【克鲁斯卡尔树定理】
TREE函数、Kruskal树定理（Kruskal's tree theorem）是逆数学的突出示例。由安德鲁·瓦兹尼推测并由first=Joseph证明。
In mathematics, Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding.
==History==
The theorem was conjectured by Andrew Vázsonyi and proved by first=Joseph; a short proof was given by  first=Crispin. It has since become a prominent example in reverse mathematics as a statement that cannot be proved within ATR0 (a form of arithmetical transfinite recursion),  and a finitary application of the theorem gives the existence of the fast-growing TREE function.
In 2004, the result was generalized from trees to graphs as the Robertson–Seymour theorem, a result that has also proved important in reverse mathematics and leads to the even-faster-growing SSCG function.
==Statement==
The version given here is that proven by Nash-Williams; Kruskal's formulation is somewhat stronger. All trees we consider are finite.
Given a tree T with a root, and given vertices v, w, call w a successor of v if the unique path from the root to w contains v, and call w an immediate successor of v if additionally the path from v to w contains no other vertex.
Take X to be a partially ordered set. If T1, T2 are rooted trees with vertices labeled in X, we say that T1 is inf-embeddable in T2 and write T1 ≤ T2 if there is an injective map F from the vertices of T1 to the vertices of T2 such that
*For all vertices v of T1, the label of v precedes the label of F(v),
*If w is any successor of v in T1, then F(w) is a successor of F(v), and
*If w1, w2 are any two distinct immediate successors of v, then the path from F(w1) to F(w2) in T2 contains F(v).
Kruskal's tree theorem then states: If X is well-quasi-ordered, then the set of rooted trees with labels in X is well-quasi-ordered under the inf-embeddable order defined above. (That is to say, given any infinite sequence T1, T2, … of rooted trees labeled in X, there is some i i ≤ Tj.)
==Weak tree function==
Define tree(n), the weak tree function, as the length of the longest sequence of 1-labelled trees (i.e. ) such that:
* The tree at position k in the sequence has no more than k + n vertices, for all k.
* No tree is homeomorphically embeddable into any tree following it in the sequence.
It is known that tree(1) = 2, tree(2) = 5, and tree(3) ≥ 844424930131960,reason=WP:RSPSOURCES; Fandom page lists YT video as reference but TREE(3) (where the argument specifies the number of labels; see below) is larger than
==Friedman's work==
For a countable label set , Kruskal's tree theorem can be expressed and proven using second-order arithmetic. However, like 古德斯坦定理 or the Paris–Harrington theorem, some special cases and variants of the theorem can be expressed in subsystems of second-order arithmetic much weaker than the subsystems where they can be proved. This was first observed by Harvey Friedman in the early 1980s, an early success of the then-nascent field of reverse mathematics. In the case where the trees above are taken to be unlabeled (that is, in the case where  has order one), Friedman found that the result was unprovable in ATR0, thus giving the first example of a predicative result with a provably impredicative proof. This case of the theorem is still provable by Πb=1-CA0, but by adding a "gap condition" to the definition of the order on trees above, he found a natural variation of the theorem unprovable in this system. Much later, the Robertson–Seymour theorem would give another theorem unprovable by Πb=1-CA0.
confirms the strength of Kruskal's theorem, with the proof-theoretic ordinal of the theorem equaling the small Veblen ordinal (sometimes confused with the smaller Ackermann ordinal).date=September 2022
===TREE(3)===
Suppose that P(n) is the statement:
There is some m such that if T1,...,Tm is a finite sequence of unlabeled rooted trees where Tk has n+k vertices, then Ti ≤ Tj for some i 1,...,Tm of rooted trees labelled from a set of n labels, where each Ti has at most i vertices, such that Ti  ≤  Tj does not hold for any i n(5)(5). A lower bound for n(4), and hence an extremely weak lower bound for TREE(3), is AA(187196)(1), where A(x) taking one argument, is defined as A(x, x), where A(k, n), taking two arguments, is a particular version of Ackermann's function defined as: A(1, n) = 2n, A(k+1, 1) = A(k, 1), A(k+1, n+1) = A(k, A(k+1, n)). 葛立恒数, for example, is much smaller than the lower bound AA(187196)(1). It can be shown that the growth-rate of the function TREE is at least  in the fast-growing hierarchy. AA(187196)(1) is approximately , where gx is Graham's function.