I am an assistant professor of mathematics, and I am quite familiar with the issues of translating mathematical concepts between languages.
A main difficulty is that there are so many concepts of a set with additional endowed structure: set, group, space, ring, field, universe, collection, family, manifold, category, bundle, sheaf, scheme…
Different languages come with different sets of available words to draw from, and some of the translations are surprising.
The key observation is that translation of such terms is not based on reason but on tradition.
The mathematical communities of different regions have settled on their terminologies, and translations must adhere to those conventions.
Therefore, when it comes to Latin, the first question is whether there is an established term.
Someone's translation on Wikipedia doesn't quite count as established, and mathematics isn't really published in Latin anymore so there is no natural force to produce established modern mathematical terminology in Latin.
If a concept is old enough to originate from the time when Latin was still in broad scientific use, then the Latin term used then is the way to go.
But set theory as the foundation of mathematics is younger, so I would opt for the same solution I would use for any modern concept:
Pick something that makes sense to you in that context and explain as necessary (maybe with a translation to a language with more established terminology) and treat it as an ad hoc translation.
Nothing is official so don't worry about getting it officially correct.
It's also not unusual that a Latin term centuries old is inconvenient for modern scientific usage.
I think Newton uses motus for momentum, and it gets confused with movement.
Finally, to answer your question:
A comment suggested the term theoria copiarum as used on Wikipedia, and it makes sense for "set theory".
With that, a "set" would be copia.
Feel free to use it, but don't treat it as the one and only truth.