Why we cannot specify sets universally

I’m reading Terrence Tao’s Analysis I and a bit of set theory recently, and a specific paradox has caught my attention. In this article, I’ll start from discussing the axioms that describes how sets are constructed, and proceed to address the dilemma: why we cannot specify sets universally?

Say now we wish to derive an axiom to construct a set. An straightforward answer comes from Cantor’s definition: any aggregation formed by combining distinct and independent objects in our minds. By this definition, it would be intuitive to consider an arbitrary property pertaining to some object and construct a collection of such that is true to form a set, as described by the axiom below (adopted from Tao’s book).

Axiom of Universal Specification. Suppose for every object we have a property pertaining to (so that for every , is either a true statement or a false statement). Then there exists a set such that for every object , It would be highly desirable that this axiom is “universally” effective. However, Russell’s paradox suggests that we cannot use universal specification as a general principle to construct sets. Heres the detailed formulation:

Define as the property that states is a set and . According to the axiom of universal specification, we can then construct a set
If , we can conclude that because property is true in this case. If is an element of , then is true and . This establishes a contradiction! The problem with the axiom is that it enables the construction of a set that is way too large. If we let denote the property “0 is a natural number,” it can be deduced that the axiom of specification literally implies the existence of a set that contains all the objects since “0 is a natural number” is universally true.

There are two ways (but not limited) to resolve this issue. We can try to establish some “hierarchy” in the construction of a set, as shown in the axiom below.

Axiom of Regularity. If is an non-empty set, then there is at least one element of which is either a set, or is disjoint from .

By the axiom of regularity, we’ll be able to show that for any arbitrary set . (Hint: construct sets and prove by contradiction.) Another way to prevent sets from being way too large is using the axiom of specification, which restrains a new set to a set that already exists.

Axiom of Specification. Let be a set and let be a property pertaining to some . If is true for some , then we can construct a set which implies is true for some .

Our discussion is now nearing its end. However, at this point, I would like to point out all the formulations above are constructed in the framework of Zermelo–Fraenkel set theory. The axioms of Zermelo-Fraenkel set theory (as we’ve already introduced) refer only to pure sets and prevent its sets from containing elements that are exactly the sets themselves. Von Neumann–Bernays–Gödel set theory is an extension of the Zermelo-Fraenkel set theory. NBG introduces the notion a class, which is a collection of sets defined by a formula whose quantifiers range only over sets. In this way, we can define classes that are larger than sets such as a the class of all sets (previously referred to as the universal set) with ease.