Set

Mathematical statement

Within a specified universe or type, a set A is characterized by which objects are its elements; sets with exactly the same elements are regarded as equal.

Representative formula (exact TeX)

x\in A

Assumptions

• The type of objects or universe of discourse has been specified.
• A is a set in that context, and x is a candidate element.

Symbol scope

• x — candidate element
• A — set
• ∈ — membership relation

Intuition, definition, and misconceptions

Sets let us discuss several objects as a single whole. For specified x and A, x∈A says that x is an element of A. This expression is an interface to set language, not a complete system of set-theoretic axioms.

The extensional view of sets says that two sets are equal when they have exactly the same elements. The order in which elements are written, and repeated occurrences in a written listing, do not create a different set.

Not every writable condition may form a set without restriction. A complete theory must specify its permitted set-construction principles; this introductory entry does not choose ZF, ZFC, or another axiomatic system.

A common mistake is to confuse an element x with the singleton set containing x, or to assume that every condition defines a set without restriction.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

Verification boundary

This entry is a conceptual candidate. Its research sources are not OCME Evidence, and no axiomatization or Lean formal proof has yet been registered.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Map the introductory semantics to Mathlib's Set α and membership notation, then decide which later axiom or theorem MKO should carry extensionality.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.