Great question—this gets right to the foundational interplay between set theory and formal logic, so let’s unpack it clearly.

ZFC里的“互异元素”是怎么保证的？ #

First off, you’re correct that ZFC is the dominant set-theoretic framework, and it already handles the "distinctness" of set elements without needing an extra ad-hoc condition. The key here is the Axiom of Extensionality, which is one of ZFC’s core axioms:

Two sets are equal if and only if they have exactly the same elements.

What this means in practice is that "duplicate" elements don’t exist in ZFC sets by definition. If you tried to write something like {a, a}, the axiom tells us this is identical to {a}—there’s no way to have multiple copies of the same element in a set because they wouldn’t contribute any new elements to the collection. So the "distinctness" is baked into the axiomatic structure, not an extra rule we add.

相等关系：是逻辑基础，不是集合论定义的 #

You mentioned framing sets as S = { some objects | no two objects in the set (object1 = object2) }, which touches on a critical point: the equality relation (=) isn’t something we build from set theory or equivalence operators—it’s a primitive concept in first-order logic, which is the formal language we use to write ZFC.

In first-order logic with equality, = is a basic binary relation that satisfies three core properties automatically:

• Reflexivity: x = x for all x
• Symmetry: If x = y, then y = x
• Transitivity: If x = y and y = z, then x = z

Plus, we have the substitutivity of equals: if x = y, any statement about x is true if and only if it’s true about y. This is all part of the logical system we use to reason about sets, not something we define using sets or equivalence operators. So when we say two objects in a set are distinct, we’re just using this built-in logical equality to say object1 ≠ object2.

布尔代数和相等关系的顺序：反过来的 #

To answer your question about Boolean algebra: no, we don’t need to build a Boolean algebra first to define equality. In fact, it’s the other way around. Boolean algebras are algebraic structures that model logical operations (AND, OR, NOT), and we can actually define Boolean algebras using set theory (for example, the power set of a set, with union, intersection, and complement as the Boolean operations).

Equality is a foundational part of the logic that underpins both set theory and Boolean algebra—you can’t even define the operations of a Boolean algebra without first having a notion of equality to say when two elements of the algebra are the same.

小疏漏：避免循环定义 #

One quick note on your proposed set definition: writing "集合定义为‘互异对象的集合’" creates a circular definition, since you’re using the word "集合" to define itself. In axiomatic set theory like ZFC, we don’t define what a "set" is in natural language—instead, we use axioms to describe the properties that sets must have, and how they interact with each other. That avoids the circularity of trying to define a set using the word itself.

内容的提问来源于stack exchange，提问作者user150203