Is there a difference between "set" and "collection"?

Question

I thought of asking this on the math stack exchange, but I think this stack exchange is better suited. Is there a difference between sets and collections? Some people say they mean the same, while other people think that all sets are collections, but not all collections are sets. The latter group of people often use "proper class" for the non-set collections. But what have philosophers of mathematics written about this topic?

Answer 1

A "collection" is a general word to refer to "some things", without specifying a formally described structure such as a set, class, type, conglomerate, etc.

Mathematicians sometimes use the word collection to denote a bunch of “things” without prejudice as to whether those things form a set, a proper class, or some other formal notion of collection such as a type. For example, the word is often used this way in the definition of a category.

https://ncatlab.org/nlab/show/collection

The reason for this is that once you try to describe "a collection of things" formally, the situation becomes more complicated. When your definition becomes more precise, you can deduce more logical consequences of such a definition. You might not want to have to deal with those questions at that moment, or, you might think that it does not matter at the moment which formalization of "a bunch of things" should be used.

It is very handy to be able to refer to a collection of things intuitively. Whereas the word "set" is bound up in a very specific theory of what "collections of things" should be like. A "proper class" is associated with a type of set theory called Von Neumann-Gödel-Bernays set theory in which they added in a second concept called a "class" which contains the sets.

This means that:

• "set" calls to mind a particular set theory, but there are multiple set theories so you should ask "which one"?
• "proper class" is more strictly associated with NGB set theory
• a "collection" is not bound to any particular formal theory

If you want to know more about the philosophy of sets, I recommend the book "Lectures on the Philosophy of Mathematics" by Joel David Hamkins.

https://mitpress.mit.edu/9780262542234/lectures-on-the-philosophy-of-mathematics/

Answer 2

To avoid the Russell antinomy modern set theory discriminates between “sets” and “proper classes”.

The generic term is “class”. A class C is

• either a “proper class”, i.e. there is no class which contains C as element,
• or a “set”, i.e. there is at least one class which contains C as element.

The intuition behind the concept of a class is often paraphrased as “classes are collections of elements.”

Answer 3

It is worth noting that in a strict mathematical sense, plenty of things that we think of as comprising a set, e.g. the members of a football team or the apples in a basket, are not actual sets. They are set-sized objects, but since formal set theories only deal with the set theoretic universe that they model, and since the only objects in pure set theories are sets, it doesn’t make sense to talk about collections of things that are not sets. The same extends to NBG/MK, where there still cannot be anything that is a member of a collection without being a set. Of course, classes can be included in other classes, but not as members.

This is all to say that there is a formal distinction between a set and a set-sized object, and as such the comparison between ‘set’ and ‘collection’ only goes so far in math and in the real world, so long as ‘set’ has is set theoretic meaning.

Answer 4

Original

A set is a collection that cannot contain duplicate elements. The order of elements in a set is not relevant. A list is an ordered collection which can contain duplicate elements. Set is a subtype of collection.

Edit

https://www.math.ucdavis.edu/~hunter/intro_analysis_pdf/ch1.pdf

We understand a set to be any collection M of certain distinct objects of our thought or intuition (called the elements of M) into a whole. (Georg Cantor, 1895)

In mathematics you dont understand things. You just get used to them. (Attributed to John von Neumann)

In this chapter, we define sets, functions, and relations and discuss some of their general properties.

A set is a collection of objects, called the elements or members of the set. The objects could be anything (planets, squirrels, characters in Shakespeares plays, or other sets) but for us they will be mathematical objects such as numbers, or sets of numbers.

If the definition of a set as a collection seems circular, that's because it is. Conceiving of many objects as a single whole is a basic intuition that cannot be analyzed further, and the notions of set and membership are primitive ones. These notions can be made mathematically precise by introducing a system of axioms for sets and membership that agrees with our intuition and proving other set-theoretic properties from the axioms.

Another Edit

Can elements in a set be duplicated?

https://math.stackexchange.com/questions/223405/can-elements-in-a-set-be-duplicated

No. Sets are collections where repetition and order are ignored.

Therefore in the word contrast we have the letters {c,o,n,t,r,a,s} and a simple observation tells us that there are exactly 7 letters in this set.

Answer 5

The term set is a technical term in the context of a modern set theory such ZF (there are of course many others). The term collection is a generic term, not technical term. Thus, if one has a (say, 1-place) predicate, the collection of constants satisfying the predicate is not a set unless one has a valid application of the Separation Axiom that would guarantee the existence of such a set.

Answer 6

In Zermelo-Fraenko Set Theory, under the standard interpretation of the Language of Set Theory - sets are elements of our domain of discourse.

What do we do with a Language of Set Theory? We build well-formed formula, a formal version of a sentence.

Atomic Well-formed formula are of the form x = y or x∈y. As our domain of discourse is one of sets, x,y here are sets. =,∈ are binary relations which act on sets.

In our minds we often interpret "x∈A" as meaning, x is an element of A. With the Axiom of Extentionality, we have that two sets are equal if they contain the same elements. In this sense, A is a collection of all of its elements. Not only does A contain elements, it is equal to anything that contains the same elements.

Note: formally, this characterization does not exist. We are welcome to think of A as a container, filled with elements. But, our theory gives no such characterization. This is not unusual, in Geoemtry we have an idea that a point is a dimensionless object, and then we draw it as a dot on a piece of paper. However, our drawing ( the image of a point in our mind) is not the same as the object defined as a point axiomatically. If there is a more specific notion of set, I believe it could only be characterized as an object in the Platonic Sense. And, if the Platonic Sets are like our ideas of Containers and Collections- I cannot speak to that.

We also have the Logical Closure of Well-formed formula. If φ,Ψ are well-formed formula, ¬φ, φ ∨ Ψ,φ → Ψ,(∃x)φ and (∀x)φ, are also well-formed formula.

Some variables might be bounded within the scope of a quantifier, and some are not. We write φ(a,b,c) to mean that a,b,c are not being acted on by a quantifier.

Once we have well-formed formula, we have a new way to work with sets, besides just being elements of our domain of discourse. Notice that for some set x, and well-formed formula φ, ∀y(y∈x ↔ φ(y)) is itself a well-formed formula.

The above, intuitvely says " y is an elment of x if and only if φ(y)" - we think of φ(y) as some property that y satisfies.

For a shorthand, we write the above formula as: x = {y| φ(y)}

This provokes the question, is {x|φ(x)} always a set? By Russell, we know this is not the case. So, there are some formula φ, s.t. (¬∃x)∀y(y∈x ↔ φ(y))

Intuively: x = {y|φ(y)} Doest not exists. However, we can still think of x, as a collection/Proper Class of all sets satsifying φ.

While working with set theory it can become usefull to have a conception of "The Universe of Sets" and objects like that. However, they are not official objects in our domain of discourse.

So then, what are they? Officially, they are just shortcuts for expressing a particular well-formed formula in our language.

If V is the universe of sets, we can write informally that x∈V, however that is not an offical formula in our language. However, notice that informally, x∈V ↔ x = x

and that x = x is a perfectly valid formula in the language of set theory.

So, whenver we write x∈V, we are using a shorthand to express the formula x = x.

This is true for all Collections/Proper Classes. When we write that a set is an "elemement" of a collection, we are just saying that it satisfies some formula.

Another example, we can write that Card ⊆ Ord, this is just a shorthand for the well-formed formula expressing that all cardinals are ordinals.

In Summary,

A set is an element of our domain of discourse in the formal language of set theory. We can interpret (informally) a set to be a collection of all of it's elements. Further, we can consider (informally) a set to be the collection of all of the elements which satisfy a certain property. It is important to note that not all collections of sets, form a set. In which case, we use such collections as a shorthand to express some longer well-formed formula.

Personally, I would use Collection/Object if I didn't wan't to commit to saying if/how my proposed object is an official element of the domain of discourse in ZFC.

I would use Proper Class, if I could show that such a collection is not an official object of ZFC.

Answer 7

A set and collection informally mean the same thing. Formally though the word 'set' has been co-opted to signify a certain mathematical object in set theories. It's an accident of history, they could concievably been called collection theories.