TITLE: Could Kronecker accept a proof of Goodstein's theorem?
QUESTION [11 upvotes]: A famous result of Goodstein asserts that the Goodstein sequence of integers terminates. 
For a precise statement and a short proof, see https://en.wikipedia.org/wiki/Goodstein%27s_theorem.
A well known proof of this theorem uses ordinal number. The idea is to associate with the Goodstein sequence a sequence of ordinal numbers that is decreasing and every decreasing sequence of ordinal numbers terminates after a finite number of steps. 
A well-known result of Kirby and Paris asserts that Goodstein's theorem cannot be proved in the Peano arithmetic. My vague understanding is (correct if I am wrong) that any proof of the Goodstein theorem would require a proof of existence of the ordinal number
$$
\large \omega^{\omega^{\omega^{\omega^{\omega^\ldots}}}}
$$
and such a large number cannot be constructed in the Peano arithmetic.
I believe this result is of wide interest in the mathematical community. While there have been a lot of discussions related to this theorem on Mathoverflow, the theorem is set deeply in foundations of mathematics and most of the discussions are so hermetic that only people with good understanding of the foundations of mathematics can follow it. I have some working knowledge in the area (as most of us do), but that restricts to a basic understanding of ZFC, practical use of the axiom of choice, and some understanding of basic results in arithmetic of ordinal numbers. With such a limited knowledge I am not able to follow the discussions.
Georg Cantor created foundations of the modern set theory. However, his theory encountered a strong resistance from prominent mathematicians including Kronecker, Poincaré, Weyl and Brouwer. Kronecker described Cantor as a "scientific charlatan", a "renegade" and a "corrupter of youth".
Without the set theory created by Cantor, the proof of Goodstein's theorem based on the arithmetic of ordinal numbers would not exist and hence the known proof would not be accepted by Kronecker, Poincaré, Weyl or Brouwer.
Therefore my question is:

Question. Is it possible to prove Goodstein's theorem in an "elementary" way so that it would be accepted by Kronecker?

I have heard that there are "elementary", but rather long proofs using second-order arithmetic (I do not even know what it means).

Question. Where can I find a proof that does not involve ordinal numbers?

Since I find these questions of great interest for everyone, I would appreciate a detailed and "elementary" explanation. For example while referring to the second-order arithmetic I would appreciate if someone could explain in details what it really means by showing some examples.
Edit: I replaced Gauss with Kronecker in the question. It seems a better choice.

REPLY [9 votes]: As a starting point, I recommend that you study the proof of Theorem 1 in my expository article on The Consistency of Arithmetic. (This is not exactly a proof of Goodstein's theorem, but for the purposes of the present discussion I think it's close enough. Start reading on page 25 where it says "Ordinals below $\epsilon_0$.")
I have tried to make this proof as "pedestrian" as possible, avoiding concepts from logic and set theory that might be unfamiliar to the "working mathematician."  In particular, my contention is that if you were to present this proof in (say) an undergraduate analysis class, it would seem completely unobjectionable and unremarkable.  True, the proof assumes that it is meaningful to talk about infinite sequences, but infinite sequences are routinely bandied about in undergraduate analysis without comment.  Also, the induction argument is trickier than most induction arguments, but I don't think that anyone would find anything suspicious about it unless they had been specifically told to look for something suspicious.
Sometimes the terms "PA proof" and "elementary proof" are used interchangeably.  I think that this practice can lead to confusion.  The class of proofs that are formalizable in PA is of course a very clearly defined class, but it does not correspond very closely to what we intuitively think of as "elementary" proofs.  In particular, the $\epsilon_0$ upper bound is, from the point of view of ordinary mathematical practice, a rather artificial bound.  For example, we know from reverse mathematics that in a certain precise sense, the Bolzano–Weierstrass theorem goes beyond PA, but in our undergraduate analysis classes we don't set out flares or start blaring emergency sirens when proving Bolzano–Weierstrass.  The exact logical strength of various assumptions that are used in ordinary mathematics is something that is not apparent on the surface, and emerges only after a careful logical analysis.
Conversely, it's possible to encode highly sophisticated concepts and proofs in PA—concepts that most mathematicians would instinctively regard as being "not at all elementary."  EDIT (in light of the comments): For example, sometimes people ask if Fermat's Last Theorem can be proved in PA.  I think that most of them are implicitly trying to ask whether there is an elementary proof of FLT.  But as Angus Macintyre has shown (see comments below), it is very likely that all the high-falutin' super-fancy "non-elementary" machinery used in the known proof of FLT can be sufficiently well approximated in PA to push the proof through.  If Macintyre is right, which I think he is, then this would show that FLT is provable in PA, while leaving completely unanswered the question of whether there is an "elementary" proof of FLT.
Returning to the question of Gauss or Kronecker, it's a little tough to guess what they would have thought about these things.  They might have affirmed that they believed in "potential infinity" and not "actual infinity," but most modern mathematicians regard these two terms as philosophical gobbledygook, making it hard to translate Gauss's beliefs directly into present-day terms.  We can probably safely say that they would have balked at ZFC, in particular the axiom of infinity, but the axiom of infinity isn't absolutely necessary for Goodstein's theorem.
Indeed, Gentzen, in his proof of the consistency of PA, made an effort to argue that the reasoning was "finitary."  Since "finitary" isn't a completely precise term either, it's hard to be sure that Gauss would have agreed with Gentzen, but it's at least plausible that he would have.  Ultimately, like all historical counterfactuals, it's a matter of speculation what Gauss would have thought.  But hopefully I've convinced you that there isn't anything more mysterious about Goodstein's theorem than many other routine mathematical theorems.