# Is a set a concept?

Since sets have both intentional and extensional definition my thought is yes they are concepts. But maybe there is a technical reason that sets aren't concepts?

• Perhaps you should explain what "concept" means to you. On conventional reading, "set" is a concept, but a set of 3 apples is not. The reason isn't technical at all, the idea that concepts have material objects, like apples, as their parts is just non-sensical. I am also not sure how intensional and extensional definitions are relevant considering that definitions are distinct from entities they define. The fact that we have to use words to refer to both sets and concepts (and anything else, for that matter) does not make them identical to each other or to the words. Oct 14, 2020 at 8:27
• During the early development of set theory, it was discovered by Cantor that the “naïve assumption that all well-defined collections, or systems, are also ‘consistent systems’ ” was wrong. This can correspond to the discovery of Russell's Paradox, according to which the view, like that of Frege, according to which to every concept there is the corresponding extension, i.e. the class of all and only those objects satisfying that concept, was mathematically untenable. Oct 14, 2020 at 8:39
• If we move to modern AST, following previous discussion, it is not true that for every formula P(x) of the language "defining a property" (and thus describing a concept) the theory proves that the corresponding "extension" is a set. For example, in usual set theory (like ZF) where there is no universal set, there is no set corresponding to the concept (x=x). Oct 14, 2020 at 8:44
• But for the other way the answer is YES. If our set theory proves that there is a set A, then this set will correspond to the "concept" x ∈ A. Oct 14, 2020 at 8:45
• @MauroALLEGRANZA Does this mean, for the linked question that the intention of Concepts can be an element but not its extension? Oct 14, 2020 at 8:49

According to the entry 'Concepts' from Stanford Encyclopedia of Philosophy:

Concepts are the building blocks of thoughts. Consequently, they are crucial to such psychological processes as categorization, inference, memory, learning, and decision-making. This much is relatively uncontroversial. But the nature of concepts—the kind of things concepts are—and the constraints that govern a theory of concepts have been the subject of much debate...

Thus, Conifold asks you what your definition of 'concept' is, because there are vastly different readings from various philosophers and linguists. Under most standard readings, a 'set' is a basic concept, but concepts tend to come in two forms, one intuitional or naive, and the other formal, rigorous, and usually axiomatic. So, while today, the word 'set' generally is a rigorously defined concept, other words in natural language like 'collection' may be used to refer to the intuitive concept to avoid confusion with the intuitional notion. For example, both axiomatic systems ZF and NBG define 'set' as a type of collection, and do so differently.

What you seem to be struggling with in this and the preceding post is what exactly a set and a concept are, and how they relate. So let's flesh out some basic ideas. First, one has an intuitional notion of 'collection', something that is just a collection of things. In modern philosophy, the study of metaphysical presumption of 'collections' is called mereology. When philosophers like Frege, Cantor, Dedekind, and Peano began examining the foundations of mathematics, they realized that something like arithmetic was intuitive and not rigorously defined. Hence the need to create and examine logical axioms of arithmetic. The question of what a 'collection' was was rocked by Russell's discovery of his infamous paradox, which in set-builder notation is quite a simple proposition:

R := {x:x∉x} -> (x∈x <-> x∉x)

Which basically says that a set which is a member of itself must not be a member of itself, hence paradox.

So, immediately, the naive presumptions of set theory needed to be explored and that wound up triggering two strategies which are known as ZF and NBG in modern parlance. Once, this process began, the definition of 'set' went from being intuitively a 'collection' (whatever that may mean) to having rigorous definitions of 'set' and 'class'. From Goldblatt's Topoi:

[NBG has a...] powerful conceptual distinction between sets and classes. All entities referred to in NBG are thought of as classes, which correspond to our intuitive notion of collections of objects. The word "set" is reserved for those classes that are themselves members of other classes. (p.10)

and:

[In ZF... t]here is only one kind of entity, the set. All sets are built up from certain simple ones (in fact one can start with Ø)... (p.11)

So, you have two issues at play, one is the notion of taking a word which expresses an intuitive concept, i.e., SET_nl (set, natural language) and then uses that intuitive idea with constraints to form a formal concept usually defined by necessity and sufficiency, i.e., SET_zf (Zermelo-Fraenkel) and SET_nbg (von Neumann-Bernays-Goedel). This is a common feature in language when common parlance is transformed into technical language using some form of formal logic such as adherence to the Laws of Thought.

Oh, and intension and extension in set theory are nice and simple. extensional definition is simply enumerating the members, i.e., D := {cat, 2, upwards, 'truth'}, where intensional definition is specifying properties and is referred to as set-builder notation. Both intensional and extensional definitions are formal methods, whereas prototype theory in linguistics purports to address intuitional definition. In the philosophy of language, there is a split between those who advocate for truth-conditional semantics based on lexical semantics and broader readings of semantics, such as cognitive semantics, which are pragmatics-oriented.

So, to review, concepts come in two broad flavors, those of natural and formal languages, and sets might either refer to the intuitive notion of a collection or be rigorously defined by a formalized system. Once, you wrap your mind around these two dichotomies, how the brain constructs mathematical categories should become a bit clearer.

• It's uncontroversial that "set" is a concept but is every set a concept? And should both the extensional and intentional definitions of Concepts be included in the set Concepts? Oct 15, 2020 at 6:37
• Oops. Yer right. Gottem backwards. I usually crank these response out on lunch. I'll edit, thanks.
– J D
Oct 15, 2020 at 15:38
• Okay, I have a response sketched out, but I want to review a monograph on concepts that Hempel has written and look over my cognitive linguistics textbook before committing to it. The short answer from my perspective is yes and yes, but I want to ensure the justification is cogent and coherent.
– J D
Oct 15, 2020 at 22:19