1. Introduction

The Continuum Hypothesis postulates that there’s no set of numbers with a size larger than that of the integers N but smaller than the real numbers R. First introduced by Cantor in 1878, it was prominently featured as the first item on Hilbert’s list of twenty-three unsolved mathematical problems. In 1938, Gödel showed that the Continuum Hypothesis (CH) is consistent with Zermelo-Fraenkel set theory combined with the Axiom of Choice. He proposed that CH was independent of ZFC, meaning it could neither be proven nor refuted using ZFC axioms. The puzzle persisted until Paul Cohen introduced the method of forcing, which he used to demonstrate the consistency of the negation of CH with ZFC. This culminated in the proof of CH’s independence from ZFC. The Continuum Hypothesis remains a significant topic of inquiry within mathematics, especially in set theory [1].

The Continuum Hypothesis holds a pivotal position in the field of mathematics, particularly within set theory. Its importance can be highlighted for several reasons: The hypothesis delves into varying magnitudes of infinity, offering a deeper comprehension of the infinite concept. As a cornerstone of set theory, which underpins the contemporary mathematical framework, understanding the Continuum Hypothesis offers insights into the foundational structure of mathematics. It remains an enigma in mathematics, with its roots tracing back to Hilbert’s foundational questions, a collection of the most pressing unsolved mathematical problems at the onset of the 20th century.

Beyond the mathematical sphere, the Continuum Hypothesis branches out into philosophical debates. The endeavors to validate or refute it have catalyzed numerous breakthroughs in logic, and some physicists contend that grasping the varying scales of infinity might be instrumental in understanding the universe’s very nature. This paper seeks to elucidate the Continuum Hypothesis, delving into its foundational premises, its historical trajectory in terms of attempted resolutions, and its interplay within the Zermelo-Fraenkel Set Theory.

2. Unpacking the Continuum Hypothesis

2.1. The Concepts of Cardinality

When it’s impossible to match every element of set A with a different element from set B, we consider set A to be larger than set B. A function f: A → B is considered injective or one-to-one if different elements from A are mapped to different elements in B. It’s considered surjective if every element in B has a corresponding image y = f(x) from some element x in A. If a function is both injective and surjective, it’s called a bijection. When there’s a bijection between sets A and B, it means they have the same size. Conversely, if there’s no surjection, |A| < |B|. This concept is evident for finite sets, and the same principle applies to infinite sets.

The quantity of elements in a set can be thought of as the cardinality of a finite set in mathematics. To clarify, the cardinality of a set S, denoted as |S|, simply corresponds to the count of elements within that set. Even though this definition appears straightforward, it is essential to set theory and other contemporary fields of mathematics. In the event that there is a bijection between two sets, they have the same cardinality. Let A and B, two sets, show a bijection. Due to this presentation, each item in set A can be matched with a distinct item in set B. However, we need to establish the existence of both surjection and injection to prove a bijection.

A surjection is present when each element in set “y” can be associated with a unique element in set “x,” while an injection or one-to-one function is present when every element in set “x” can be linked to a distinct element in set “y.”

Given Function f and finite sets A and B:

If f is injective, then ∀a1, a2∈A, f(a1) =f(a2) ⇒ a1 = a2.

If f is surjective, then ∀b∈B, ∃a∈A, such that f(a)=b.

If and only if |A| ≦ |B|: There is an injection from A to B.

If and only if |A| ≧ |B|: There is a surjection from A to B.

If and only if |A| = |B|: There is a bijection from A to B.

2.2. The Concepts of power set

The power set of a set A, denoted as P(A), encompasses all possible subsets of A, including the empty set and A itself. It’s worth noting that the power set of a set is invariably more extensive than the original set. This can be proven using Cantor’s theorem. Here’s a simple proof:

Let’s take a set A. The power set of A, denoted by P(A), is the set of all possible subsets of A.

Now, let’s assume for contradiction that there exists a function f: A → P(A) that is surjective, i.e., for every subset B of A, there exists an element a in A such that f(a) = B.

Let’s construct a new set C = {a in A | a not in f(a)}. This means C consists of elements in A that are not in their image under f.

Since f is assumed to be surjective, there must exist an element c in A such that f(c) = C.

Now there are two possibilities:

If c is in C, then by the definition of C, c must not be in f(c). But f(c) = C, so this is a contradiction.

If c is not in C, then by the definition of C, c must be in f(c). But f(c) = C, so this is also a contradiction.

Therefore, our assumption that there exists a surjective function f: A → P(A) must be false. This means that the power set P(A) must be larger than the original set A.

This proof is a version of Cantor’s theorem and it shows that the power set of any set (whether finite or infinite) always has strictly more elements than the set itself.

2.3. The Continuum Hypothesis in Relation to Real and Natural Numbers

Aleph numbers, denoted ℵ₀, ℵ₁, ℵ₂, etc., are used to denote cardinalities in set theory. ℵ₀ is the smallest infinite cardinal number, while ℵ₁ is commonly linked to the size of the set of real numbers. The cardinality of N, represented as א₀ (aleph null), is countably infinite, meaning you can list them in a sequence. In contrast, R has an uncountably infinite cardinality, making it impossible to enumerate all of them in a sequence. Because there is no one-to-one correspondence between R and N, we conclude that the cardinality of R surpasses that of N. This observation stems from Cantor’s theorem, which asserts that the cardinality of the power set of any set A is strictly greater than that of A itself. Real numbers can be shown to be equivalent to the power set of natural numbers, establishing their larger cardinality.

So, is there a set of size between N and R? The question is surprisingly independent of the broader field of mathematics. If either possibility were proven, it could be transformed into a proof of contradiction using a complex but well-defined algorithm. It becomes evident that there is no mathematically describable set, demonstrably positioned between N and R. In this sense, such a set cannot be said to exist. Nevertheless, one can formulate a new axiom regarding the existence or non-existence of such a set without introducing any new contradictions to established axioms, and this axiom would modify the properties of R [2].

3. Attempts to Prove or Disprove the Continuum Hypothesis

3.1. Kurt Gödel’s work towards proving the continuum hypothesis

Before discussing his work on the continuum hypothesis, it’s important to mention Gödel’s first incompleteness theorem. In this theorem by Gödel, it was demonstrated that within any consistent formal system capable of expressing fundamental arithmetic, there exist mathematical truths that cannot be proven using that system. This theorem highlighted the inherent boundaries of what can be established through formal mathematical techniques. In 1940, Gödel authored a paper where he made noteworthy contributions to the examination of the continuum hypothesis and set theory [3]. The first aspect is the concept of relative consistency, as demonstrated by Gödel. He established that the continuum hypothesis (CH) does not lead to contradictions within the framework of set theory; in fact, it’s possible to construct a model of set theory where CH holds true. This didn’t settle the truth of CH but showed that it’s not inherently contradictory within set theory. Additionally, Gödel introduced the notion of constructible sets. He proved that assuming the Axiom of Constructability, which posits the existence of a specific, well-defined hierarchy of sets, leads to CH being true within this constructible universe. This result is known as Gödel’s Constructible Universe, denoted as L. Gödel’s most renowned contribution concerning CH is his proof of its independence from the standard axioms of set theory. He demonstrated that CH cannot be proven either true or false using ZFC alone. This breakthrough established that CH is undecidable within ZFC [4].

Kurt Gödel’s exploration of the continuum hypothesis revealed the boundaries of set theory and the intricate nature of inquiries about the sizes of infinite sets. Moreover, his contributions served as a cornerstone for subsequent developments in set theory. Gödel’s work on the continuum hypothesis illustrated that CH aligns with set theory, introduced the notion of constructible sets, and confirmed CH’s unprovability within the conventional set theory axioms, solidifying its status as one of the most renowned undecidable questions in the realm of mathematics.

3.2. Paul Cohen’s work on disproving the continuum hypothesis

Paul Cohen’s groundbreaking contribution to disproving the continuum hypothesis (CH) represents a significant milestone in the domain of set theory. In 1963, Cohen introduced a novel technique known as “forcing,” which enabled him to establish the independence of CH from the conventional set theory axioms, particularly Zermelo-Fraenkel set theory with the Axiom of Choice. This outcome demonstrated that within the scope of ZFC, CH is neither provable as true nor false, ultimately resolving one of the most renowned mathematical conundrums.

Cohen introduced the concept of forcing, a technique used to extend models of set theory. The central idea behind forcing is to construct a new model of set theory by adding generic sets to a given model in such a way that it satisfies certain desired properties. This method allowed Cohen to manipulate set-theoretic concepts and demonstrate the independence of various statements, including CH. Using forcing, Cohen was able to construct two models of set theory. In one model, CH was true, meaning that there were no sets of real numbers with cardinality strictly between that of the integers and the real numbers. In the other model, CH was false, meaning that there were such sets. This showed that CH is independent of ZFC, meaning that ZFC cannot settle the question of whether CH is true or false. Cohen’s work had a profound impact on the philosophy and methodology of mathematics. It demonstrated that there are mathematical statements that are undecidable within the existing axiomatic framework, challenging the traditional view that mathematics could provide definitive answers to all well-posed questions. His work spurred further research into set theory, model theory, and the study of large cardinals [5].

4. Situating the Continuum Hypothesis within Zermelo-Fraenkel Set Theory

Defining the concept of a set can be quite challenging. In 1901, Bertrand Russell presented his well-known paradox, known as “the set of all sets that do not contain themselves.” Let’s refer to this set as “x.” The question arises: Is x a member of itself? Upon careful consideration, it becomes apparent that x can only be a member of itself if it is not a member of itself, creating a contradictory situation. It is evident that the definition of a set cannot be left without limitations [6].

In 1908, Ernst Zermelo introduced the first axiomatic set theory as a solution to address such paradoxes. The proposed axioms imposed limitations on the concept of sets, effectively preventing self-referential paradoxes. However, it was later discovered, primarily by Abraham Fraenkel and others, that these axioms were insufficient to establish the existence of certain sets that mathematicians commonly assumed. To address this limitation, additional axioms were introduced by Fraenkel, John von Neumann, and others. This comprehensive framework, known as Zermelo-Fraenkel set theory with the Axiom of Choice, was established. ZFC is designed to be sufficiently powerful to demonstrate the existence of various sets that mathematicians typically rely on while avoiding constructs that lead to contradictions [7]. In the subsequent exposition we will assume that every item in the mathematical world is a set. Terms of ZFC are very simple: they are just the variables. Formulas are defined recursively to be either a ground formulas x ∈ y and x = y where x, y are terms, or composites φ ∧ ψ, ¬φ, ∀x φ and so on, where φ, ψ are formulas and x is a variable.

The axioms of ZFC include: (1) the standard first order logical axioms; (2) two axioms for equality: x = y → (x ∈ z → y ∈ z) and x = y → (z ∈ x → z ∈ y); (3) the domain-specific axioms [8].

ZFC, or Zermelo-Fraenkel set theory with the Axiom of Choice, can be expressed through various equivalent formulations. Here are some examples: (1) Two sets are considered equal if and only if they contain the same elements. (2) For any set, the union of all the sets it contains is itself a set. (3) For any set, there exists a set comprising all its subsets. (4) There is a set with an infinite number of elements. (5) When provided with a collection of nonempty sets, there exists a function capable of selecting one element from each set, which is known as the Axiom of Choice (AC) [9].

It’s important to observe that most of the axioms postulate the existence of a certain set. The goal here is to define the concept of a set in a way that encompasses a wide range of possibilities while preventing self-referential contradictions. Actually, ZFC can be likened to a simplistic programming language, deliberately designed with minimal syntax to facilitate precise reasoning. While composing formal proofs within ZFC can be exceedingly laborious, it is typically unnecessary in practical mathematics [10].

However, ZFC, despite being a powerful foundational theory for set theory and mathematics in general, does not provide a definitive answer to the Continuum Hypothesis. In fact, ZFC allows for multiple models of set theory, and in some models, CH is true, while in others, it is false. Kurt Gödel and Paul Cohen showed that CH is independent of the axioms of ZFC. The independence of CH from ZFC implies that mathematicians have some degree of freedom in accepting CH as an axiom or rejecting it. Depending on which model of set theory they work in, they may choose to accept CH as true (assuming it) or false (assuming its negation). Mathematicians can choose to work in set theories where CH is true, false, or undecidable based on their preferences and research goals. This independence has led to a rich field of study exploring different set-theoretic axioms and their consequences.

5. Conclusion

Throughout the exploration of the continuum hypothesis, this paper has delved into the rich history of mathematics and the profound questions it raises. The paper began by understanding the concepts of power set and cardinality. The hypothesis posits that there exists no set with cardinality sitting between that of the natural numbers and the real numbers. Its pivotal role in set theory and its influence on the evolution of mathematical thought were explored. The pioneering contributions of Gödel and Cohen were also highlighted; they underscored that the continuum hypothesis eludes definitive resolution within the confines of standard ZFC set theory. This revelation ushered in an era of renewed mathematical exploration and recognition of its undecidability as a core axiom.

The horizon of the continuum hypothesis remains rich with possibilities and questions. While the work of Gödel and Cohen conclusively indicated that the hypothesis can’t be pinned down within ZFC’s framework, alternative set theories and mathematical paradigms might offer fresh perspectives. Mathematicians, eager to break new ground, are on the hunt for different axioms or methodologies that might illuminate the continuum hypothesis or even chart a path to its conclusive understanding. Furthermore, advancements in diverse mathematical domains and their synergy with set theory present thrilling opportunities. With technological advancements, computational methodologies might unlock previously impenetrable mathematical domains, potentially offering novel insights into the continuum hypothesis. Within the vast realm of mathematics, numerous enigmas remain. The intractability of the continuum hypothesis serves as a poignant reminder of the boundaries of current mathematical understanding and the endless potential for exploration and discovery.