The Mystery of Zero and the Empty Set in Set Theory

Set theory forms the foundation of modern mathematics, with core concepts such as subsets and the empty set playing critical roles. In this article, we will explore why zero is considered a subset of the empty set, as well as delve into different interpretations and models of set theory.

Definition of a Subset

In set theory, a set A is a subset of a set B, denoted as A ? B, if every element of A is also an element of B. This definition is pivotal in understanding the relationship between the empty set and zero.

The Empty Set

The empty set, denoted as ? or null set, contains no elements. Since it has no elements, there are no elements in the empty set that could violate the subset condition. Therefore, the empty set ? is considered a subset of every set, even itself.

Subset of Itself

By the definition of a subset, the empty set is a subset of itself. This is due to the fact that there are no elements in the empty set that need to be checked against any other set, including itself. This makes the condition ? ? ? vacuously true.

Universal Case

The empty set being a subset of every set, including itself, is a universal truth in set theory. In the context of zero, if we represent zero as the empty set in a particular model, then zero would also be a subset of the empty set. In this model, zero is not a proper subset of the empty set because the empty set has no proper subsets.

Conventional Set Theory

In conventional set theory, including naive set theory, ZFC (Zermelo–Fraenkel set theory with the axiom of choice), and Peano arithmetic, zero (0) is not a subset of the empty set ?. Zero is not even a set—it is an element. The empty set ? consists of no elements, and its only subset is the empty set itself.

Von Neumann Construction of Ordinals

In the von Neumann construction of ordinals, the number zero is defined as the empty set: 0 ?. This construction follows the rule that the successor of a set a is defined as Sa a ∪ {a}. In this sense, if a ?, then 0 ? ? ?. Similarly, in the Zermelo-constructed ordinals, where Sa {a}, zero is also a subset of the empty set.

No Successor Set

It's important to note that in these constructions, the empty set or zero does not succeed any other set. In Peano arithmetic, the essential property that no set a exists such that Sa ? holds true. This means that zero (or the empty set) is not the successor of any set in Peano arithmetic.

Conclusion

The concept of zero as a subset of the empty set is a fascinating and complex topic in set theory. Depending on the model of set theory used, the relationship can vary. The key is to clearly define the nature of an object (whether it's a set or an element) to avoid confusion. Understanding these distinctions is crucial in maintaining consistency in mathematical reasoning.