I've recently been studying set theory from some introductory textbooks (like Steinhart's "More Precisely" or Open Logic Project's "Sets, Logic, Computation"). I'm interested in the notion of a set.

In set theory textbooks the answer usually is "a set is a collection of objects" or something along these lines. The problem is with the empty set. Surely, it is a set, but it has no members, i.e., it is not a collection of objects. So saying that a set is a collection of objects must simply be a shorthand way of explaining the notion of a set to the non-specialist, but it can't be technically precise. But what is the more technical definition of a set? Perhaps, of course, sets are simply assumed as primitive within the theory and hence cannot be defined. In that case, the question can be put more broadly: what is a set?

Perhaps someone can direct me towards relevant literature regarding this topic?

  • Can we define it like this: "A set is a collection that helps rational thinking. The collection can be anything including objects."? Can the usages -- 'rational thinking' and 'anything' deny the entry-pass to null-set? – SonOfThought Oct 24 at 7:18
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    The problem with "a set is a collection of objects" is not the empty set, it is that "collection" is a synonym, so read as definition this is circular. The meaning of "set" or "collection" is rather fixed by manipulations we can do with them and sentences we can use them in, and the sketch of it is distilled into axioms of set theory(ies). So the answer is that there is no definition of "set", there can be no definition of "set", and "set" is what the axioms, or their intuitive counterparts, describe. Alternatively, one can choose alternative primitive notions and define "set" in those terms. – Conifold Oct 24 at 7:29
  • Maybe it would help to compare sets with numbers and ask whether zero is really a number. If I said I have a number of coins in my pocket, and you asked, "How many?" and I said I don't have any coins but zero is a number, you would rightly think I was messing with you. But at the very least, it is convenient to treat zero as a number, and it is convenient to treat the empty set as a set. Sets and numbers can be constructed recursively, and the operations on them require that { } and 0 qualify. Perhaps you could think of them as degenerate cases. – Bumble Oct 24 at 9:43
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    A "set" is a group of games in tennis. – Hot Licks Oct 25 at 2:40

We can compare the issue regarding the definition of set with Geometry.

Euclid's Elements opens with five definitions :

  1. A point is that which has no part.

  2. A line is breadthless length. [...]

  3. A surface is that which has length and breadth only.

They can be of some help in grasping the basic concepts, but hardly they can be conceived as real definitions at all.

In 1899 David Hilbert's published his groundbraking book on the axiomatization of geometry : Grundlagen der Geometrie, based on previous lectures. These are the first paragraphs (page 3):

Let us consider three distinct systems of things. The things composing the first system, we will call points and designate them by the letters A, B, C,...; those of the second, we will call straight lines and designate them by the letters a, b, c,...; and those of the third system, we will call planes and designate them by the Greek letters alpha, beta, gamma. [...]

We think of these points, straight lines, and planes as having certain mutual relations, which we indicate by means of such words as “are situated,” “between,” “parallel,” “congruent,” “continuous,” etc. The complete and exact description of these relations follows as a consequence of the axioms of geometry.

Hilbert's work on foundations of mathematics and logic has been called Formalism and it is still the prevailing philosophical view between "working" mathematicians.

For set we can consider Georg Cantor's mature definition of set in "Beiträge zur Begründung der transfiniten Mengenlehre", Mathematische Annalen (1895-97, Engl.transl.1915 - Dover reprint), §1, page 85 :

By an "aggregate" (Menge) we are to understand any collection into a whole (Zusammenfassung su einem Ganzen) M of definite and separate objects m of our intuition or our thought. These objects are called the "elements" of M.

Compare it with a modern textbook on set theory : Nicolas Bourbaki, Elements of Mathematics : Theory of sets (1968 - 1st French ed : 1939-57), page 65 :

From a "naive" point of view, many mathematical entities can be considered as collections or "sets" of objects. We do not seek to formalize this notion, and in the formalistic interpretation of what follows, the word "set" is to be considered as strictly synonymous with "term". In particular, phrases such as "let $X$ be a set" are, in principle, quite superfluous, since every letter is a term. Such phrases are introduced only to assist the intuitive interpretation of the text.

Thus, from a mathematical perspective, points and lines are "things" satisfying the axioms of geometry; in the same way, sets are "objects" satisfying the axioms of set theory.

Of course, also if a definition "inside" set theory of the notion of set is impossible, we can still have attempts to elucidate the notion of set from a philosophical perspective.

See e.g. Paul Benacerraf & Hilary Putnam (editors), Philosophy of Mathematics: Selected Readings, (2nd ed : 1983), Part IV. The concept of set.

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One current working definition of Set is provided by the Zermelo-Fraenkel axioms, usually with the Axiom of Choice.

There is plenty of debate about whether these axioms capture all there is to say about sets (both specific to set theory and generally around mathematical completeness), and about whether some axioms are necessary or correct, but proofs using ZFC are generally accepted.

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  • There are set theoretical systems beyond ZF and ZFC such as NBG, no? – J D Oct 24 at 19:06

I would push back on the notion that an empty set can't be a collection of objects because it has no elements in it. That's like saying a chest of drawers stops being a chest of drawers if there's nothing inside of it. Aside from this, it is really necessary for us, technically and formally, to have the notion of an empty set, because:

  1. We want the intersection of two sets to always be a set. For any two sets A, B, we would like for their intersection A ⋂ B to also be a set. In order for this to hold true even when A, B have no elements in common, we need to consider a set with no elements--an empty set--as being a valid set.

  2. We want to use hypothetical properties to define sets. For instance, I am used to thinking of the "solution set" of an equation as being the set of all values that make the equation true. If I ask for all real-number solutions to the equation x = x + 1, there are no numbers which make this equation true. But we still need to consider the set {x: p(x) = q(x)} as being a set, even if it happens that p(x) =/= q(x) for every x. In general, the axiom schema of comprehension means that, given any set A, I should be able to exhibit a subset B ⊆ A, where B is the set of all elements of A with a certain property. I need B to be a set even if there are no elements of A with that property.

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  • "That's like saying a chest of drawers stops being a chest of drawers if there's nothing inside of it." On the other hand, as a mathematical object, the empty set cannot change and suddenly have elements in it. It is necessarily empty. Taking the empty inset notation and compassing about another element with it, decides the case otherwise not quite at all, as the empty notation is not the empty set itself, is it? – Kristian Berry Oct 24 at 18:45
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    @KristianBerry I don't know what would be weird about the fact that, once you add elements to the empty set, it's no longer the empty set. Every set becomes a different set once you add more elements to it. This is not some unique property of the empty set. – Rivers McForge Oct 24 at 18:58
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    My point is that the natural-language description of the empty set is weird. Maybe not false, but weird, and perhaps absurd... I'm reaffirming the intuition behind the question we're replying to above. – Kristian Berry Oct 27 at 18:17
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    @KristianBerry I agree to an extent that "the natural-language description of the empty set is weird" and counterintuitive to a beginner. But insofar as I agree, I consider that to be more an artifact of our natural language than the empty set's intrinsic weirdness. – Rivers McForge Oct 28 at 0:35
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    @KristianBerry Like, imagine an alternate history of math where instead of the word "set", we ended up using the term "conceptual receptacle" to refer to the same concept (which would be hilarious, btw: imagine if math people constantly had to use jargon like "Euclid proved the conceptual receptacle of all primes is infinite"). If we had called "sets" "conceptual receptacles" instead, nobody would be surprised that "conceptual receptacles" could be empty. An "empty set" mostly sounds weird because we chose to call these things "sets" in the first place. – Rivers McForge Oct 28 at 0:38

A set is something that can have elements. Now, the allegedly empty set is necessarily empty, i.e. nothing can be an element of it, for it is an eternal, abstract object (let us suppose), and these don't change. Since the empty set cannot be a set of anything, it can only be a set of nothing. But if it can't be a set of anything, is it really a set? I would argue that no, it isn't. For example, if I have a cube that automatically pushes objects out of itself, any time I try to put an object into it, would I be right to call this cube a box, or rather a vicious cube? Perhaps a subconscious appreciation of this detail is what lead Peano to originally commence the natural ordinals from 1, not 0.

(Tangentially, I would even argue that 0 is not empty, but contains itself, and is not contained by the rest of the universe of sets, but is "quarantined" due to the weird arithmetical results that come from various non-wellfounded expressions involving it. But that's a tangent.)

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  • Doesn't the empty set contain itself? Intuitionally therefore, doesn't a subregion of a volume contain a smaller subregion itself devoid? – J D Oct 24 at 19:02
  • The empty set does not contain itself. { } is not the same as { ∅ }. Indeed, with von Neumann ordinals, { } is the number 0 and { ∅ } is the number 1. – Bumble Oct 24 at 19:48
  • Zermelo's (or von Neumann's) zero doesn't contain itself, but if there's a real zero aside from theirs, maybe it does contain itself... And we haven't even gotten to Conway's surreal zero, which albeit is still empty, but in a surreal way. – Kristian Berry Oct 27 at 18:22

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