TREE 函式、Kruskal 樹定理是逆數學的突出示例。由推測並由約瑟夫克魯斯卡爾證明。 In Kruskals tree theorem states that the set of finite s over a set of labels is itself wellquasiordered under embedding 在數學中，克魯斯卡樹定理指出，在準有序（英文：wellquasiordering）標籤集上的有限樹集合本身在同胚下也是良準有序的（英文：同胚（圖論））嵌入。 歷史 該定理由安德魯瓦茲索尼（Andrew Vázsonyi）猜想並由約瑟夫克魯斯卡爾（Joseph Kruskal，1960）證明； Crispin NashWilliams（1963）給了一個簡短的證明。它從此成為逆向數學（英語：reverse math）中的一個突出例子，作為在 ATR0 內無法證明的陳述（算術超限遞迴（英語：arithmetic transfinite recursion）的一種形式）和有限（英語：finitary）應用該定理給出了快速增長的 TREE 函式的存在性。 2004 年，這個結果被從樹推廣到圖，成為羅伯遜 西摩定理（英語：RobertsonSeymour theorem），這一結果在逆向數學中也被證明很重要，並導致了增長更快的 SSCG 函式（英語： Friedman 的 SSCG 函式）。 陳述 這裡給出的版本是由 NashWilliams 證明的；克魯斯卡爾的表述要強一些。我們認為的所有樹木都是有限的。 給定一棵有根的樹 T，並給定頂點 v、w，如果從根到 w 的唯一路徑包含 v，則稱 w 為 v 的後繼；如果從 v 到 w 的路徑另外不包含 v，則稱 w 為 v 的直接後繼其他頂點。 取 X 為偏序集（中文：partiallyorderedset）。如果 T1、T2 是頂點標記為 X 的有根樹，我們說 T1 可 infembeddable 在 T2 中，並且如果存在從 T1 的頂點到 T2 的頂點的單射對映 F，則寫 T1 T2，使得 弗裡德曼的工作 對於可數標籤集 X X 克魯斯卡樹定理可以用二階算術來表達和證明（中文：二階算術）。然而，就像古德斯坦定理或巴黎 哈靈頓定理（英語：ParisHarrington theorem）一樣，該定理的一些特殊情況和變體可以用比可以證明它們的子系統弱得多的二階算術子系統來表達。這是哈維弗裡德曼（Harvey Friedman）在 20 世紀 80 年代初首次觀察到的，這是當時新興的逆向數學領域的早期成功。在上面的樹被認為是未標記的情況下（即，在 X X 有順序一），Friedman 發現結果在 ATR0（英語：算術超限遞迴）2 從而給出了第一個謂詞（英語：impredicativity）結果的例子，並帶有可證明的謂詞證明3 此定理的這種情況仍可由 Π1 證明 1CA0，但透過在上述樹的順序定義中新增「間隙條件」4，他發現該定理的自然變體在該系統中無法證明。56 很久以後，羅伯森 西摩定理給了另一個無法由 Π1 證明的定理 1CA0。 序數分析（中文：序數分析）證實了克魯斯卡爾定理的強度，該定理的證明論序數等於小維勃倫序數（英文：小維勃倫序數）（有時與較小的阿克曼序數（英文：阿克曼序數）相混淆） 來源請求 樹 3 假設 P n 是這樣的語句： 存在一些 m，如果 T1Tm 是未標記有根樹的有限序列，其中 Tk 有 nk 個頂點，則對於某些 i j，Ti Tj。 根據克魯斯卡定理和柯尼格引理，所有陳述 P n 都是正確的。對於每個 n，皮亞諾算術（英語：Peano axioms）可以證明 P n 為真，但皮亞諾算術不能證明 P n 對於所有 n 都為真 這一命題。7 此外，皮亞諾算術中 P n 的最短證明的長度作為 n 的函式增長得非常快，遠遠快於任何原始遞迴函式（英語：primitive recursive function）或阿克曼函式。 P n 所適用的最小 m 同樣隨 n 增長得非常快。 透過合併標籤，弗裡德曼定義了一個成長速度更快的函式。8 對於正整數 n，取 TREE n 為最大的 m，這樣我們有： 有一個由 n 個標籤集標記的有根樹序列 T1Tm，其中每個 Ti 最多有 i 個頂點，使得 Ti Tj 對於任何 i j m 不成立。 TREE 序列開始為 TREE 1 1 TREE 2 3，然後突然 TREE 3 爆炸到非常大的值，以至於許多其他「大」組合常數，例如 Friedman 的 n 4 相較之下， 非常小。事實上，它比 nn 55 大得多。 n 4 的下界（因此 TREE 3 的下界極弱）為 AA 18719619，其中 A x 採用一個引數，定義為 A x x，其中 A k n 有兩個引數，是阿克曼函式（英語：Ackermanns function）的一個特定版本，定義為： A 1 n 2n A k1 1 A k 1 A k1 n1 A k A k1 n。例如，葛立恆數遠小於下限 AA 1871961。可以證明函式 TREE 的成長率至少為 F θ （ Ω ω ω ） displaystyle f_theta Omega omegaomega 中的快速成長層次結構（中文：快速成長層次結構）。 AA 1871961 大約為 G 3 187196 3 displaystyle g_3uparrow 187196 3，其中 gx 是格雷厄姆函式（中文：格雷厄姆數）。 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 fastgrowing 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 evenfastergrowing Statement The version given here is that proven by NashWilliams Kruskals 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 infembeddable in and write if there is an injective map from the vertices of to the vertices of such that For all vertices of the label of precedes the label of If is any successor of in then is a successor of and If are any two distinct immediate successors of then the path from to in contains Kruskals tree theorem then states If is then the set of rooted trees with labels in is wellquasiordered under the infembeddable 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 1labelled trees ie such that The tree at position in the sequence has no more than vertices for all 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 but where the argument specifies the number of labels see is larger than mathrm treemathrm treemathrm treemathrm treemathrm tree877777 Friedmans work For a countable label set X Kruskals 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 secondorder 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 thennascent field of reverse mathematics In the case where the trees above are taken to be unlabeled that is in the case where X has order one Friedman found that the result was unprovable in Simpson 1985 Theorem 18 thus giving the first example of a result with a provably impredicative proofFriedman 2002 p 60 This case of the theorem is still provable by ΠCA0 but by adding a gap conditionSimpson 1985 Definition 41 to the definition of the order on trees above he found a natural variation of the theorem unprovable in this systemSimpson 1985 Theorem 514Marcone 2001 p 89 Much later the RobertsonSeymour theorem would give another theorem unprovable by ΠCA0 confirms the strength of Kruskals theorem with the prooftheoretic ordinal of the theorem equaling the sometimes confused with the smaller TREE 3 Suppose that P n is the statement There is some m such that if T1Tm is a finite sequence of unlabeled rooted trees where Tk has nk vertices then Ti Tj for some i j All the statements P n are true as a consequence of Kruskals 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 nSmith 1985 p 120 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 阿克曼函式 for example The least m for which P n holds similarly grows extremely quickly with n By incorporating labels Friedman defined a far fastergrowing function For a positive integer n take TREE n to be the largest m so that we have the following There is a sequence T1Tm 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 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 Friedmans n 4 are extremely small by comparison In fact it is much larger than nn 55 A lower bound for n 4 and hence an extremely weak lower bound for TREE 3 is AA 1871961 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 2n A k1 1 A k 1 A k1 n1 A k A k1 n 葛立恆數，for example is much smaller than the lower bound AA 1871961 It can be shown that the growthrate of the function TREE is at least f_theta Omegaomega omega in the AA 1871961 is approximately g_3 uparrow187196 3 where gx is 參見 註釋 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 kletter alphabet such that no block of letters xix2i is a subsequence of any later block xjx2j n 1 3 n 2 11 textrm and n 3 2 uparrow7197 158386 Category 數理邏輯 Category 離散數學定理 Category 序理論 Category 良基性 參考 Citations Bibliography