Reading online, it generally seems accepted that TREE(n) where n >= 3 is a finite number, but large enough to be incomputable and only has extremely loose lower bounds today. TREE(n) is the function defined by Harvey Friedman, based on Joseph Kruskal's tree theorem. A simplified definition:
"TREE(n) = the maximum length of a set of rooted trees Tm with n possible vertex labels, where Tm has m or fewer vertices, and subsequent trees do not homeomorphically embed a preceding tree."
We can trivially show that TREE(1) = 1 and TREE(2) = 3.
My question is, how can it be known that for values of n greater than 2, the series of possible trees is finite? If we can't compute the solution, how do we know that it ends? If there is an existing write up online describing proof that these numbers are finite, please link me to it, I'm finding limited reading material on the TREE function. Thanks.