Proving the Existence of Unique Cardinal Numbers Without the Axiom of Choice

The Axiom of Choice (AC) is a fundamental principle in set theory, but in settings where AC does not hold, the concept of cardinality can require a more nuanced approach. This article explores how to establish the existence of unique cardinal numbers for non-empty sets without relying on AC or its equivalents, such as Scott's trick and concepts from the Von Neumann universe.

Cardinality and Equivalence Classes

Traditionally, cardinality is defined as the equivalence class of a set where sets are equivalent if and only if there exists a bijection between them. This definition allows us to prove that cardinality is well-defined, even without AC. However, this approach requires the ability to work with equivalence classes, which can become problematic when AC is not assumed.

Von Neumann Ordinals and Equinumerous Sets

One way to define cardinality in Zermelo-Fraenkel set theory (ZFC) is through the use of Von Neumann ordinals. In ZFC, you can define the cardinality of a set (X) as the least ordinal that is equinumerous with (X). This works because, in ZFC, every set can be well-ordered, allowing for a natural correspondence with ordinals. However, this approach relies on the Axiom of Choice, which may not be available in general.

When the Axiom of Choice Fails

If the Axiom of Choice (AC) is false, some sets fail to be well-ordered. Consequently, not every set admits an ordinal representation. This presents a challenge in defining the cardinality of sets that cannot be well-ordered.

Scott's Trick and the Von Neumann Hierarchy

Scott's trick provides a workaround for this issue. It involves using the concept of the von Neumann universe, a cumulative hierarchy of sets, to define cardinal numbers even when the Axiom of Choice is absent. The key idea is to consider the sets in the lowest rank of the von Neumann hierarchy that are equinumerous with a given set (X).

In the von Neumann universe, denoted (V), every set belongs to a certain rank (alpha). For a set (X), the least rank (V_alpha) that contains a set equinumerous with (X) is considered. The cardinal number of (X) is then defined as the set of all sets in (V_alpha) that are equinumerous with (X).

This approach relies on the Axiom of Regularity (AR), which ensures that every set has a rank in the von Neumann hierarchy. The Axiom of Regularity combined with Scott's trick ensures that the equivalence classes (now equivalence sets) are actually sets, thus avoiding the issues related to proper classes.

Conclusion

While the Axiom of Choice is a powerful tool for establishing the existence of unique cardinal numbers, it is not always available in all mathematical contexts. By utilizing Scott's trick and the von Neumann universe, we can still show that every non-empty set has a unique cardinal number, even in the absence of AC. This approach provides a robust and consistent framework for understanding cardinality in more general set-theoretic settings.