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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 Joseph Kruskal (1960); a short proof was given by Crispin Nash-Williams (1963). It has since become a prominent example in reverse mathematics as a statement that cannot be proved in ATR0 (a second-order arithmetic theory with a form of arithmetical transfinite recursion). 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, which dwarfs TREE ( 3 ) {\displaystyle {\text{TREE}}(3)} . A finitary application of the theorem gives the existence of the fast-growing TREE 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 T 1 ≤ T 2 {\displaystyle T_{1}\leq T_{2}} 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 ) {\displaystyle F(v)} ; If w is any successor of v in T1, then F ( w ) {\displaystyle F(w)} is a successor of F ( v ) {\displaystyle F(v)} ; and If w1, w2 are any two distinct immediate successors of v, then the path from F ( w 1 ) {\displaystyle F(w_{1})} to F ( w 2 ) {\displaystyle F(w_{2})} in T2 contains F ( v ) {\displaystyle 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 < j {\displaystyle i<j} so that T i ≤ T j {\displaystyle T_{i}\leq T_{j}} .) Friedman's work For a countable label set X, Kruskal's tree theorem can be expressed and proven using second-order arithmetic. However, like Goodstein's theorem 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 X has size 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 Π11-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 Π11-CA0. Ordinal analysis 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). Weak tree function Suppose that P ( n ) {\displaystyle P(n)} is the statement: There is some m such that if T1, ..., Tm is a finite sequence of unlabeled rooted trees where Ti has i + n {\displaystyle i+n} vertices, then T i ≤ T j {\displaystyle T_{i}\leq T_{j}} for some i < j {\displaystyle i<j} . All the statements P ( n ) {\displaystyle P(n)} are true as a consequence of Kruskal's theorem and Kőnig's lemma. For each n, Peano arithmetic can prove that P ( n ) {\displaystyle P(n)} is true, but Peano arithmetic cannot prove the statement " P ( n ) {\displaystyle P(n)} is true for all n". Moreover, the length of the shortest proof of P ( n ) {\displaystyle P(n)} in Peano arithmetic grows phenomenally fast as a function of n, far faster than any primitive recursive function or the Ackermann function, for example. The least m for which P ( n ) {\displaystyle P(n)} holds similarly grows extremely quickly with n. Define tree ( n ) {\displaystyle {\text{tree}}(n)} , the weak tree function, as the largest m so that we have the following: There is a sequence T1, ..., Tm of unlabeled rooted trees, where each Ti has at most i + n {\displaystyle i+n} vertices, such that T i ≤ T j {\displaystyle T_{i}\leq T_{j}} does not hold for any i < j ≤ m {\displaystyle i<j\leq m} . It is known that tree ( 1 .... Discover the Andrew Harvey Seymour Bernstein popular books. Find the top 100 most popular Andrew Harvey Seymour Bernstein books.