# Pardon my ignorance, but isn't TREE(3) a finite number?

Pardon my ignorance, but isn't TREE(3) a finite number? -Dylan Thurston

It is my understanding as well that TREE(3) is finite (Proof that TREE(n) where n >= 3 is finite?).

However, I have seen statements such as:

TREE(3) is known to exceed the $$\Gamma_0$$-level, which is much higher than the $$\epsilon_0$$-level. -Source

It is also my understanding that $$\omega < \epsilon_0 < \Gamma_0$$ (where $$\omega$$, $$\epsilon_0$$ & $$\Gamma_0$$ are transfinite).

If $$\Gamma_0, wouldn't that imply TREE(3) is transfinite?

• I think this refers not to the size of $\operatorname{TREE}(3)$ itself, but of the power of the formal system required to show that the expression “$\operatorname{TREE}(3)$” is meaningfully defined. Recall that the definition of $\operatorname{TREE}$ begins “the maximum value such that…”. In general there may not be such a maximum value; its existence requires a proof, typically an inductive proof. Some axiomatic systems are only strong enough to provide induction over a set of size $\epsilon_0$; others, more powerful, can induct over larger sets. – MJD Dec 18 '18 at 22:14
• I have no idea! – MJD Dec 19 '18 at 0:29
• The claim is made in the context of a specific fast growing hierarchy of functions $f_\alpha\!:\mathbb N\to\mathbb N$. I assume that what they mean is that $f_{\Gamma_0}(3)<\mathrm{TREE}(3)$. – Andrés E. Caicedo Dec 19 '18 at 2:05
• (The question on MO linked to at the beginning explains this for a different $\alpha$.) – Andrés E. Caicedo Dec 19 '18 at 2:09
• @AndrésE.Caicedo I think your comments are very nearly the answer to the question, so feel free to expand them to an answer. – Mark S. Dec 19 '18 at 2:42

This is indeed a confusing bit of language. $$TREE(3)$$ is indeed finite. I think the later comment by Peter clears things up:
"For example, Graham's number is approximately $$f_{\omega+1}(64)$$, so we say that Graham's number is at level $$f_{\omega+1}$$, or [for] short at the $$\omega+1$$-level."
Basically, we want to say that a number is "at level $$\kappa$$" if it is $$f_\kappa(s)$$ for some "small" $$s$$ (e.g. here $$64$$ is considered small). Note that this is a subjective distinction - what's the least non-small number? :P - but it still gives a sense of the size of the number involved. The point, roughly, is to say: $$TREE(3)$$ is so huge that, even with a special symbol for $$f_{\Gamma_0}$$, it is still infeasible to express $$TREE(3)$$.
There are various precise versions of this - e.g. the statement "$$f_{\Gamma_0}(3)" (per Andres) is a perfectly precise statement, and (I believe) is known to be true - but I think it's better to view this as a general piece of descriptive language