Is there a precise meaning of the word 'exist', what does it mean for a set to exist?
And what does it mean for a set to 'not exist' ?
And what is a set, what is the precise definition of a set?
There is no definition of what a set is. The reason is simple: it is hard to imagine anything more fundamental than a set that you could use in order to say what a set is. And if there were such a thing, let's call it a tet, then one would ask "but what exactly is a tet? can we define it in terms of something more fundamental, perhaps in terms of a uet or a vet or a wet or xet or a zet?". This never ends.
So, what we do is we take one of two approaches. The naive approach resorts to trying to brush the vagueness away by saying things like "a set is a collection of things where order is not important, and repetition is not possible". While not incorrect, what the hell is a collection then? Well, in the naive approach we don't care and just assume we all share the same intuition as to what sets are. This works up to a certain degree.
The other approach is the axiomatic one. Instead of saying what sets are, we say what we can do with them and which laws govern them. There are many different axiomatizations of set theory, none of which is particularly simple (e.g., requires a bit of effort to understand). Fundamental results in axiomatic set theory include such results as "it is impossible to prove the consistency of set theory", essentially saying you can't be sure that a universe of sets as we pretend to exist really exists. Other results show that certain fundamental axioms are independent of others, showing that what one may consider 'obvious' about her idea of sets need not be so obvious for others.
Sets are objects in the universe of [pure] set theory.
Informally, sets are formalization of the idea of a collection of mathematical objects.
As for the existence, existence [of a set] in the pure mathematical sense means that in a mathematical universe there is a set with particular properties. So the real question is what is a mathematical universe? It is a collection of mathematical objects which obey particular laws which we call axioms.
So does a mathematical universe exist? This is a borderline theological question. It requires belief and personal conviction. But one can do mathematics in either case. It is possible to think of mathematics as a formal manipulation of strings; or to make-believe that these objects exists, for the sake of argument; or to believe that there is some idealized universe in which the mathematical objects reside; or even believe that our universe is a mathematical one.
Each approach has its merits and its downsides, and I do not wish to discuss those. One should follow their heart and mind, and decide for themselves.
Let me add a bit on the difficulty that people have with this concept when they meet it.
What is a real number? Well, it is the limit of rational numbers. What are rational numbers? Fractions, ratios between integers. What are integers? Integers are all the things you can have by adding or subtracting $1$ from itself indefinitely.
What is $1$? Well, I have one head. That's one.
That was relatively easy, because we have a firm grasp about what is "one thing", and what is "addition of two things", so we can understand the natural numbers naturally, and then we can construct other systems. Furthermore in modern times we all hear about physics about how the time is a continuum, and it is engraved into the common knowledge of mankind. This is why many, if not all, people think about the time line when they think about real numbers.
In fact once I sat with a few folks that were engineering students, and they were told that a four-dimensional cube is best thought as a regular cube moving through time (and collecting its "trail" somehow). I don't think any of my teachers would have dared to tell students something like that.
But sets, what are sets? We don't have sets in real life, at least not in the mathematical sense, and even the collections we do have in real life are not often identified with sets. This causes a mental strain when we try to give sets some physical manifestation.
If $\pi$ is the length of half a circle with radius $1$; and if $\frac23$ is the number of "sons/children" my parents have; and if $1$ is the loneliest number... then what is a "set"?
If we consider sets as collections without order or repetition, then we can think of sets as families, or classrooms, or nations, or the clowder of alley cats living around the local dumpster. These are collections of objects, without repetition and without a particular order.
But here's the kicker. We are used to mathematical objects signifying quantity, this much is obvious from how the common person often mistakes mathematics to a bunch of equations we "solve for $x$". If that is the case, what quantity do we measure with sets?
The answer may shock you, we don't measure quantities with sets, and not every mathematical object is a number or signifies quantity. Rather mathematical objects represent structure, and quantity can be thought of as a form of structure. In that case, sets represent the minimal structure possible, on which we add more. But let me stop here, this post is long enough as it is.
The concept of set has evolved during the history of mathematics. The general idea is that we have some kind of a property (for example described by a logical formula) and we want to capture all items of our mathematical universe that satisfy this property into some kind of a collection.
However, it turned out soon that it's very tricky to define sets right. The simple definition saying for any formula $\phi$ we have a set $S_\phi$ such that $x\in S_\phi \;\leftrightarrow\; \phi(x)$ turns out to be contradictory: If we pick $\phi(x)\equiv x\not\in x$ then we get the contradiction $S_\phi\in S_\phi \;\leftrightarrow\; S_\phi\not\in S_\phi$.
This led to a precise formulation of the concept using axiomatic theories, such as ZFC. Such theories axiomatize precisely what sets exist (in such a theory) and give you tools to prove existence of certain sets. The idea is to be able to define all sets that we naturally feel should exist, but to avoid contradictions such as the Russel paradox.
For example, let's prove in ZFC that for any given set $x$ there is a set of all unordered pairs picked from $x$:
But in general you don't (and can't) have tools to decide existence of sets - for some sets you can prove their existence, for some not. For example in ZFC you can't prove (or disprove) the continuum hypothesis that states
It turns out that it is both consistent with ZFC that such a set exists or that it doesn't.
In mathematics, you do not simply say, for example, that set $S$ exists. You would add some qualifier, e.g. there exists a set $S$ with some property $P$ common to all its elements.
Likewise, for the non-existence of a set. You wouldn't simply say that set $S$ does not exist. You would also add a qualifier here, e.g. there does not exist a set $S$ with some property $P$ common to all its elements.
How do you establish that a set exists? It depends on your set theory. In ZFC, for example, you are given the only the empty set to start, and rules to construct other sets from it, sets that are also presumed to exist.
In other set theories, you are not even given the empty set. Then the existence of every set is provisional on the existence of other set(s). You cannot then actually prove the existence of any set.
To prove the non-existence of a set $S$ with property $P$ common to all its elements, you would first postulate its existence, then derive a contradiction, concluding that no such set can exist.
In many contexts, the word 'exist' has a precise meaning. If the context is omitted, its meaning becomes vague. The situation is more complicated for existence of a set, because its existence ("supposedly") requires the existence of its elements, but the existence of its elements is not always enough to ensure existence of the set itself.
One common situation for a set to 'not exist' is that the "abstract collection" is a proper class instead of a set. Slightly less common is the situation that the "abstract collection" contains some "provably" non-existent elements.
I would say that a 'set' should "model" an "extensional collection of existing objects". This is to be contrasted with an "intensional collection of potential objects", which we might call a 'proper intensional class'. The collection of all sets with two elements is a typical example of such a 'proper intensional class'. On the other hand, every specific two element set in this collection is a perfectly valid existing 'set'. However, I should add that there is a grey zone between these two extremes, for example an "intensional collection of existing objects" will often be a set, but I'm not sure that this is always the case.
Note: By "extensional", I mean that a set is (or can be) defined by the elements it contains. By "intensional", I mean the intention behind the definition of a collection. I'm thinking of polymorphic functions in modern programming languages (for example "templates" in C++) when I write this.
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