Can There Be an Infinite Number of Infinite Sets?

The concept of infinity in mathematics has fascinated mathematicians for centuries. One of the most intriguing aspects of infinity is the idea that there can be different sizes of infinite sets, each with a higher degree of infinity than the last. This article explores this concept deeply, examining the foundational theories and recent insights.

Georg Cantor and Different Levels of Infinity

The notion that there are different levels of infinity was established by the illustrious mathematician Georg Cantor. Cantor's work on set theory has paved the way for a deeper understanding of the nature of infinity.

Cantor introduced the concept of the cardinality of a set, which measures the size of a set. He showed that the set of natural numbers (1, 2, 3, ...) and the set of real numbers are both infinite but have different cardinalities. The cardinality of the natural numbers is denoted by #955;0, and the cardinality of the real numbers is denoted by #955;1, indicating that the set of real numbers is uncountably infinite.

Limits to the Collection of Infinite Sets

Despite the existence of different sizes of infinite sets, modern set theory asserts that there is no set containing all infinite sets. In other words, there is no set S that includes every infinite set. The reason behind this is rooted in both the axiom of regularity and the union axiom, which we will discuss.

Axiom of Regularity

The axiom of regularity, also known as the axiom of foundation, is a principle in Zermelo-Fraenkel set theory (ZF) that ensures no set contains itself and there are no infinite descending chains of membership. This axiom helps us understand why there cannot be a set of all infinite sets.

Union Axiom

Another way to approach this is through the union axiom. The union axiom states that for any set of sets, there exists a set that contains all the elements of the sets in the original collection. Suppose there is a set S such that every infinite set is a member of S. By the union axiom, the union of S would contain every element of the infinite sets in S. This would imply that the union of S is a set that contains all elements of all infinite sets, which contradicts the assumption that no such set exists.

New Foundations and Sets of All Sets

In the context of New Foundations (NF), a different set theory that strives to avoid some of the paradoxes of traditional set theory, the idea of a set that contains all sets, including infinite ones, can be more intuitively understood. In NF, the set of all sets (including infinite sets) is allowed to exist without leading to contradictions, as long as no type loops are formed. The comprehension principle in NF guarantees that a set defined by any property (x: P(x)) exists, provided no type loops are created.

Implications and Conclusion

The existence of different levels of infinity and the limitations on collections of infinite sets are fundamental to modern set theory. From the ZFC perspective, the well-founded nature of sets precludes the existence of a set of all infinite sets, while from the New Foundations perspective, such a set can exist without paradox.

Understanding these concepts is crucial for those working in fields like theoretical mathematics, philosophy, and computer science, as they provide the foundational groundwork for advanced mathematical structures and algorithms.

In conclusion, while it is possible to have different levels of infinity, the collection of all infinite sets is restricted by the principles of set theory. This complexity reflects the rich and profound nature of mathematical inquiry and the ongoing exploration of the infinite.