Confusion in the definition of set

Question

Which of the following is the correct definition for set? Set is a well defined collection of objects. Set is a collection of well defined objects.

Answer 1

Neither is the correct definition of set, but the first one is closer to the truth. And as Bhaskar points out, an even better definition (which is still wrong) would combine the two: a well-defined collection of well-defined objects.

Answer 2

Set is a well defined collection of objects. But obviously objects must be well defined too. You can not talk about a set consisting of three smiles and two fears and infinitely many other emotions.

Answer 3

Neither is formally correct. To properly define sets, you need to choose an axiomatic system - almost always ZF or ZFC (adding Choice), but it is occasionally interesting to consider what would happen using slightly different axioms.

Note however, that there are collections of well defined things that are not sets (these are called proper classes) - e.g. the collection of sets, or the collection of ordinals.

However, if you take a set and have a function from that set, then the image is a set (by the axiom of replacement), so in this sense a "well defined collection of objects" is a set - but only subject to the restriction that you defined the collection by looking at a set.

To define sets in ZF, we have the following axioms (expressed mostly in words here for more explanatory context):

• The empty set is a set
• A subset of a set is a set
• The power set of a set is a set (i.e. the collection of all subsets of a set is a set)
• Given two sets, there is a set that contains exactly those two sets as its elements
• Given a set $x$, the union of $x$ (i.e. collection of all elements of elements of $x$) is a set
• There is a set $\omega$ that contains $\emptyset$ and for every $x \in \omega$, $\omega$ also contains $x \cup \{x\}$
• Given a set $x$ and a function taking elements of $x$ to other sets (a function, so each is sent to only one image), the collection of images of this function is a set
• Sets are equal iff they have the same elements (i.e. $x = y$ if every element of $x$ is in $y$ and every element of $y$ is in $x$)
• Every set $x$ contains some element $z$ such that $x$ and $z$ are disjoint

You then know that something is a set if you can infer it's existance from these axioms. (Note that here elements of sets are sets, and there is a difference between, for example, $\emptyset$ and $\{ \emptyset \}$.)

Answer 4

Neither, really, as was remarked by others.

We want to talk about collections of mathematical objects, but we wanted to promote these collections from abstract notions to actual mathematical objects as well. Namely, sets come to serve as mathematical objects, which are themselves collections of other mathematical objects.

But that is just the idea behind the definition of a set. It's not the definition of a set. In fact, the problem with defining a set, is that nowadays we can think about the entire mathematical universe as being composed of sets, just like a molecule is composed of atoms.

Namely, sets serve as a primitive concept, in some sense. This means that they don't have an actual mathematical "definition". But it's not a horrible thing, "number" does not have a mathematical definition either, it can be a natural number, or a rational number, or a complex number, or an ordinal number, or a surreal number, or a $p$-adic number, or many many other types of "number".

To overcome this, we formally define a set as an object inside a mathematical universe of set theory. This seems as somewhat circular, but the term "set theory" does not depend on the term "set", it comes first. So what is a "set theory" from a mathematical point of view? It is a theory that we, the people who use it, decide that it is good enough for our purposes for saying what properties the notion of a set has. For example, if we have two collections with the same elements (objects inside them), then these collections should be equal; or if we have a collection, and there is some property we can describe, then those elements in our collection with the said property should also form a collection.

So let's get this in order. First we decided that there should be a formal mathematical object which will be a collection of other mathematical objects. We decided to call that notion set. Then we [formally] described properties expected from such object, and called the resulting axioms "set theory". And finally we say that a set is a mathematical object inside a universe of set theory.

Answer 5

As @meta writes, these days a set is properly defined in an axiomatic system.

I should take a small objection to @meta's comment that ZF is almost always the axiomatic system. NBG is another axiomatic system for set theory and is also used a fair amount, even if less than ZF. A famous theorem basically states that the sets in ZF and NBG are the same, but NBG also has other well-defined collections of well-defined objects that are not sets. These are "too large" to be sets and are called proper classes. Classes are the basic notion in NBG, and some classes have a property that allow them to be called sets. (If you were wondering, a class is called a set if it is a member of another class.) One class that is well-defined and is not a set is "the class of all sets." Such a thing does not exist in ZF but it does in NBG.

So, even approximately, it is not true a Set is a well defined collection of objects. That is a near-definition of a Class, not a set. At least, in NBG.