**TREE函數**、**Kruskal樹定理**（英語：）是逆數學的突出示例。由安德魯·瓦茲尼（英语：）推測並由約瑟夫·克魯斯卡爾證明。

已隱藏部分未翻譯内容，歡迎參與翻譯。
In mathematics, **Kruskal's tree theorem** states that the set of finite trees over a well-quasi-ordered（英语：） set of labels is itself well-quasi-ordered under homeomorphic（英语：） embedding.

History
-------

The theorem was conjectured by Andrew Vázsonyi（英语：） and proved by Joseph Kruskal （1960）; a short proof was given by Crispin Nash-Williams （1963）. It has since become a prominent example in reverse mathematics（英语：） as a statement that cannot be proved within ATR0 (a form of arithmetical transfinite recursion（英语：）), and a finitary（英语：） 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（英语：）, a result that has also proved important in reverse mathematics and leads to the even-faster-growing SSCG function（英语：）.

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（英语：）. If *T*1, *T*2 are rooted trees with vertices labeled in X, we say that *T*1 is inf-embeddable in *T*2 and write *T*1 ≤ *T*2 if there is an injective map F from the vertices of *T*1 to the vertices of *T*2 such that

* For all vertices v of *T*1, the label of v precedes the label of *F*(*v*),
* If w is any successor of v in *T*1, then *F*(*w*) is a successor of *F*(*v*), and
* If *w*1, *w*2 are any two distinct immediate successors of v, then the path from *F*(*w*1) to *F*(*w*2) in *T*2 contains *F*(*v*).

Kruskal's tree theorem then states: 
> If X is well-quasi-ordered（英语：）, 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 *T*1, *T*2, … of rooted trees labeled in X, there is some *i* < *j* so that *Ti* ≤ *Tj*.)
> 
> 

Weak tree function
------------------

Define tree(*n*), the weak tree function, as the length of the longest sequence of 1-labelled trees (i.e. *X* = {1}) 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,[需要較佳来源] but TREE(3) (where the argument specifies the number of *labels*; see below（英语：）) is larger than {\displaystyle \mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{8}(7)}(7)}(7)}(7)}(7)}

Friedman's work
---------------

For a countable label set {\displaystyle X}, Kruskal's tree theorem can be expressed and proven using second-order arithmetic（英语：）. However, like 古德斯坦定理 or the Paris–Harrington theorem（英语：）, 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（英语：） 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 {\displaystyle X} has order one), Friedman found that the result was unprovable in <span class="ilh-all "data-orig-title="ATR0" data-lang-code="en" data-lang-name="英语" data-foreign-title="arithmetical transfinite recursion">[[:ATR0|ATR0]]（英语：）, thus giving the first example of a predicative（英语：） result with a provably impredicative proof. This case of the theorem is still provable by Π1  
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 Π1  
1-CA0.

Ordinal analysis（英语：） confirms the strength of Kruskal's theorem, with the proof-theoretic ordinal of the theorem equaling the small Veblen ordinal（英语：） (sometimes confused with the smaller Ackermann ordinal（英语：）).[來源請求]

### TREE(3)

Suppose that *P*(*n*) is the statement:

There is some *m* such that if *T*1,...*T**m* is a finite sequence of unlabeled rooted trees where *T**k* has *n*+*k* vertices, then *T**i* ≤ *T**j* for some *i* < *j*.
All the statements *P*(*n*) are true as a consequence of Kruskal's theorem and 柯尼格引理. For each *n*, Peano arithmetic（英语：） 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 primitive recursive function（英语：） or the 阿克曼函數 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:

There is a sequence *T*1,...*T**m* of rooted trees labelled from a set of *n* labels, where each *T**i* has at most *i* vertices, such that *T**i*  ≤  *T**j* does not hold for any *i* < *j*  ≤ *m*.
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 Ackermann's function（英语：） 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*)). 葛立恆數, 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 {\displaystyle f\_{\theta (\Omega ^{\omega }\omega )}} in the fast-growing hierarchy（英语：）. *A**A*(187196)(1) is approximately {\displaystyle g\_{3\uparrow ^{187196}3}}, where *g**x* is Graham's function（英语：）.

参见
--

* Paris–Harrington theorem（英语：）
* Kanamori–McAloon theorem（英语：）
* Robertson–Seymour theorem（英语：）

注释
--

**^ \*** Friedman originally denoted this function by *TR*[*n*].
**^ \*\*** *n*(*k*) is defined as the length of the longest possible sequence that can be constructed with a *k*-letter alphabet such that no block of letters x*i*,...x2*i* is a subsequence of any later block x*j*,...x2*j*. {\displaystyle n(1)=3,n(2)=11,\,{\textrm {and}}\,n(3)>2\uparrow ^{7197}158386}.


Category:Trees (graph theory)（英语：）

參考
--

Citations

1. ↑ . Googology Wiki | Fandom.  [9 July 2020]. （原始内容存档于2020-01-09）.
2. ↑  Simpson 1985, Theorem 1.8
3. ↑  Friedman 2002, p. 60
4. ↑  Simpson 1985, Definition 4.1
5. ↑  Simpson 1985, Theorem 5.14
6. ↑  Marcone 2001, p. 8–9
7. ↑  Smith 1985, p. 120
8. ↑  Friedman, Harvey.. Ohio State University Department of Maths. 28 March 2006  [8 August 2017]. （原始内容存档于2022-12-07）.
9. ↑  Friedman, Harvey M.  (PDF). Ohio State University. 1 June 2000  [8 August 2017]. （原始内容存档 (PDF)于2016-10-18）.
10. ↑  Friedman, Harvey M.  (PDF). Ohio State University Department of Mathematics: 5, 48 (Thm.6.8). 8 October 1998  [8 August 2017]. （原始内容存档 (PDF)于2017-08-09）.

Bibliography
* Friedman, Harvey M. Lect. Notes Log. **15**, Urbana, IL: Assoc. Symbol. Logic: 60–91, 2002, MR 1943303
* Gallier, Jean H.  (PDF), Ann. Pure Appl. Logic, 1991, **53** (3): 199–260  [2022-11-07], MR 1129778, doi:10.1016/0168-0072(91)90022-E 可免费查阅, （原始内容存档 (PDF)于2023-02-03）
* Kruskal, J. B.  (PDF), Transactions of the American Mathematical Society (American Mathematical Society), May 1960, **95** (2): 210–225  [2022-11-07], JSTOR 1993287, MR 0111704, doi:10.2307/1993287, （原始内容存档 (PDF)于2021-10-21）
* Marcone, Alberto.  (PDF). Reverse Mathematics. 2001, **21**: 303–330  [2022-11-07]. （原始内容存档 (PDF)于2022-05-08）.
* Nash-Williams, C. St.J. A. Proc. Camb. Phil. Soc. 1963, **59** (4): 833–835, Bibcode:1963PCPS...59..833N, MR 0153601, S2CID 251095188, doi:10.1017/S0305004100003844
* Rathjen, Michael; Weiermann, Andreas.. Annals of Pure and Applied Logic. 1993, **60** (1): 49–88. doi:10.1016/0168-0072(93)90192-g 可免费查阅.
* Simpson, Stephen G. Harrington, L. A.; Morley, M.; Scedrov, A.; et al (编), Studies in Logic and the Foundations of Mathematics, North-Holland: 87–117, 1985
* Smith, Rick L. Harrington, L. A.; Morley, M.; Scedrov, A.; et al (编), Studies in Logic and the Foundations of Mathematics, North-Holland: 119–136, 1985

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