ZFC set theory has the surprising property of being able to represent many mathematical objets that have intuitively nothing to do with sets. For example, a couple is represented by a Kuratowski pair, a function is represented by a set of Kuratowski pairs, natural numbers are represented by the finite Von Neumann's ordinals. In particular, the number 2 is represented by the set $\{\emptyset, \{\emptyset\} \} $.

These representations made some mathematicians declare that everything is a set in mathematics. The representations then became dogmatic equalities, sometimes even called definitions. This had devastating effects in the teaching of mathematics, at least in France in the 1960s, under the authority of Bourbaki. Some philosophers even claimed that everything in reality is a set. That makes me wonder every morning when I put on my socks, what ZFC axioms they use.

I won't speak long about other possible foundations of mathmematics: category theory, (dependently typed) lambda calculus, univalent foundations... Instead, I am wondering, within the ZFC theory itself, how the claim that everything is a set is compatible with the existence of various models of ZFC.

In each ZFC model $(M, E)$, everything that can be proven by ZFC has a representation. For example, the empty set $\emptyset$ is represented by a set $\emptyset^M \in M$, which has no reason to be equal to $\emptyset$. Likewise we have representations in $M$ of couples, functions, natural numbers and all the usual mathematics that ZFC handles.

Then if representations are taken as definitions, what is really the number 2? Is it $\{\emptyset, \{\emptyset\} \} $ or $\{\emptyset^M, \{\emptyset^M\}^M \}^M $? Is it still possible today, after all we know about ZFC, to support that it must be the unique foundation of mathematics?

Don't get me wrong, I think ZFC is a magnificent theory, especially to study infinity. I just don't understand how it drifted towards the foundations of mathematics.


closed as primarily opinion-based by Lord Shark the Unknown, ArsenBerk, Henno Brandsma, Asaf Karagila Oct 14 '18 at 14:15

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    $\begingroup$ Yes. Assuming ZFC as a foundation everything is a set. Otherwise, your question is more of an opinion and philosophy based. It suits more for a tea time, or over some beers, less so to the StackExchange Q&A format. $\endgroup$ – Asaf Karagila Oct 14 '18 at 14:17
  • $\begingroup$ @AsafKaragila I always appreciate a tea or a beer. Here I am also interested in the answer, and I guess you have part of it. If you know other websites to discuss this I can post on them instead. $\endgroup$ – V. Semeria Oct 14 '18 at 22:35
  • $\begingroup$ Not really, sorry... $\endgroup$ – Asaf Karagila Oct 15 '18 at 6:58