Understanding Cardinalities Greater than Continuum

Understanding Cardinalities Greater than Continuum

When delving into the realm of infinite cardinals, one can quickly get overwhelmed with the abstract nature of these concepts. Alon Amit’s approach towards building intuition for lower infinite cardinals, such as (aleph_0) and (aleph_1) (assuming the Continuum Hypothesis), is commendable. In this article, we will explore the idea of extending this understanding to higher cardinalities, specifically (aleph_2), and discuss the role of Dedekind cuts in characterizing cardinalities.

Cardinals and Linear Orders

One key concept in understanding cardinalities is the idea of linear orders. A linear order is a binary relation that is reflexive, antisymmetric, and transitive, and for any two elements, one is either less than or greater than the other. The continuum, or (aleph_0), corresponds to the cardinality of the set of natural numbers, whereas (aleph_1) under the Continuum Hypothesis (CH) corresponds to the cardinality of the real numbers.

Alon Amit suggests defining a function (ded) (Dedekind density) to describe the relationship between cardinality and the structure of linear orders. For a cardinal (kappa), the function (ded(kappa)) is defined as:

[ded(kappa) sup { lambda : text{there exists a linear order of size } leq kappa text{ with } lambda text{ Dedekind cuts} }]

This means that (ded(kappa)) gives the supremum of the number of Dedekind cuts that can be created using a linear order of size (leq kappa). This function reveals interesting connections between the size of a set and the complexity of its order structure.

The Role of Dedekind Cuts

Dedekind cuts are a method of constructing the real numbers from the rational numbers. A Dedekind cut is a partition of the set of rational numbers (mathbb{Q}) into two non-empty subsets (A) and (B) such that every element of (A) is less than every element of (B), and (A) has no greatest element. The real numbers can be defined as the set of all Dedekind cuts of the rationals.

Alon Amit notes that the function (ded(kappa)) can be bounded by (2^{kappa}). Specifically, (ded(aleph_0) 2^{aleph_0}), which aligns with the cardinality of the continuum. Assuming the Generalized Continuum Hypothesis (GCH), (ded(kappa) 2^{kappa}) for all cardinals (kappa).

These observations suggest a relationship between the cardinality of a set and the complexity of its order structure, beyond just well-orderings. This relationship is encapsulated in the Dedekind density function, which helps to understand the nature of cardinalities greater than the continuum.

Further Theorems and Implications

Interestingly, there is an intriguing theorem by Chernikov and Shelah (referenced in the slides here) which states that for any cardinal (kappa), not assuming the Continuum Hypothesis, we have:

[2^{kappa} leq ded(ded(ded(ded(kappa))))]

This result provides a tighter bound on the cardinality of sets and their order structures, demonstrating that the complexity of linear orders can significantly affect the cardinality of a set. This theorem is particularly fascinating as it shows how deeply intertwined these concepts are.

Properties of Larger Cardinalities

The study of larger cardinals, such as (aleph_2) and beyond, is often considered more abstract, and many properties of these cardinals depend on set-theoretic assumptions. While these assumptions can sometimes take the form of supposing the existence of certain kinds of large cardinals, many properties can be proven within the framework of Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC).

This is explored in depth in a survey article by Shelah called "Cardinal Arithmetic for Skeptics" (article link). This paper is particularly readable and is aimed at a general mathematical audience. One of the notable theorems from this paper is:

[aleph_omega^{aleph_0} leq 2^{aleph_0}^{aleph_{omega_4}}]

This theorem provides a constraint on the cardinal exponentiation of (aleph_omega) and the continuum, showing how these cardinalities are interrelated within the framework of ZFC.

Conclusion

In conclusion, the study of larger cardinalities, such as (aleph_2) and beyond, involves complex and abstract concepts. The Dedekind density function offers a new perspective on understanding these cardinalities in terms of order structures. Theorems and results from set theory, such as those by Chernikov and Shelah, and Shelah himself, provide valuable insights into these cardinals and their properties. While many properties depend on set-theoretic assumptions, much can be understood within the framework of ZFC, making the study both fascinating and rigorous.