How much should an average mathematician not working in an area like logic, set theory, or foundations know about the foundations of mathematics?
The thread Why should we believe in the axiom of regularity? suggests that many might not know about the cumulative hierarchy, which is the intended ontology that ZFC describes. (Q1) But is this something everyone should know or does the naive approach of set theory suffice, even for some mathematical research (not in the areas listed above)?
EDIT: Let me add another aspect. As Andreas Blass has explained in his answer to the MO-question linked above, when set theorists talk about "sets", they mean an entity generated in the following transfinite process:
Begin with some non-set entities called atoms ("some" could be "none" if you want a world consisting exclusively of sets), then form all sets of these, then all sets whose elements are atoms or sets of atoms, etc. This "etc." means to build more and more levels of sets, where a set at any level has elements only from earlier levels (and the atoms constitute the lowest level). This iterative construction can be continued transfinitely, through arbitrarily long well-ordered sequences of levels.
Now, for foundational purposes it may be prudent to take the "none atoms" (pure) version of set theory. But in mathematical practice, one is working with many different types of mathematical objects, and many of them are not sets (even though one might often encode them as such). But even if one adds some atoms/non-sets – for example the real numbers –, one does not get a universe which is satisfactory for the practice of mathematics. This is because notions like "functions" or "ordered tuples" are from a conceptual perspective not sets; but we can't take them as our atoms for the cumulative hierarchy – the set of all "functions" or "ordered pairs" … leads to paradoxes (Russell). Now I wonder:
(Q2) What should mathematicians not working in an area like logic, set theory, or foundations understand by "set"?
Note that there is also the idea of structural set theory (see the nlab), and systems such as ETCS or SEAR (try to) solve these "encoding problems" and remove the problem with "junk theorems" in material set theories. One can argue that these structural set theories match (more than material set theories do) with the mathematical practice. But my personal problem with them is that they have no clear ontology. So this approach doesn't answer Q2, I think.
Or, (Q3) do mathematicians not working in foundational subjects not need to have a particular picture in mind of how their set-theoretic universe should look like? This would be very unsatisfactory for me since this would imply that one had to use axioms that work – but one doesn't understand why they work (or why they are consistent). The only argument I can think of that they are consistent would be "one hasn't found a contradiction yet".
Best Answer
The answer is essentially the same as how much should the average mathematician know about combinatorics? Or group theory? Or algebraic topology? Or any broad area of mathematics... It's good to know some, it's always helpful to know more, but only really need the amount that is relevant to your work. Perhaps a small but significant difference with foundations is that there is a natural curiosity about it, just like I'm naturally curious about the history and geography of where I live, even though I only need minimal knowledge in day-to-day life.
One does need to know where to go when deeper foundational questions arise. So one should keep a logician colleague as a friend, just in case. This entails knowing enough about foundations and logic to engage in casual conversation and, when the need arises, to be able to formulate the right question and understand the answer you get. This is no different than any other area of mathematics.
Some mathematicians may have more than just a casual curiosity about foundations, even if they work in a completely different area. In that case, learn as much as your time and curiosity permits. This is great since, like other areas of mathematics, foundations needs to interact with other areas in order to advance.
So, what do you need to have a casual conversation with a logician? Adjust to personal taste:
To address the additional questions regarding sets. Personally, I don't think it's right to say that the notion of set is defined by foundations. It's a perfectly fine mathematical concept though it (sometimes confusingly) has two distinct and equally important flavors.
The main evidence for this point of view is that the notion of set existed well before Cantor and their use was common. Here is one of my favorite early definitions due to Bolzano (Paradoxien des Unendlichen, 1847):
(See this MO question for additional early occurrences of sets of various shapes and forms.)
What Bolzano describes is the combinatorial flavor of sets: it's a basic container in which to put objects, a container that is so basic and plain that it has no structure of its own to distract us from the objects inside it. There is another flavor to sets in which they are used to classify objects, to put into a whole all objects of the same kind or share a common feature. This usage is also very common and also prior to foundational theories.
Mathematicians use both flavors of sets, often together. For a variety practical reasons, foundational theories tend to focus on one, and formalize standard (albeit awkward) ways to accommodate the other.
So-called "material" set theories (ZFC, NBG, MK) focus on the combinatorial flavor of sets. To accommodate classification, these theories allow for as many collections of objects as possible to be put together in a set (with little to no concern whether this is motivated, necessary, or even useful).
So-called "structural" set theories (ETCS, SEAR, many type theories) focus on the classification flavor of sets. To accommodate combinatorics, these theories include a lot of machinery to relate sets and identify similar objects across set boundaries (with little to no concern about the nature of elements within sets).
Both of these approaches are viable and they both have advantages over the other. However, it's plainly wrong to think that working mathematicians have to choose one over the other, or even worry about the fact that it's difficult to formalize both simultaneously. The fact is that the sets mathematicians use in their day-to-day work are just as suitable as containers as they are as classifiers.