Here is Juliet Floyd stating "Wittgenstein's non-extensionalism, like Russell's in Principia, precluded development of an extensional theory of the infinite (set theory). https://youtu.be/WGv8pqzHfoE?t=1094
So what has happened since? Is there still this view of Wittgenstein's that collections of infinite objects can't have extensional definitions?
With the term “extension” Wittgenstein has two things in mind. First, he will strictly distinguish between sequences of numbers that the extensionalist considers to be, in Cantor’s sense, “finished” [fertig] entities or sets – these are the “extensions” – from the techniques or rules by means of which such entities may be produced, assessed, or accessed. If there are such techniques, the extensionalist’s interest is ultimately only in their results, the produced sequences, and not the possible processes or conceptual motifs or definitions leading to them.
Wittgenstein’s criticism is directed at the common view of “modern” mathemati- cians that mathematics is about extensions. We call this the extensionalist point of view and the contrary view proffered by Wittgenstein the non-extensionalist one. We do not use the term “intensionalist” because it would suggest that words get their meaning via certain entities individually attached to them, entities called “intensions”, and this is a conception totally contradicting Wittgenstein’s stance. According to him, words (linguistic expressions, signs) get their meaning – their “life”, as he sometimes says – through the uses we make of them within language, and for him it is a sort of category mistake to base or ground this so-called meaning in immediate relations to certain entities accompanying the words.6
For the non-extensionalist, on the other hand, it is the processes and structured conceptual motifs, the grammar or logic of the notions, we should be concerned with. The most important cases discussed in Wittgenstein’s texts are given by the conception of a real number as a rule-governed calculational procedure for which we can see that for any given n it will generate n digits: for example, a recipe for generating more and more successive digits of √2. About the number√2 Wittgenstein explicitly says that “we [as extensionalists] have a tendency to think that there is one result produced by√2, viz., an infinite decimal fraction”. But on his view √2 produces a series of results, but no single result: “√2 is a rule for producing a fraction, not an extension” (AWL, p. 221) cover
The second context in which Wittgenstein speaks of “extensions” is the context of sets, paradigmatically sets of numbers: N, Q, R, and subsets of them. Considered extensionally, the laws or rules or techniques through which we may approach them are taken as irrelevant, as their identity is only determined by the elements of which they consist. From Wittgenstein’s non-extensionalist view, however, it is precisely these laws, rules or techniques we should take as primary.
Juliet Floyd, Felix Mühlhölzer Nordic Wittgenstein Studies 7 Springer International Publishing; Springer, Year: 2020 9783030484804
Doesn't this at least put people in a tough position? Reject the "greatest philosopher of the 20th century" or reject set theory.