1. Introduction

The Continuum Hypothesis stands as a cornerstone query in set theory, the mathematical discipline that delves deep into the nature and intricacies of sets. First postulated in 1878 by the pioneering mathematician, Georg Cantor, the CH seeks to bridge the gap in our understanding of the size and scope of certain infinite sets.

The hypothesis dwells at the heart of mathematical foundations and has, for well over a century, captivated the minds of mathematical luminaries and enthusiasts alike. Its complexities and ramifications have beckoned some of the most brilliant minds of the last century to grapple with its challenges. Noteworthy among these are Kurt Gödel, whose incompleteness theorems shook the very foundations of formal systems; Paul Cohen, who introduced the technique of forcing in set theory; and Saharon Shelah, known for his profound contributions to combinatorial set theory. Each of these mathematicians, in their unique ways, sought to decipher the enigma that is the Continuum Hypothesis. In contemporary mathematical discourse, the CH still stands tall, its solution tantalizingly out of reach. Its elusive nature, far from deterring interest, has only amplified its allure. The CH serves not just as a specific question about the sizes of infinity, but as a testament to the enduring challenges and mysteries that mathematics, as a field, offers to the world. As we continue our journey into the 21st century, the Continuum Hypothesis remains a poignant reminder of the vast terrains of knowledge still awaiting exploration.

2. Foundational Information

2.1. Cardinality

Cardinality quantifies the number of elements present in a set. For finite sets, determining cardinality is as simple as counting the elements. For instance, the set {1, 2, 3, 4} has a cardinality of 4 [1].

When comparing two sets, A and B, they possess equal cardinality if there exists a bijection between them. Such a bijection is present when both sets exhibit injective (one-to-one) and surjective (onto) mappings between each other. For infinite sets, the notion of cardinality takes on a more nuanced meaning. We often represent the cardinality of such sets using cardinal numbers like ℵ₀, symbolizing countable infinity [2]. Nonetheless, the foundational idea remains the same: two infinite sets have equivalent cardinality if a bijection can be established between them. An illustrative example pairs each natural number, denoted as ‘n,’ with its double, ‘2n,’ thereby forging a bijective link between the two sets.

2.2. Levels of Infinity

There are many different levels of infinity, and some of the key levels of infinity are: countable infinity (ℵ₀), uncountable infinity (ℵ₁) [3], higher levels of infinity (ℵ₂, ℵ₃, etc.) As an instance, the natural numbers have a cardinality characterized as countable infinity, whereas the real numbers have a cardinality characterized as uncountable infinity.

The uncountable infinity represents sets that are known to be strictly larger than countable infinity [4]. The cardinality of R is uncountable infinity as it is impossible to establish a one-to-one correspondence between the real numbers and either the natural numbers or the set of positive integers.

To illustrate this, suppose there exists a list that claims to contain all the real numbers. However, it is always possible to create a new real number by changing the nth digit of the nth number to something different [5]. Therefore, the original list cannot contain all real numbers because a real number that is not in the list has just been constructed. Using a binary expansion could better illustrate this proof. Suppose there exists a list that claims to list all real numbers between 0 and 1 in their binary expansions:

R1 = 0.101010101010...

R2 = 0.101100110011...

R3 = 0.011000100101...

R4 = 0.101011100100...

…

Now, a new binary number can be constructed by flipping the nth digit of the nth number, as follows:

0.0101...

The constructed number is guaranteed to be different from every number in the list because it varies from each number by at least one decimal place [6].

Higher levels of infinity are represented by cardinal numbers like ℵ₂, ℵ₃, and so on, each corresponding to larger and larger sets. These cardinalities are used to describe various mathematical objects and concepts, including larger sets of real numbers, uncountable sets of sets, and more.

2.3. Georg Cantor and the Continuum Hypothesis

Georg Cantor, the renowned German mathematician, is hailed for spearheading advancements in set theory. During the waning years of the 19th century, Cantor’s revolutionary insights radically reshaped the mathematical landscape [7]. He stands out as the trailblazer who meticulously dissected the enigma of infinity, positing the intriguing idea of multiple tiers of infinity.

|N| < |R|

This result, known as Cantor’s theorem, paved the way for the formulation of the Continuum Hypothesis. In other words, another level of infinity in between ℵ₀ and ℵ₁.

|N| <|X| < |R|

Given its monumental implications, it’s unsurprising that the Continuum Hypothesis was anointed as the foremost enigma on Hilbert’s roster of the 20th century’s unresolved mathematical conundrums [8].

3. Independence of the Continuum Hypothesis

3.1. Zermelo-Fraenkel Set Theory with the Axiom of Choice (ZFC)

An axiom represents a foundational, self-evident truth or principle that underpins a system of beliefs or logical reasoning. Examples of axioms include: the sun rising in the east, the equation 1+1=2, and the principle that the whole surpasses its individual parts.

The concept of independence pertains to the inability to prove a particular statement based on other given statements. A statement is considered independent if scenarios exist where it can be both true and false. In set theory, these scenarios or “universes” are termed “models.” To illustrate, consider three axioms: X, Y, and Z. One could conceive a model where both X and Y are true, and Z is likewise true. Yet, another model might affirm the truth of X and Y but negate Z. This scenario would establish the independence of axiom Z from the others. The abbreviation ZF in ZFC denotes Zermelo–Fraenkel, referencing the two mathematicians who made significant contributions to set theory. Meanwhile, the “C” in ZFC signifies the integration of the axiom of choice [9].

3.2. Gödel’s Proof

In the world of set theory, Kurt Gödel made a monumental contribution by illustrating the feasibility of a universe in which both Zermelo–Fraenkel set theory with the axiom of choice (ZFC) remains consistent and the Continuum Hypothesis (CH) is validated. He pioneered the construction of a unique model referred to as the “constructible universe,” symbolically represented as “L.” This model, underpinned by rigidly defined parameters, provides a stage for every set to be articulated through a concise set of deterministic rules.

Gödel’s genius lay not just in the creation of “L” but in his ability to harness its properties. Within the confines of this universe, he incontrovertibly demonstrated the truth of the Continuum Hypothesis. His meticulous proofs within “L” emphasized that if ZFC is accepted without contradictions, then the addition of CH doesn’t introduce any either. In layman’s terms, Gödel’s work meant that, within the specific parameters of the constructible universe “L”, the Continuum Hypothesis could coexist harmoniously with ZFC. However, while Gödel’s accomplishment was remarkable, it didn’t ascertain CH’s truth in every conceivable mathematical universe, only its consistency with ZFC. His groundbreaking work, rather than sealing the fate of CH, enriched the discourse, highlighting the intricate dance between axioms and hypotheses in set theory [10]. In doing so, Gödel underscored the beauty and complexity inherent in the foundational questions of mathematics.

3.3. Cohen’s Proof

In 1963, Paul Cohen established that the statement “not CH” (the negation of the Continuum Hypothesis) is also in accordance with set theory. To achieve this, Cohen developed a technique known as “forcing” in the early 1960s. The concept of forcing was introduced to add new sets to the universe of sets without breaking any of the usual rules. He created a special kind of list, called “conditions,” that describes how these new sets can be added. These conditions follow the usual set theory rules. In addition, Cohen also introduced the idea of a “filter,” which is like a sieve for picking out the best conditions. These filters help to choose a specific way to add the new sets. By using different filters, Cohen built two different mathematical worlds. In one world, CH is true, and in the other, CH is false. Because he could make models where CH is both true and false, he proved that CH can’t be decided using the standard set theory rules.

Cohen’s method of forcing is a big deal in mathematics because it showed how to create different mathematical worlds with different properties, and it settled a long-standing question about the Continuum Hypothesis.

4. Significance of the Continuum Hypothesis

The continuum hypothesis is one of the fundamental questions in set theory as it is one of the earliest challenges mathematicians in this field have faced. In the early 20th century, resolving CH was seen as a crucial step in establishing a solid foundation for mathematics.

This outcome had a significant and far-reaching influence on the field of mathematics. For instance, a level of uncertainty was introduced to this field, the independence of CH meant that different, equally valid mathematical universes that can be constructed based on different choices regarding CH.

The continuum hypothesis brings forth important philosophical questions regarding the nature of infinity, mathematical reality, and the boundaries of human comprehension in mathematics. It has sparked debates among philosophers of mathematics about the nature of mathematical truth.

5. Conclusion

In conclusion, the continuum hypothesis remains a profound enigma within the world of mathematics. Its independence from the Zermelo–Fraenkel set theory with the axiom of choice (ZFC) injects an undeniable element of ambiguity into mathematics. It brings to light the intriguing possibility of parallel mathematical realities, each offering its unique perspective on the CH. This realization has spurred intense philosophical discussions, compelling us to reflect on the very nature of mathematical truths. Throughout history, mathematics has been a beacon of certainty, a realm where truths were absolute. However, the CH underscores a fascinating paradox in this domain. It highlights that there exist questions so complex and deep that they might forever remain outside the scope of our full comprehension. In a discipline defined by precision and definitive answers, the continuum hypothesis emerges as a symbol of enduring challenges and the uncharted territories within the mathematical world.

Serving as a humbling reminder of limitations, no matter how advanced the progress in mathematical endeavors, enigmas will always tantalizingly elude grasp. Yet, these very mysteries make the journey captivating. The CH forces a grappling with profound notions of truth, infinity, and the foundational pillars of knowledge. A testament to the endless allure of mathematical conundrums, it weaves an intricate web of questions about reality, understanding, and the infinite. Standing on the precipice of mathematical discovery, the continuum hypothesis acts as both a beacon and a challenge, beckoning deeper exploration, challenging perceptions, and embracing uncertainties. Whether the CH will ever unveil its secrets remains unknown. However, its existence ensures that successive generations of mathematicians and philosophers remain engaged, driven by curiosity to navigate the limitless expanses of mathematical exploration.