Friedman's finite ordered tree problem

The finite ordered tree problem was researched by Harvey Friedman.

Friedman defines an ordered tree as a triple (V,≤,<') where (V,≤) is a finite poset with a least element (root) in which the set of predecessors under ≤ of each vertex is linearly ordered by ≤, and where for each vertex, <' is a strict linear ordering on its immediate successors. He also defines the following:

• Vertex x ≤* y if and only if x is to the left of y, or if x ≤ y.
• d(v) is the position of v in counting from 1.

He then defines T[k] to be the tree of height k such that every vertex v of height ≤k - 1 has exactly d(v) children, and |T[k]| to be number of children.

Friedman has proven that |T[k]| has a similar growth rate to that of the Ackermann function. The first few values are as follows:

• |T[0]| = 1
• |T[1]| = 2
• |T[2]| = 4
• |T[3]| = 14
• |T[4]| = 82*2³⁸-2
• |T[5]| > 2↑↑2²⁹⁵

References

• Friedman, Harvey. Enormous Integers in Real Life

See also

Googology by Harvey Friedmann

Main article: Harvey Friedman

Mythical tree problem · Friedman's vector reduction problem · Friedman's finite ordered tree problem · block subsequence theorem n(4) · Friedman's circle theorem · TREE sequence TREE(3) · subcubic graph number SCG(13) · transcendental integer · finite promise games · Friedman's finite trees · Greedy clique sequence · Bop-counting function