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Probably everyone of us has seen set-theoretic encodings of mathematical objects which we wouldn't naturally consider to be sets. May it be the "definition" of a function from $A$ to $B$ as a relation $f\subseteq A\times B$ with the property $$\text{for every single $a\in A$ one can find exactly one $b\in B$ with $(a, b)\in f$ }$$ or the construction of natural numbers due to von Neumann in which we consider $0$ to be the empty set, $1$ to be the set $\{0\}$, $2$ to be the set $\{0, 1\}$, and so on.

But why do we bother about set-theoretic encodings? When did mathematicians historically felt that we need to code non-set entities as sets? What's the purpose of doing so?

user98612
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    It is the Grand Unified Theory of mathematics. – Goldstern Sep 18 '16 at 11:44
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    Nota bene: There are other foundations of mathematics where not everything is a set, notably ETCS, SEAR and HTT. References: https://ncatlab.org/nlab/show/ETCS + https://ncatlab.org/nlab/show/SEAR + https://ncatlab.org/nlab/show/homotopy+type+theory . One might argue that ZFC is a relict from the past, the first serious attempt to axiomatize mathematics. – HeinrichD Sep 18 '16 at 11:50
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    @HeinrichD: Just out of curiosity, why do you think that "ZFC is a relict from the past"? Certainly there are foundations other than ZFC, but have they been successfully replacing ZFC (say, in the textbooks and math education)? – Burak Sep 18 '16 at 12:25
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    @HeinrichD: ZFC formalizes what set-theorists mean by "set", as Andreas Blass explains here: http://mathoverflow.net/a/219647/98612. ZFC is the axiomatic system which is most used by set-theorists. That's why a consider your comment "ZFC is a relict from the past" a bit arrogant. Also, see http://mathoverflow.net/a/7241/98612, where Mike Shulman says: "I view topos-like set theory and ZF-like set theory as exposing two faces of the same subject." – user98612 Sep 18 '16 at 12:41
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    I don't really see why this question was closed... anyway here are my two cents: we code objects as sets because it's harmless (once we make the definition we forget about it) and it makes the theory somewhat slicker. If you want, you could work in "ZFC + urelements", where we allow objects other than sets to exist, but typically all this does is complicate set-theoretic arguments (since you have to take urelements into account). There are exceptions, for example in Barwise's "Admissible sets and structures" it's actually more convenient to use urelements, and so he does. – Danielle Ulrich Sep 24 '16 at 18:23

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Modern mathematicians have mostly settled on a rough consensus as to what methods are valid. This consensus includes Aristotelian logic, the use of objects of infinite size, and nonconstructive as well as constructive methods of proof. ZFC encodes this consensus, and it also suffices to encode the mathematical objects used in virtually all mathematics. The fact that it can encode, for example, all the objects used in real analysis tells a couple of useful things about real analysis. It tells us that real analysis can be carried out using consensus methods. It also tells us that real analysis is consistent if ZFC is consistent.