Sets, collections and semantics

Question

Sets are not defined and are a primitive notion in mathematics. In introductory books the intuition of a collection is presented when talking about sets. However, not any such collection can be considered a set if one wants to be consistent. This is where axiomatic set theory comes into play and states axioms that sets satisfy, meaning that any set behaves in the way the axioms state. When finding out about this seeming difference between collections and sets (can we even say there is any? Any inconsistency about sets should also lead to one about collections, shouldn't it?) made me think whether there are differences between sets and collections or whether they are still the same concept, even in axiomatic set theory.

One way to not view sets as collections is by simply thinking of sets as objects that are characterized by the property of being a set, without any intuition in the background whatsoever - they are just sets. Then one can work with them by using the axioms and derive statements about those abstract objects. This would, however, be really far away from the initial intuition of a collection, since one treats sets without any intuition here whatsoever. One could compare it to me presenting an assumption that there exist objects called "abcdefg" that obey some properties. Here no one would know what they are, but one could still derive results (depending on the axiom), analogous to treating sets as abstract.

The other one is to view sets as collections that obey the axioms that one agreed upon. In this way sets are collections, but we dont know if collections are always sets (as described above, they may coincide I guess?). One then has intuition for sets and the axioms that are presented and would know what sets are.

Question: Which is the usual path that is taken? Is there even one, or are both present (are there perhaps others?). Does one lead to problems?

This also made me think that when we have symbols like "$\in$" which is supposed to represent set membership, can we sure that it really represents membership, or could it also represent another property that just follows the same rules? However, when we use words to describe things I guess we essentially do the same thing and just agree on a fixed meaning. If we do the same for "$\in$" in the view of the third abstract one should be fine and it should actually mean set membership.

Answer 1

An example of a collection that is not a set is a class. A difference between classes and sets is that any collection $$\{x : \phi(x)\}$$ of everything that satisfies a given formula $\phi(x)$, expressed in terms of $\in$, is a class. However, the Axiom of Comprehension warrants building a set $$\{x \in X : \, \phi(x)\},$$, by consider only those sets $x$ that satisfy $\phi(x)$ that are elements of a particular fixed set $X$. We proscribe the collection of everything that satisfies $\phi(x)$ from being a set.

The set-theoretic axioms were developed to avoid certain inconsistencies about collections. For example, allowing classes results in Russell's Paradox by taking $\phi(x)$ as $x \notin x$. But $\{x \in X : x \notin x\}$ does not face the same paradox.

The axioms of set theory formalises intuitions we have about collections. When faced with inconsistencies, we modify these axioms. Yet, upon choosing a standard axiomatisation of set theory, say the Zermelo-Fraenkel axioms, were a theorem could be proven from these axioms that conflicts our intuition of collections, we then need to alter our intuition. Alternatively, we could also modify the axioms slightly and study properties of other set-theoretic universes that satisfy the alternative axiomatic systems.

In set theory, there is only one sort of objects, namely sets. So in the relation $$x \in y,$$ both $x$ and $y$ are sets. However, this presentation conflicts our original conception of collections, where the objects in the collection need not be collections themselves. For instance the set of natural numbers $\mathbb{N} = \{1,2, 3\}$ contain numbers, which we don't originall think of as sets. So alternative presentation develop, such as type theory, where a type is analogous to a set, while a term is analogous to an element. And terms are not types, but are another kind of object.