Intensional set as human signal

Grok

An intensional mathematical set is fundamentally a shorthand verbalization. It is a human communication signal.

Grok: Worked for 5s
Yes. An intensional mathematical set is, at root, a compressed human speech-act that points at a property rather than enumerating members.

Extensional vs. Intensional

• Extensional presentation: the set is its members.
  {0, 1, 2, 3}
  The notation simply lists what is already there; no further verbal work is required once the list is given.
• Intensional presentation: the set is introduced by a predicate or defining condition.
  { n ∈ ℕ | n is even }
  or, more loosely, “the set of all even natural numbers.”

The curly-brace notation plus the vertical bar (or colon) is a conventional shorthand that stands in for a longer English (or other natural-language) sentence: “the collection of those objects that satisfy the following property…” That sentence is a communicative signal between mathematicians. It relies on shared background knowledge of the ambient universe, the meaning of the predicates, and the conventions of the formal language in which the property is written.

Why “Shorthand Verbalization”

• The intension (the property) is primary; the extension (the actual collection of objects) is secondary and may even be empty, infinite, or undecidable.
• Different intensions can determine the same extension (e.g., “even numbers” versus “numbers congruent to 0 mod 2”), so the linguistic form is not uniquely determined by the set of objects.
• In ordinary mathematical practice the intensional description is almost always what is written and spoken; the full extension is rarely exhibited.

Thus the intensional set-notation functions exactly as you said: a compact, conventionalized human communication signal that lets one interlocutor direct another’s attention to a class of objects via a shared property rather than by pointing at each member individually.

You: Finite sets can be extensional of course. And in a loose way we may term some infinite sets extensional. But really any infinite set is intensional, since devices like ellipses are surreptitious conditions.

Grok: Worked for 8s
Yes — that is essentially correct, and the distinction sharpens once we separate the object from its mode of presentation.

Finite Sets

A finite set can be given purely extensionally:
{0, 1, 2, 3}
Every element is exhibited. No residual rule, pattern, or predicate is left unspoken. The notation is just a finite list.

Infinite Sets and the Ellipsis

Once the collection is infinite, any concrete linguistic presentation must stop short of listing. The devices we then employ — ellipses, “and so on,” recursive clauses, inductive definitions, or set-builder notation — all smuggle in a condition:

• {0, 1, 2, 3, …} relies on the tacit rule “continue by adding 1.”
• {2, 4, 6, 8, …} relies on “the even numbers” or “2n for n ∈ ℕ.”

Even a recursive definition such as:
ℕ = {0} ∪ {n + 1 | n ∈ ℕ}
is an intensional package: it encodes a generative property rather than an exhaustive inventory.

In that precise sense every infinite set, as it appears in mathematical discourse, is intensional. The ellipsis is not a harmless abbreviation; it is a surreptitious comprehension principle.

Foundational Caveat

In the formal ontology of ZFC (or any other standard set theory) every set is an extensional object: two sets are identical precisely when they have the same elements (Axiom of Extensionality). The intension/extension contrast therefore lives at the level of how we refer to the set, not at the level of what the set is. For infinite sets, the only practical referring devices available to us are intensional.

So your formulation is accurate: finite sets can be given extensionally; infinite sets, in actual mathematical communication, cannot. The ellipsis (and its more formal cousins) is the tell-tale sign that an intension is doing the work.