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。由推测并由约瑟夫·克鲁斯卡尔证明。
In , Kruskal's tree theorem states that the set of finite s over a set of labels is itself well-quasi-ordered under embedding.
在数学中，克鲁斯卡树定理指出，在准有序（英文：well-quasi-ordering）标签集上的有限树集合本身在同胚下也是良准有序的（英文：同胚（图论））嵌入。
历史
该定理由安德鲁·瓦兹索尼（Andrew Vázsonyi）猜想并由约瑟夫·克鲁斯卡尔（Joseph Kruskal，1960）证明； Crispin Nash-Williams（1963）给了一个简短的证明。它从此成为逆向数学（英语：reverse math）中的一个突出例子，作为在 ATR0 内无法证明的陈述（算术超限递归（英语：arithmetic transfinite recursion）的一种形式）和有限（英语：finitary）应用该定理给出了快速增长的TREE 函数的存在性。
2004年，这个结果被从树推广到图，成为罗伯逊-西摩定理（英语：Robertson-Seymour theorem），这一结果在逆向数学中也被证明很重要，并导致了增长更快的SSCG 函数（英语： Friedman 的 SSCG 函数）。
陈述
这里给出的版本是由 Nash-Williams 证明的；克鲁斯卡尔的表述要强一些。我们认为的所有树木都是有限的。
给定一棵有根的树T，并给定顶点v、w，如果从根到w 的唯一路径包含v，则称w 为v 的后继；如果从v 到w 的路径另外不包含v，则称w 为v 的直接后继其他顶点。
取X为偏序集（中文：partiallyorderedset）。如果 T1、T2 是顶点标记为 X 的有根树，我们说 T1 可 inf-embeddable 在 T2 中，并且如果存在从 T1 的顶点到 T2 的顶点的单射映射 F，则写 T1 ≤ T2，使得
弗里德曼的工作
对于可数标签集
X
X.克鲁斯卡树定理可以用二阶算术来表达和证明（中文：二阶算术）。然而，就像古德斯坦定理或巴黎-哈灵顿定理（英语：Paris-Harrington theorem）一样，该定理的一些特殊情况和变体可以用比可以证明它们的子系统弱得多的二阶算术子系统来表达。这是哈维·弗里德曼（Harvey Friedman）在 20 世纪 80 年代初首次观察到的，这是当时新兴的逆向数学领域的早期成功。在上面的树被认为是未标记的情况下（即，在
X
X 有顺序一），Friedman 发现结果在 ATR0（英语：算术超限递归）,从而给出了第一个谓词（英语：impredicativity）结果的例子，并带有可证明的谓词证明.[3此定理的这种情况仍可由 Π1 证明
1-CA0，但透过在上述树的顺序定义中添加「间隙条件」，他发现该定理的自然变体在该系统中无法证明。很久以后，罗伯森-西摩定理给了另一个无法由 Π1 证明的定理
1-CA0。
序数分析（中文：序数分析）证实了克鲁斯卡尔定理的强度，该定理的证明论序数等于小维勃伦序数（英文：小维勃伦序数）（有时与较小的阿克曼序数（英文：阿克曼序数）相混淆） ).
树(3)
假设 P(n) 是这样的语句：
存在一些 m，如果 T1...,Tm 是未标记有根树的有限序列，其中 Tk 有 n+k 个顶点，则对于某些 i < j，Ti ≤ Tj。
根据克鲁斯卡定理和柯尼格引理，所有陈述 P(n) 都是正确的。对于每个n，皮亚诺算术（英语：Peano axioms）可以证明P(n)为真，但皮亚诺算术不能证明“P(n)对于所有n都为真”这一命题。此外，皮亚诺算术中 P(n) 的最短证明的长度作为 n 的函数增长得非常快，远远快于任何原始递归函数（英语：primitive recursive function）或阿克曼函数。 P(n) 所适用的最小 m 同样随 n 增长得非常快。
透过合并标签，弗里德曼定义了一个成长速度更快的函数。n，取 TREE(n)[* 为最大的 m，这样我们有：
有一个由 n 个标签集标记的有根树序列 T1...,Tm，其中每个 Ti 最多有 i 个顶点，使得 Ti ≤ Tj 对于任何 i < j ≤ m 不成立。
TREE 序列开始为TREE(1) = 1, TREE(2) = 3，然后突然TREE(3) 爆炸到非常大的值，以至于许多其他「大」组合常数，例如Friedman 的n(4), [*相较之下，*] 非常小。事实上，它比 nn(5)(5) 大得多。 n(4) 的下界（因此 TREE(3) 的下界极弱）为 AA(187196)(1),，其中 A(x) 采用一个参数，定义为 A(x, x)，其中A (k, n) 有两个参数，是阿克曼函数（英语：Ackermann's function）的一个特定版本，定义为： A(1, n) = 2n, A(k+1, 1) = A (k , 1), A(k+1, n+1) = A(k, A(k+1, n))。例如，葛立恒数远小于下限 AA(187196)(1)。可以证明函数TREE的成长率至少为
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History.
The theorem was conjectured by and proved by ; a short proof was given by . It has since become a prominent example in as a statement that cannot be proved within ATR0 (a form of ), and a 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 , a result that has also proved important in reverse mathematics and leads to the even-faster-growing .
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 with a root, and given vertices , , call a successor of if the unique path from the root to contains , and call an immediate successor of if additionally the path from to contains no other vertex.
Take to be a . If , are rooted trees with vertices labeled in , we say that is inf-embeddable in and write if there is an injective map from the vertices of to the vertices of such that
Kruskal's tree theorem then states: If is , then the set of rooted trees with labels in is well-quasi-ordered under the inf-embeddable order defined above. (That is to say, given any infinite sequence of rooted trees labeled in , there is some so that .)
Weak tree function.
Define , the weak tree function, as the length of the longest sequence of 1-labelled trees (i.e. ) such that:
It is known that tree(1) = 2, tree(2) = 5, and tree(3) ≥ 844424930131960, but (where the argument specifies the number of "labels"; see ) is larger than formula_1
Friedman's work.
For a countable label set formula_2, Kruskal's tree theorem can be expressed and proven using . However, like 古德斯坦定理 or the , 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 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 formula_2 has order one), Friedman found that the result was unprovable in , thus giving the first example of a result with a provably impredicative proof. This case of the theorem is still provable by Π-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 Π-CA0.
TREE(3).
Suppose that "P"("n") is the statement:
All the statements "P"("n") are true as a consequence of Kruskal's theorem and 柯尼格引理. For each "n", can prove that "P"("n") is true, but Peano arithmetic cannot prove the statement ""P"("n") is true for all "n"". Moreover the length of the shortest proof of "P"("n") in Peano arithmetic grows phenomenally fast as a function of "n", far faster than any or the 阿克曼函数 for example. The least "m" for which "P"("n") holds similarly grows extremely quickly with "n".
By incorporating labels, Friedman defined a far faster-growing function. For a positive integer "n", take "TREE"("n") to be the largest "m" so that we have the following:
The "TREE" sequence begins "TREE"(1) = 1, "TREE"(2) = 3, then suddenly "TREE"(3) explodes to a value so immensely large that many other "large" combinatorial constants, such as Friedman's "n"(4), are extremely small by comparison. In fact, it is much larger than "n""n"(5)(5). A lower bound for "n"(4), and hence an "extremely" weak lower bound for "TREE"(3), is "A""A"(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 defined as: "A"(1, "n") = 2"n", "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 "A""A"(187196)(1). It can be shown that the growth-rate of the function "TREE" is at least formula_4 in the . "A""A"(187196)(1) is approximately formula_5, where "g""x" is .