Certainly, when we apply set theory, we consider collections of concrete objects as sets.

For example, when I count 5 apples, I establish a bijection between the number 5 ( which is defined as the set { 0,1,2,3,4} ) and my collection of apples. At the end of my counting I say : the collection has cardinality 5, or, rather " there are 5 apples here".

My question is: in set theory proper ( not applied) , would such a collection be considered as a set?

Or , rather, need not every element of a set also be a set? or at least an abstract object?

In " The 5 WH of set theory" I read that sets are collections of mathematical objects. ( https://plato.stanford.edu/entries/set-theory/#UniVAllSet)

Maybe there is no unique answer due to the plurality of set theories?

What is the answer for ZFC?

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    Consider set-theory with Urelements, i.e. objects that are not set themselves. But the "usual" point of view about them is to consider as urelements mathematical objects, like e.g. natural numbers. – Mauro ALLEGRANZA Jan 3 at 17:34
  • See also Finitist set theory. – Mauro ALLEGRANZA Jan 3 at 17:36
  • Set theory proper, such as ZFC, does not "consider" any examples of sets, "is a set" is not even a predicate in it. It has a single non-logical predicate ∈ and axioms involving it, and derives theorems based on that, nothing else. It is only in external interpretations ("applications") that the question can be meaningfully asked, but what counts as a set then depends on interpretation, not on ZFC. And in popular structuralist interpretations it makes no difference whether sets consist of "mathematical objects" or not, since the theorems hold regardless of that. – Conifold Jan 3 at 19:09

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