Introduction to Infinite Cardinal Numbers and ZFC Axioms

In the realm of set theory, the Zermelo-Fraenkel axioms with the Axiom of Choice (ZFC) provide a foundation for understanding infinite cardinal numbers. One of the cardinal numbers of particular interest is (2^{aleph_0}), the cardinality of the continuum, which is often denoted as (c).

The question of whether any infinite cardinal numbers have been ruled out by the axioms of ZFC, specifically for (2^{aleph_0}), has sparked considerable debate among mathematicians. This article delves into the intricacies of this question, drawing from contemporary set theory and theorems related to the Continuum Hypothesis.

Why (2^{aleph_0}) Cannot Be (2^c) or (c^beta)

A popular notation for (2^{aleph_0}) is (c). However, (2^c) is a cardinality that is not used in this context, as it refers to the power set of (c). Similarly, (c^beta) denotes the smallest cardinality greater than (c), which is not the same as (2^{aleph_0}).

If (alpha) is the smallest ordinal having cardinality (c^beta), then (2^{aleph_0}) cannot be (aleph_alpha), since (aleph_alpha) would be a larger cardinality. This relationship introduces a complex interplay between ordinals and cardinals within the framework of ZFC.

The Continuum and the Cardinality of (alpha)

There exists an ordinal number (alpha) such that (c aleph_alpha). This relationship is crucial and is only established with the Axiom of Choice. Thus, ZFC ensures that the continuum (c) is indeed (aleph_alpha) for some ordinal (alpha).

However, one of the significant results in ZFC is due to K?nig's Theorem, which states that the continuum is not the union of countably many sets of smaller cardinality. Therefore, (c) cannot be (aleph_alpha) for ordinals (alpha) with countable cofinality or be (aleph_0).

Solovay's Theorem and the Possible Values of (alpha)

Robert M. Solovay's contributions to set theory are pivotal. Specifically, Solovay's theorem demonstrates that, if ZFC is consistent, then it is consistent for each of the statements (c aleph_1, aleph_2, ldots) and (c aleph_{omega_1}, aleph_{omega_2}, ldots) as well as (c aleph_{omega_1}), where (omega_1) is the smallest uncomputable ordinal.

Solovay's approach involves forcing extensions to achieve such models. An ordinal (alpha) is chosen that does not have countable cofinality, and a forcing extension is used to make (aleph_alpha) the continuum. However, the forcing extension can change the identity of (alpha), particularly not in cases like (aleph_n) where (n) is an integer.

The challenge lies in the fact that the continuum cannot be expressed as (aleph_alpha) for ordinals (alpha) with countable cofinality. Additionally, (c) is not ruled out by the successor ordinal (c^ ), as expressed in the context of (alpha).

Consistency and the Identity of (alpha)

It is an open question in set theory whether it is consistent that (alpha) is the smallest ordinal having cardinality (c). This open problem underscores the complexity and depth of the issues surrounding the Continuum Hypothesis and the relationship between cardinals.

The conclusion of any proof in the form "it has been proven that (c eq X)" is an intentional statement. The identity of (X) can affect the truth of the claim, creating a potential pitfall in addressing the continuum cardinality question.

Conclusion

The study of infinite cardinals, particularly (2^{aleph_0}), remains an intricate and evolving field within set theory. While ZFC provides a robust framework, questions and open problems persist, particularly around the continuum and the smallest ordinal having cardinality (c).

The implications of Solovay's results and K?nig's theorem highlight the nuanced nature of cardinal arithmetic and the limitations of ZFC in fully determining the continuum. Further exploration into these areas continues to enrich our understanding of the foundational aspects of mathematics.