Kruskal's tree theorem

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.

A finitary application of the theorem gives the existence of the fast-growing TREE function. TREE(3) is largely accepted to be one of the largest simply defined finite numbers, dwarfing other large numbers such as Graham's number and googolplex.^[1]

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 ${\text{TREE}}(3)$ .

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 $T 1$ , $T 2$ are rooted trees with vertices labeled in $X$ , we say that $T 1$ is inf-embeddable in $T 2$ and write $T_{1}\leq T_{2}$ 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, 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 $T 1, T 2, \dots$ of rooted trees labeled in $X$ , there is some $i<j$ so that $T_{i}\leq T_{j}$ .)

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,^[2] thus giving the first example of a predicative result with a provably impredicative proof.^[3] This case of the theorem is still provable by Π¹
₁-CA₀, but by adding a "gap condition"^[4] to the definition of the order on trees above, he found a natural variation of the theorem unprovable in this system.^[5]^[6] 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).^[7]

Weak tree function

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 i

has

i+n

vertices, then

T_{i}\leq 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 $n$ ".^[8] 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.^{[citation needed]} The least $m$ for which $P(n)$ holds similarly grows extremely quickly with $n$ .

TREE function

By incorporating labels, Friedman defined a far faster-growing function.^[9] For a positive integer $n$ , take ${\text{TREE}}(n)$ ^[a] 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}\leq T_{j}

does not hold for any

i<j\leq m

.

The TREE sequence begins ${\text{TREE}}(1)=1$ , ${\text{TREE}}(2)=3$ , before ${\text{TREE}}(3)$ suddenly explodes to a value so large that many other "large" combinatorial constants, such as Friedman's $n(4)$ , $n^{n(5)}(5)$ , and Graham's number,^[b] are extremely small by comparison. A lower bound for $n(4)$ , and, hence, an extremely weak lower bound for ${\text{TREE}}(3)$ , is $A^{A(187196)}(1)$ .^[c]^[10] Graham's number, for example, is much smaller than the lower bound $A^{A(187196)}(1)$ , which is approximately $g_{3\uparrow ^{187196}3}$ , where $g_{x}$ is Graham's function.

Notes

^{^ 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.^[11]

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)).

References

Citations

^ "The Enormity of the Number TREE(3) Is Beyond Comprehension". Popular Mechanics. 20 October 2017. Retrieved 4 February 2025.
^ Simpson 1985, Theorem 1.8
^ Friedman 2002, p. 60
^ Simpson 1985, Definition 4.1
^ Simpson 1985, Theorem 5.14
^ Marcone 2005, pp. 8–9
^ Rathjen & Weiermann 1993.
^ Smith 1985, p. 120
^ Friedman, Harvey (28 March 2006). "273:Sigma01/optimal/size". Ohio State University Department of Maths. Retrieved 8 August 2017.
^ Friedman, Harvey M. (1 June 2000). "Enormous Integers In Real Life" (PDF). Ohio State University. Retrieved 8 August 2017.
^ Friedman, Harvey M. (8 October 1998). "Long Finite Sequences" (PDF). Ohio State University Department of Mathematics. pp. 5, 48 (Thm.6.8). Retrieved 8 August 2017.

Bibliography

Friedman, Harvey M. (2002). "Internal finite tree embeddings". In Sieg, Wilfried; Feferman, Solomon (eds.). Reflections on the foundations of mathematics: essays in honor of Solomon Feferman. Lecture notes in logic. Vol. 15. Natick, Mass: AK Peters. pp. 60–91. ISBN 978-1-56881-170-3. MR 1943303.
H. Gallier, Jean (September 1991). "What's so special about Kruskal's theorem and the ordinal Γ₀? A survey of some results in proof theory" (PDF). Annals of Pure and Applied 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. 95 (2). American Mathematical Society: 210–225. doi:10.2307/1993287. JSTOR 1993287. MR 0111704.
Marcone, Alberto (2005). Simpson, Stephen G. (ed.). "WQO and BQO theory in subsystems of second order arithmetic" (PDF). Reverse Mathematics. Lecture Notes in Logic. 21. Cambridge: Cambridge University Press: 303–330. doi:10.1017/9781316755846.020. ISBN 978-1-316-75584-6.
Nash-Williams, C. St. J. A. (October 1963). "On well-quasi-ordering finite trees" (PDF). Mathematical Proceedings of the Cambridge Philosophical Society. 59 (4): 833–835. Bibcode:1963PCPS...59..833N. doi:10.1017/S0305004100003844. ISSN 0305-0041. MR 0153601. S2CID 251095188.
Rathjen, Michael; Weiermann, Andreas (February 1993). "Proof-theoretic investigations on Kruskal's theorem" (PDF). Annals of Pure and Applied Logic. 60 (1): 49–88. doi:10.1016/0168-0072(93)90192-G. MR 1212407.
Simpson, Stephen G. (1985). "Nonprovability of certain combinatorial properties of finite trees". In Friedman, Harvey; Harrington, L. A.; Scedrov, A.; et al. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Amsterdam ; New York: North-Holland. pp. 87–117. ISBN 978-0-444-87834-2.
Smith, Rick L. (1985). "The Consistency Strengths of Some Finite Forms of the Higman and Kruskal Theorems". In Friedman, Harvey; Harrington, L. A. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Vol. 117. Amsterdam ; New York: North-Holland. pp. 119–136. doi:10.1016/s0049-237x(09)70157-0. ISBN 978-0-444-87834-2.

[1] "The Enormity of the Number TREE(3) Is Beyond Comprehension". Popular Mechanics. 20 October 2017. Retrieved 4 February 2025.

[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 2005, pp. 8–9

[FOOTNOTERathjenWeiermann1993-7] Rathjen & Weiermann 1993.

[SmiA-8] Smith 1985, p. 120

[9] Friedman, Harvey (28 March 2006). "273:Sigma01/optimal/size". Ohio State University Department of Maths. Retrieved 8 August 2017.

[10] Friedman, Harvey M. (1 June 2000). "Enormous Integers In Real Life" (PDF). Ohio State University. Retrieved 8 August 2017.

[11] Friedman, Harvey M. (8 October 1998). "Long Finite Sequences" (PDF). Ohio State University Department of Mathematics. pp. 5, 48 (Thm.6.8). Retrieved 8 August 2017.

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