You're not thinking about this properly. There are some facts about finite objects which simply cannot be proved without reasoning about infinite objects - for example, the well-definedness of the TREE function. These facts are true if you have access to infinite objects, but not necessarily if you don't.
This paper shows that a certain class of statements about finite objects, which we knew were true by virtue of reasoning about a certain infinite object, in fact remain true if you're restricted only to reasoning about finite objects.
I tried to give my reasons by using only appeals to objects of our common understanding. You have countered my claim by appealing to "the well-definedness of the TREE function". Do you expect me (and everyone else) to know what that is? In order for us to be able to follow your argument you would need to spell that out in a bit more detail.
If I'm not thinking about this properly, which of the assertions I made was incorrect? Where is the error in my reasoning? Is there an error in how I am conceiving things?
> This paper shows that a certain class of statements about finite objects
From what I can gather, it's not that simple. For a start, the initial set is the set of natural numbers. This is not finite. The procedure for generating/enumerating them is. The paper deals with pairs of inequalities based on the natural numbers and sub-sequences to be found therein. This set of pairs is also non-finite (but I'm happy with the assertion that in some sense it is a different order of infinity from the natural numbers). We are now trying to reason about the nature of the sequencing of these sub-sequences.
Are you saying that sometimes the sub-sequences are finite? If so, their complement would be infinite. And it is the partitioning that makes this so.
Your understanding of the point of the theorem is very different to mine, and I'm moderately sure my understanding is pretty close to correct.
It is a fact of mathematics that there are some statements which are solely about finite objects, but to prove them requires reasoning about an infinite object. For a more accessible example than TREE, I think the Ackermann function falls into this category. The Ackermann function A(n+1, m+1) = A(n, A(n+1, m)) is well-defined for all n and m (we prove this by induction over NxN), but the proof relies on considering the lexicographic order on NxN which is inherently infinite. (I'm not totally certain that all proofs of Ackermann's well-definedness rely on an infinite object, but the only proof known to me does.) Ackermann's function itself is in some sense a "finite" object, but the proof of its well-definedness is in some sense "infinite". Whatever the status of my conjecture that "you can't prove that Ackermann's function is well-defined without considering an infinite object", it is certainly a fact that Ackermann is not primitive-recursive, and "primitive-recursive functions" corresponds to the lowest level of the five "mysterious levels" the article talks about.
So the analogy is as follows. Imagine that we knew of this "infinitary" proof that Ackermann is well-defined, but we hadn't proved that no "finitary" proof exists. (So finitists are not happy to use Ackermann, because it might not actually be well-defined according to them: any known proof requires dealing with an infinite object.) Now, this paper comes along and proves that actually a finitary proof exists. Suddenly the finitists are happy to use the Ackermann function.
The actual definition of TREE is a bit too long for me to explain here, but it is an example of a function like Ackermann, which is well-defined, but in fact if you're not allowed to consider infinite objects during the proof then it is provably impossible to prove that TREE is well-defined. So the statement "TREE is well-defined" is, in some sense, "less constructive" or "more infinitary" than R_2^2.
This paper shows that a certain class of statements about finite objects, which we knew were true by virtue of reasoning about a certain infinite object, in fact remain true if you're restricted only to reasoning about finite objects.