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Generally, the structures that people study in every-day modern mathematics can be seen as defined on sets, so far as I know. (In the sense that they are collections of objects that don't give rise to russel's paradox).

E.g. the collection of real numbers, or of $5$-dimensional manifolds or of symmetry maps on the unit circle or whatever, are all sets.

However, I recall that sometimes structures cannot be assumed to be sets. E.g. we cannot speak of the Category of Categories apparently (unless we mean the category of small categories), since this would not be a set, and would give rise to russel's paradox (if I underatand correctly)

My question is: is it generally required for any mathematical theory that the domain of any arbitrary structure that satiafies it is a set, rather than generally a collection? E.g. do we require that a group not only satiafies the axioms, but implicitly require also that the domain of a particular group is a set?

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  • $\begingroup$ I think the question is too broad since the word "domain" in "domain of a mathematical theory" does not have a meaning that is consistent and precise enough across different example theories to answer the question. Nevertheless, typically structures are built form pre-existing structures and paradoxes like "set of all sets" do not arise when we do so. $\endgroup$ – Michael May 1 '18 at 15:33
  • $\begingroup$ Another thought: We can also have theorems that say: “If a particular structure satisfies property P, then it must also satisfy property Q.” This does not require us to talk about whether or not the “set” of all possible things that satisfy property P is really “too large” to be a set. It just means that if we happen to encounter something new that indeed has property P, we can apply our theorem to it. So that “something new” might be built from structures that were invented after the theorem, but the theorem still applies. $\endgroup$ – Michael May 1 '18 at 15:58
  • $\begingroup$ @Michael, why doesn't "domain" have a consistent meaning? $\endgroup$ – user56834 May 1 '18 at 16:06
  • $\begingroup$ Related: math.stackexchange.com/questions/1259892/… $\endgroup$ – Mike Earnest May 1 '18 at 16:45
  • $\begingroup$ @Programmer2134 : The "domain" of a function means the set over which the function is defined. On the other hand "domain of a mathematical theory" seems a non-mathematical use of "domain" to refer to the general topics, and possibly also the existing literature, related to that theory. I don't think it makes sense to place specific requirements on something as broad as a "mathematical theory." If a smart group of people want to develop a new mathematical theory, there is no reason to place artificial restrictions on what they are allowed to do. $\endgroup$ – Michael May 1 '18 at 19:12
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No. For one thing a set can generally be defined without use of the words group, or collection, as being: a mathematical object, containing other objects, without repeat,or ordering. Order only comes into play in ordered-set related things. Repeats usually only come into play, when generalizing to multisets. To set-theory $2^4$ has 1 prime factor, to the theory of multisets it has 4 (as represented by the exponent), as it counts the multiplicities (repeats) as distinct.

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