By a linear spread we understand a spread which is a chain.It is easy to see that the linear spreads are precisely the infinite chains.Clearly a linear spread has exactly one element of any given length.

Infinite paths
The following definitions will be used throughout this section.Let T be a tree.We consider the predicate P(u, k) ≡ ∀w |w| = |u| + k → w ∈ T → u w , for which P(u, k) implies P(u, ) whenever k .We further set S = {u ∈ A * : ∃k P(u, k)} .
If T is detachable, and A is finite, then P is decidable, and thus S simply existential (Σ 0 1 ).