What is a Set … Really?

Sometimes the more basic an idea is, the harder it is to define it. It is also very hard to understand a definition in English when you are not a native speaker! We have had some interesting discussions of such issues recently with a student who asks very basic and yet very challenging questions of this sort; some have just been too long to post, but I thought this one would be interesting.

Defining “Set”

The question came to us at the end of July from Shaurya:

Respected maths doctor,

In my school mathematics textbook set is defined as follows:

a set is a well defined collection of objects.

Then what I understood was like that if I talk about set of planets in our solar system then anyone would collect planets in solar system from their actual position and place  it in front  of me so that I could say now I have collection of all planets in my solar system but that would not make any sense.

Then I studied further and found an extract in Wikipedia. I think that it could solve my doubt but I am not able to understand what it means to say and it is as following:

a set is a gathering together into a whole of definite, distinct objects of our perception or thought which are called  elements of the set.

Please explain it in simplified form and tell me that whether there is any flaw in definition of set in my textbook.

Thank you\

Can I define sets as follows?

a set is the gathering of definite quantity of words or symbols that triggers our abstraction  for particular object.

The first definition is very common, and very basic. The second, from Wikipedia, tries to say more, and therefore raises more questions! And Shaurya, in trying to put this in his own words, helps us to see a little of where he is struggling. Paraphrasing can be a good way to demonstrate your understanding (as I recall being told to do in studying poetry in English class long ago). He has chosen to use “gathering” rather than “collection”, and “words or symbols” to extend “objects”.

What’s an “object”?

Though I have had most of our discussions with Shaurya, Doctor Rick answered this one, starting from another of our standard sources for definitions:

Hi, Shaurya.

Here’s the definition given in Wolfram MathWorld:

A set is a finite or infinite collection of objects in which order has no significance, and multiplicity is generally also ignored (unlike a list or multiset).

Is it the word “object” that bothers you? MathWorld’s definition, like your textbook’s definition, uses that word.

This definition lacks the term “well-defined”, but introduces two additional issues, order and multiplicity.

For now, we focus on the word “object”:

Doctor Peterson has answered your questions about the history of set theory, and one thing he mentioned was “abstraction”. From other questions, we see that you have learned something about group theory, which is all about abstraction – that is, ignoring a lot of details (about addition of whole numbers, for instance) and focusing only on a few key points, then seeing what you can do with those key points alone.

Sets are sort of an extreme abstraction. We don’t care about what objects are in the set – they could be numbers, they could be words, they could be colors, they could be chemical elements … they could be planets! It is not senseless to talk about the set of planets. The word “object” in definitions of “set” is deliberately vague, even undefined, because it makes no difference what kind of thing is in a set – we essentially ignore everything about it, except that it is different from the other members of the set.

The idea here is that we use the word “object” in the broadest possible way. It does not convey any idea of solidity or reality; I sometimes use the word “thing” in the same way, mostly to emphasize that I don’t care what sort of “thing” it is. The second definition Shaurya quoted, which we’ll get back to soon, seems to be trying to say the same thing in more words rather than fewer: These “objects” can be anything.

What’s a “collection”?

Perhaps you are concerned about the word “collection“, which again is in both MathWorld’s and your textbook’s definitions. From your proposed definition, it appears that you prefer the term “gathering together”. The word “collection” means either the act of gathering things together, or the result of that gathering — and the latter of these is what we have in mind here. However, “gathering” does not avoid the problem you seemed to have with a “set of planets”: namely that we can’t actually bring the planets together in one place.

So you propose that the things “gathered” are not the objects themselves, but words or symbols representing the objects. That makes some sense … but I think it’s still too concrete. A set doesn’t have to be written down in any form. Note that the MathWorld definition says a set can be infinite! It is not possible to collect an infinite number of objects in any physical sense, no matter what they are.

I am wondering if Shaurya’s suggestion of “words or symbols” might be motivated by one of the ways to define a set, called “roster notation”, in which we simply list the elements – using symbols like \(\{1,2,3,\dots\}\), or using words like \(\{\text{Mercury},\text{Venus},\text{Earth},\dots\}\). Here we are “gathering together” not the “objects” themselves, but representations of them. The problem is that this is not the set itself, but a representation of the set! The set does not consist of the words or symbols, but of the “things” they refer to.

My answer to your objections (as I’m interpreting them) is simply to take the word “collection” in a purely conceptual sense. If I can imagine all even numbers on my left and all odd numbers on my right, then I have created two sets, the set of even numbers and the set of odd numbers (both infinite, by the way).

The word “collection”, as I think of it, is better than “gathering” simply because the latter is concrete, while the former is commonly used both ways.  For example, here is a definition of the word from Merriam-Webster:

Definition 2a is a literal, physical gathering; 2b is what we have in mind for sets; the example is not physical “objects”, but concepts that are thought of as a single group.

Another way to define a set, called “set-builder notation”, may better represent this idea. Using Doctor Rick’s example, we can define the set \(\{2,4,6,8,\dots\}\) by describing the elements: \(\{x\in\mathbb{N} : x\text{ is an even number}\}\). Here, the statement following the colon is a way to test a potential element to see if it belongs; we don’t need to actually “collect” it, but just know how to identify it.

Focusing on the parts that matter most

We must not focus so much on questionable details that we overlook what parts of the definition are important:

Let me point out the phrases in MathWorld’s definition that really matter. They are not the words “collection” and “objects”, but the descriptions “order has no significance” and “multiplicity is (generally) ignored”. The second of these corresponds to the word “distinct” in the extract you found on Wikipedia: you can’t have two objects that are the same — all objects in the set can be distinguished one from another. The first phrase means that we ignore any sort of relation among the objects — even if there is a “first object” or “biggest object” or anything like that, we don’t care.

So we don’t distinguish between the sets \(\{1, 2\}\) and \(\{2, 1\}\); we can’t think of one element as first. We also can’t distinguish between the sets \(\{1\}\) and \(\{1, 1\}\); an element is either in the set or not, never “in twice”.

On the other hand, your textbook’s definition and the words from Wikipedia have something in common that MathWorld didn’t emphasize: “well-defined” or “definite“. You put “definite quantity” in your proposed definition. The idea is that there must be no question whether any particular object is in the set or not! We can’t have a “set of all tall people” unless we have a very precise definition of “tall”. (As there are “multisets” in which there can be more than one of the same object in the set, there are also “fuzzy sets” in which a given object has some probability, between 0 and 1, of being in the set. These are different things from sets!)

Those are my thoughts. The important thing here isn’t coming up with exactly the correct way to define a set, but developing a clear sense of what is a set and what is not a set.

This is important: It is easy to try to hard to make a perfect definition, and miss the whole point. The concept of “set” is hard to pin down, yet extremely simple.

Translating the original definition

I had noticed something in Shaurya’s question, so I jumped in:

I’d like to add a couple little thoughts to what Doctor Rick said.

First, I observe that the definition you quoted from Wikipedia,

a set is a gathering together into a whole of definite, distinct objects of our perception or thought which are called  elements of the set

is actually a translation into English of what the originator of the concept, Georg Cantor, said (in German) in introducing his idea. They also show, as an image, another translation of the same German:

When we translate from one language into another, there are frequently multiple words we could choose, which may have slightly different connotations (or sometimes entirely different meanings). Here, the English word “collection” comes from a Latin word meaning “gathering together” (con-, “together”, plus legere, “select”); the German word “Zusammenfassung” (which today is translated as “summarizing” or “compilation”) comes from Zusammen, “together”, plus fassen, “grasp”. Both historically could be thought of as literally bringing things together, but both can be used figuratively, especially the German. “Gathering together” is too literal a translation.

Note that they use different English words here both for “set” (“aggregate”), “gathering together” (“collection”), and “distinct” (“separate”). This suggests that the precise words are not the important thing, and also that you should be careful not to read too much into the words. I have wondered if part of your difficulty could be the language.

“Collection” commonly means a group of things actually “collected” (gathered) together in one place, but that is not what is meant here. For that reason, I think “gathering together” is a poor translation. He is talking about a sort of mental “gathering”, simply thinking of a group of things as a single entity – what he calls a “whole”. When we think of all the integers as a set Z, that is what we are doing: bring them together in our minds, thinking of all of them together as forming a single thing.

This will hopefully deal with the idea of having to literally bring the planets together (which was a wonderful example to use!).

So Cantor’s definition is simply saying that we can consider any bunch of things we think about (“objects of our intuition/perception or thought”) and think about them together (“collection”). That’s all it is. And it’s very vague because it is meant to allow these “objects” to be anything at all, tangible or not.

I deliberately use the phrase “bunch of things” in talking about sets, to convey the informality of the definition. And I think Cantor’s explanation of the “objects” he was referring to had this same goal: to take attention away from anything concrete, and focus on the fact that they can be anything we can imagine.

What goes into a set?

Shaurya responded:

Respected Math doctor,

So does it mean that a set is a sort of mental gathering of the abstract description of object that we can perceive please help me to conclude by providing confirmation.

Please tell me whether the definition of set in my textbook is correct or not.

And please elaborate the meaning of the phrase object of our intuitions in sir Cantor’s Definition.

Thank you

On his first sentence, I replied:

The trouble with this description is that it sounds more psychological than mathematical!

As Doctor Rick said, it is not words or symbols that constitute a set, nor is it “descriptions“. And a set is not thought of as “mental” (existing in one’s mind). In some sense they exist in some imagined “world” of mathematical objects.

The point of all these extra words is merely to emphasize that the idea of sets is entirely abstract. Nothing here should be taken as concrete. The elements of a set can be any “things” we can think about.

So the contents of a set are not “descriptions”, but the “things” themselves. But the “things” (objects) may be “perceived” merely by imagination. The important thing is that we don’t care what they are, so we shouldn’t put too much effort into being specific about it!

As to his book’s definition,

That definition was

a set is a well defined collection of objects.

That is an appropriate definition, to the extent that it can really be defined at all. You just have to keep in mind that an “object” can be anything you can imagine, and that “collection” doesn’t mean you are literally gathering them together in one place, but merely that you are mentally tagging them as belonging to one set.

The MathWorld definition Doctor Rick quoted at the top is identical apart from adding qualifiers and dropping the term “well-defined”, which as he said is useful:

A set is a finite or infinite collection of objects in which order has no significance, and multiplicity is generally also ignored (unlike a list or multiset).

Nothing is wrong with your book’s definition.

These are just slight variations on the same definition, emphasizing different details. But …

Is this really a definition at all?

I should explain my comment, “to the extent that it can really be defined at all“. In formal mathematics, we treat “set” as an undefined term; the “definition” we are discussing is really a description of how to think of a set, informally. As it says here, for example,

“… this is a basic, undefined word in mathematics. Other things are defined in terms of it, but it is not defined in terms of other mathematical words. Maybe imagine any collection of objects, which can be physical objects, numbers, or other sets. I sometimes think of sets as paper grocery bags, with the paper bag denoted by { and }.”

Similarly, as explained here in far more detail,

“The notion of set is so simple that it is usually introduced informally, and regarded as self-evident. In set theory, however, as is usual in mathematics, sets are given axiomatically, so their existence and basic properties are postulated by the appropriate formal axioms.”

So your questions about words like “collection” and “object” and “intuition” are not really about the mathematical definition of a set, but just about how we think of sets. It’s precisely because we can’t clearly define what a set actually is, that we ultimately “define” a set only in terms of axioms. So don’t worry that the definition is unclear to you. It is meant to be!

I often use the bag analogy myself. But in using such an image, we are not defining, just waving our hands and saying “A set is sort of like this.” This informal approach is what is called Naïve Set Theory, which is sufficient for most applications. As I indicated, mathematicians, in Axiomatic Set Theory, recognize that it is impossible to make a fully clear definition – and since definitions have to refer back to some previous, more basic concepts, something must ultimately be undefined, as discussed in the post Why Does Geometry Start With Unproved Assumptions?

As for the phrase “object of our intuitions”:

This, again, merely emphasizes that an “object” can be anything we can think about. I’m not sure of the German word that was translated either as “perception” or as “intuition”, but it would appear to mean something we “perceive” either by literally sensing it (e.g. seeing) or by imagining it. I can’t see a “3”, but I can perceive the “threeness” of a group of three ducks, so that the idea of “3” is in my mind, and I can think of it as existing in some ideal world.

But that is largely philosophy or psychology. In mathematics, we don’t worry about what a 3 actually is! We just know what we can do with it (axioms).

I did later find the exact quote in German, here:

“Unter einer ‘Menge’ verstehen wir jede Zusammenfassung M von bestimmten wohlunterschiedenen Objekten unserer Anschauung oder unseres Denkens (welche die ‘Elemente’ von M genannt werden) zu einem Ganzen.” – Georg Cantor

Google translates this as

“By a ‘set’ we understand every combination M of certain well-differentiated objects of our intuition or our thinking (which are called the ‘elements’ of M) into a whole.” – Georg Cantor

The word here translated as “intuition” is also used to mean “view” or “opinion”; he is simply referring to “things we can think about”.

One final thought from me:

All we are doing with a set is thinking of some things as a single entity.

If I talk about the set of all mammals, I am not collecting anything together, or even trying to imagine all mammals in the world being in one place. I am merely describing how to identify members of the set. If someone were to point to a mountain goat and ask if it is in my set, I would say yes, because mountain goats are mammals. That is all the set means.

I don’t need to gather all mammals, or to see them, or to list them, or to name them, but only to think of them as one kind of thing. They are a set.

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