In his expository article, "The Consistency of Arithmetic," Professor Chow has the following theorems:

Theorem 1. yes $ a_1 $, $ a_2 $, $ a_3 $, … is a sequence of ordinals and $ a_i $ $ ge $ $ a_j $ when $ i $ $ lt $ $ j $, then the sequence stabilizes; that is, there is $ i_0 $ $ ge $ 1 such that $ a_i $ = $ a_0 $ for all $ i $ $ ge $ $ i_0 $.

Theorem 2. yes $ M $ It is a Turing machine that given. $ i $ as input, it outputs an ordinal $ M (i) $Y $ M (i) $ $ ge $ $ M (i + 1) $, then the sequence stabilizes.

Note that Theorem 2 "is a weak corollary of Theorem 1." Additional note on what Prof. Chow writes $ PA $ and its relation to Theorem 1 as found in its answer to IamMeeoh's question about mathematical overflow, "Understanding the accounting ordinals up to $ epsilon_0 $ (56062)

It seems to me that after understanding this test [of Theorem 1–my comment]The hard part of wrapping my head is how it can be true that $ PA $ make *do not* prove that there is no infinite descending sequence. My current intuition is that $ PA $ as strangely weak, because he can not even formalize a test as simple as this one.

With respect to Theorem 2, write (on page 22 of his expository article):

… In fact, Theorem 2 can almost be proven in $ PA $.[Keepinmindthatonthelegofpage7onthepageyouwritethat[Notethatinfootnote7onpg26hewritesthat[Tengaencuentaqueenlanotaapiedepágina7enlapág26escribeque[Notethatinfootnote7onpg26hewritesthat$ PRA $ + *Theorem 2* implies that $ PA $ It's consistent – my comment.]

How does Professor Chow justify this? Consider the following, again form pg. 26 of his expository article:

First, we can formulate a theorem – call it $ 1 ^ {& # 39; $ $ theorem– it is an intermediate force between Theorem 1 and Theorem 2, which restricts Theorem 1 to weakly diminishing ordinal sequences that can be defined by a first-order formula $ phi $. To prove this version of the theorem, suppose we have a formula $ phi $ which defines a sequence of ordinals that weakly diminishes and states that everyone has at least one height $ H $ [seethedefinitionoftheheightoftheteacherChowandhissystemofkeymarksbelow[seeProfChow'sdefinitionofheightandhissystemofordinalnotationsbelow[vealadefinicióndealturadelprofesorChowysusistemadenotacionesordinalesacontinuación[seeProfChow’sdefinitionofheightandhissystemofordinalnotationsbelow$ epsilon_0 $ on pg.25 – my comment]. So we can imitate the proof of Theorem 1 to build a $ PA $ proof of theorem $ 1 ^ {& # 39;} $ for $ phi $. The only downside is that we need, as building blocks, P$ A $ proof of theorem $ 1 ^ {& # 39;} $ for smaller formulas that $ H $– but we can assume by induction that they are available. Keep in mind that this is an inductive procedure for construction $ PA $ Tests of individual instances of the theorem. $ 1 ^ {& # 39;} $ and can not be converted to $ PA $ proof of theorem $ 1 ^ {& # 39;} $ itself; however, it illustrates that each instance of Theorem $ 1 ^ {& # 39;} $ It can be proved without assuming the existence of infinite sets.

Interesting so far … but there are questions (for example, the question I asked in the title is still not answered by Professor Chow's quote cited above). Why? Well, according to Professor Chow, Theorem 1 "presupposes the concept of an arbitrary infinite set and, therefore, is not finite." From the theorem $ 1 ^ {& # 39;} $ is "intermediate in force between Theorem 1 and Theorem 2, the order of" force "in this case refers to (for example) Theorem 1 is" more infinite "than the theorem $ 1 ^ {& # 39;} $ (because "each instance of the theorem $ 1 ^ {& # 39;} $ it can be proved without assuming the existence of infinite sets "), and the theorem $ 1 ^ {& # 39;} $ it's more infinite & # 39; that Theorem 2 (but that's exactly the question I asked in the title, since "Theorem 2 can almost be shown in $ PA $"must, in a certain sense, be & # 39; infinite & # 39 ;, that is, your proof must" assume the existence of infinite sets ", but how? … also, given the" list "notation of" ordinals " "from Professor Chow below $ epsilon_0 $"How can you extend that to include $ epsilon_0 $ as a "special type of finite list of finite lists of finite lists of … finite lists" [this from his answer to IamMeeoh’s mathoverflow question–my comment])?

Finally, it may seem to the reader of this question to read the article by Maria Hameen-Antila, ordinal nominalistas, recursion in the superior types and finitismo. *Symbolic logic bulletin*, 25 (1): 101-124 (2019) because it provides the historical context in which to understand Prof. Chow's expository article, his notation list system (which would be an example of a nominal noun representation of transfinite ordinals) and its theorems 1, $ 1 ^ {& # 39;} $and 2 (and a possible financial interpretation of Theorems 1, $ 1 ^ {& # 39;} $, and 2).

Any help in this matter would be greatly appreciated. Thanks in advance.