Kruskal's Tree Theorem - Friedman's Finite Form

Friedman's Finite Form

Friedman (2002) observed that Kruskal's tree theorem has special cases that can be stated but not proved in first-order arithmetic (though they can easily be proved in second-order arithmetic). Another similar statement is the Paris–Harrington theorem, but Friedman's finite form of Kruskal's theorem needs a much stronger fragment of second-order arithmetic to prove than the Paris-Harrington principle.

Suppose that P(n) is the statement

There is some m such that if T₁,...,T_m is a finite sequence of trees where T_k has k+n vertices, then T_i ≤ T_j for some i < j.

This is essentially a special case of Kruskal's theorem, where the size of the first tree is specified, and the trees are constrained to grow in size at the simplest non-trivial growth rate. 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". Moreover 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.

Friedman also proved the following finite form of Kruskal's theorem for labelled trees with no order among siblings, parameterising on the size of the set of labels rather than on the size of the first tree in the sequence (and the homeomorphic embedding, ≤, now being inf- and label-preserving):

For every n, there is an m so large that if T₁,...,T_m is a finite sequence of trees with vertices labelled from a set of n labels, where each T_i has at most i vertices, then T_i ≤ T_j for some i < j.

The latter theorem ensures the existence of a rapidly growing function that Friedman called TREE, such that TREE(n) is the length of a longest sequence of n-labelled trees T₁,...,T_m in which each T_i has at most i vertices, and no tree is embeddable into a later tree.

The TREE sequence begins TREE(1) = 1, TREE(2) = 3, then suddenly TREE(3) explodes to a value so enormously large that many other "large" combinatorial constants, such as Friedman's n(4), are extremely small by comparison. A lower bound for n(4), and hence an extremely weak lower bound for TREE(3), is A(A(...A(1)...)), where the number of A's is A(187196), and A is a version of Ackermann's function: A(x) = 2↑x-1x in Knuth's up-arrow notation. Graham's number, for example, is approximately A64(4) which is much smaller than the lower bound AA(187196)(1). It can be shown that the growth-rate of the function TREE exceeds that of the function f_Γ0 in the fast-growing hierarchy, where Γ₀ is the Feferman–Schütte ordinal.

The ordinal measuring the strength of Kruskal's theorem is the small Veblen ordinal (sometimes confused with the smaller Ackermann ordinal).