TLDR at the end. Hi, I recently saw this comment given by "completely-ineffable" on the r/badmathematics subreddit. And I just wanted to make sure if I understand it correctly and wanted to know it's relation to the work by Benacerraf, Van Bendegem and Allis & Koetsier. All the articles mentioned can be found here (By Ross-Littlewood I'm refering to the version of the paradox where in step $n$ you introduce balls $10n-9$, $10n-8$, $10n-7$, $...$, $10n$, and substract ball $n$.)
About the comment:
What I understood is that in Set Theory, there's a way to describe a problem using transfinite numbers and transfinite recursion, and this problem could be translated into simple words as the Ross Littlewood Pradox. The balls represent an Aleph 0 amount of elements, the urn represents a set where we will add and substract elements with a transfinite recursion/induction, the clock before noon represents a finite step, while the clock at noon represents a step $\omega$, a step taken after an infinite amount of steps. If in step $x$ we substract element $x$ from the set, then at step $\omega$ the urn is empty. So the problem represented in the context of set theory can be solved without contradiction. Arguing that Ross Littlewood has no solution in any context would then require to deny the existence of transfinite concepts, and therefore, being a finitist. Is my understanding correct?
About Benacerraf, Van Bendegem, and Allis & Koetsier:
Articles here.
What I understand is that Benacerraf published an article about supertasks in 1962, where he analized supertasks, in particular Thompson's Lamp, and concluded that in general supertasks have no answer because they are do not specify correctly what happens when you reach noon. I noticed that Ross Littlewood was not mentioned anywhere on the article.
Then Allis and Koetsier published their first article about Ross-Littlewood in 1991, where they conclude that Benacerraf was right that many supertasks had no answer, but in the case of Ross Littlewood in particular there was an answer, and showed why it made sense that the urn is empty at noon.
Then in 1994, Van Bendegem concludes that Allis & Koetsier did not succeed in showing that Ross Littlewood isn't impossible.
Then in 1995, Allis and Koetsier point out mistakes made by Van Bendegem in his article and show his reasoning is flawed, therefore Ross Littlewood is a supertask that makes sense
What confuses me is the connection between Allis & Koetsier's work with the Set Theory approach that I mention at the beginning. It looks like what Allis & Koetsier did is not the same as using transfinite numbers and transfinite recursion. So, are the Set Theory approach and the AllisKoetsier approach different ways to analize Ross Littlewood in a way that it does have an answer (the urn is empty)? Or are they really the same approach but written in a different way?
TL;DR: I want to know if my understanding of the Set Theory version of the Ross Littlewood paradox is correct, and it's relation to the approach proposed by Allis & Koetsier