Kruskal's tree theorem

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). A finitary application of the theorem gives the existence of the fast-growing TREE function.

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 < j so that T_i ≤ T_j.)

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 order one), Friedman found that the result was unprovable in ATR₀,[1] thus giving the first example of a predicative result with a provably impredicative proof.[2] This case of the theorem is still provable by Π¹
₁-CA₀, but by adding a "gap condition"[3] to the definition of the order on trees above, he found a natural variation of the theorem unprovable in this system.[4][5] Much later, the Robertson–Seymour theorem would give another theorem unprovable by Π¹
₁-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).

Suppose that P(n) is the statement:

There is some m such that if T₁,...,T_m is a finite sequence of unlabeled rooted trees where T_i has i + n 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 Kőnig's lemma. For each n, Peano arithmetic can prove that P(n) is true, but Peano arithmetic cannot prove the statement "P(n) is true for all n".[6] 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 or the Ackermann function for example. The least m for which P(n) holds similarly grows extremely quickly with n.

Define tree(n), the weak tree function, as the largest m so that we have the following:

There is a sequence T₁,...,T_m of unlabeled rooted trees, where each T_i has at most i + n vertices, such that T_i ≤ T_j does not hold for any i < j  ≤ m.

It is known that tree(1) = 2, tree(2) = 5, and tree(3) ≥ 844424930131960, tree(4) > Graham's number (by a lot) but TREE(3) (where the argument specifies the number of labels; see below) is larger than ${\displaystyle \mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{8}(7)}(7)}(7)}(7)}(7)}$.

To differentiate the two functions, TREE with all letters capitalized is the big TREE function; tree with all letters in lowercase is the weak tree function.

A sequence of rooted trees labelled from a set of 3 labels (blue < red < green). The n^th tree in the sequence contains at most n vertices, and no tree is inf-embeddable within any later tree in the sequence. TREE(3) is defined to be the longest possible length of such a sequence.

By incorporating labels, Friedman defined a far faster-growing function.[7] For a positive integer n, take TREE(n)^[a] to be the largest m so that we have the following:

There is a sequence T₁,...,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 that is so big that many other "large" combinatorial constants, such as Friedman's n(4), nⁿ⁽⁵⁾(5), and Graham's number,^[b] are extremely small by comparison. A lower bound for n(4), and hence an extremely weak lower bound for TREE(3), is A^A(187196)(1).^[c][8] Graham's number, for example, is much smaller than the lower bound A^A(187196)(1). It is approximately ${\displaystyle g_{3\uparrow ^{187196}3}}$, where g_x is Graham's function.

• Paris–Harrington theorem
• Kanamori–McAloon theorem
• Robertson–Seymour theorem

^{^ a} Friedman originally denoted this function by TR[n].

^{^ b} 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,...,x_2i is a subsequence of any later block x_j,...,x_2j.[9] ${\displaystyle n(1)=3,n(2)=11,\,{\textrm {and}}\,n(3)>2\uparrow ^{7197}158386}$.

^{^ c} 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)).

Citations

Bibliography

• Friedman, Harvey M. (2002), Internal finite tree embeddings. Reflections on the foundations of mathematics (Stanford, CA, 1998), Lect. Notes Log., vol. 15, Urbana, IL: Assoc. Symbol. Logic, pp. 60–91, MR 1943303
• Gallier, Jean H. (1991), "What's so special about Kruskal's theorem and the ordinal Γ₀? A survey of some results in proof theory" (PDF), Ann. Pure Appl. Logic, 53 (3): 199–260, doi:10.1016/0168-0072(91)90022-E, MR 1129778
• Kruskal, J. B. (May 1960), "Well-quasi-ordering, the tree theorem, and Vazsonyi's conjecture" (PDF), Transactions of the American Mathematical Society, American Mathematical Society, 95 (2): 210–225, doi:10.2307/1993287, JSTOR 1993287, MR 0111704
• Marcone, Alberto (2001). "Wqo and bqo theory in subsystems of second order arithmetic" (PDF). Reverse Mathematics. 21: 303–330.
• Nash-Williams, C. St.J. A. (1963), "On well-quasi-ordering finite trees", Proc. Camb. Phil. Soc., 59 (4): 833–835, Bibcode:1963PCPS...59..833N, doi:10.1017/S0305004100003844, MR 0153601, S2CID 251095188
• Rathjen, Michael; Weiermann, Andreas (1993). "Proof-theoretic investigations on Kruskal's theorem". Annals of Pure and Applied Logic. 60 (1): 49–88. doi:10.1016/0168-0072(93)90192-g.
• Simpson, Stephen G. (1985), "Nonprovability of certain combinatorial properties of finite trees", in Harrington, L. A.; Morley, M.; Scedrov, A.; et al. (eds.), Harvey Friedman's Research on the Foundations of Mathematics, Studies in Logic and the Foundations of Mathematics, North-Holland, pp. 87–117
• Smith, Rick L. (1985), "The consistency strengths of some finite forms of the Higman and Kruskal theorems", in Harrington, L. A.; Morley, M.; Scedrov, A.; et al. (eds.), Harvey Friedman's Research on the Foundations of Mathematics, Studies in Logic and the Foundations of Mathematics, North-Holland, pp. 119–136