The Continuum Hypothesis (CH) is a problem in set theory that was proposed by mathematician George Cantor in 1878. Many mathematicians, set theorists, philosophers, and logicians have attempted to provide an answer to the CH question, including Cantor himself, but though some progress has been made in certain areas as a result of these attempts, ultimately these investigations have failed to resolve CH in a satisfactory manner. CH is thus still an open question in professional set theory, and has been for almost 150 years.

    This paper, Resolution of the Continuum Hypothesis and Related Commentary, provides a satisfactory resolution to CH. The first part of this paper discusses various attempts, both historical and modern, that have been made to resolve CH, as well as why these attempts are inadequate. It is here that we discuss the inherent limitations of intuitionistic logic, a tool that has been used by some practitioners to try to find a resolution to CH, that prevent it from being a useful tool in resolving CH. In the second part, we discuss and clarify the basic concepts necessary to understand the resolution presented in this paper - specifically, cardinals and ordinals, the real line, measure, and power set. On the basis of these clarified ideas, we then present the resolution of CH at the end of the section.

    The final section of the second part provides a detailed explanation of the flaw in Cantor's famous diagonal argument, the argument which is the sole basis for the belief in multiple levels of infinity, and discusses numerous ways in which this flaw can be understood. It is not necessary to understand the flaw in the diagonal argument in order to understand the resolution to CH provided in the previous section, but given how important the diagonal argument is to the idea of multiple levels of infinity, and given that CH directly touches upon the relationship between different levels of infinity, understanding this flaw provides important context for the resolution of CH. Understanding the flaw in the diagonal argument also provides essential context for a proper interpretation of many of the results in modern set theory, which results make use of the diagonal argument and the idea of uncountable infinity which this argument implies. Without a proper understanding of the flaw in the diagonal argument we cannot obtain a sound and correct understanding of the results of modern set theory.

    The third part of this paper provides detailed discussion of a variety of the implications of the resolution of CH and the flaw in the diagonal argument presented earlier in the paper. CH is a foundational problem, and the diagonal result is a foundational idea in set theory, so there are many and diverse implications of the ideas and conclusions presented earlier in the paper. In this part, we discuss the following topics, among others:

• A proper definition of the continuum (the real number line).
• The definition of rational and irrational numbers in relation to a valid understanding of the concept of the real number line.
• The nature and meaning of continuity.
• The idea of infinitesimals, as well as certain non-standard concepts such as the hyperreals and the extended real line.
• The multiverse of set theories idea.
• The debate over whether infinite sets can exist as completed entities, and how to satisfactorily resolve this debate.
• The Cantor set, and, in particular, the claim of its uncountability.
• Whitehead's point-free geometry.
• The Banach-Tarski paradox, and, in particular, the critical role that the concept of uncountability and the full Axiom of Choice play in producing this paradoxical result.
• Goodstein sequences, and the flaw in the standard set theoretical argument that uses ω, the symbol that in set theory represents the totality of the natural numbers, to prove the Goodstein result. We discuss the nature and pattern of Goodstein sequences in detail, and then provide a legitimate proof of the Goodstein result that does not contain this flaw.
• Skolem's paradox, and why even though it is not considered an "actual" paradox by today's researchers, it is still considered an anomaly and its existence is still not satisfactorily understood. A resolution to this anomaly is provided, as well as commentary on the legitimacy of higher-order logic systems.
• The axiom of constructibility and the motivation behind its creation. Here we discuss, and critique, certain aspects of Gödel's proof of the consistency of generalized CH with his extension of ZFC into the transfinite.
• The nature of and motivations behind the axiom of choice. The paper provides a detailed assessment of the legitimacy and proper bounds of applicability of the axiom of choice, in both its countable and full versions, and critiques various aspects of its current interpretation. The paper demystifies the axiom of choice, shows its inner connection with the axiom of infinity in ZFC, and discusses various claims made about the axiom of choice in relation to its meaning and its usefulness or necessity in proving ideas both in set theory and in broader mathematics.
• The idea of inaccessible cardinals, and the extent to which this idea is a valid one.
• Suslin's problem, which is an unresolved question about lines that are isomorphic to the real line but which do not have countable dense subsets. The paper provides a resolution to Suslin's problem, and shows the inner connection between Suslin's problem and CH, which inner connection explains why both problems have resisted many serious attempts at resolution over many years.
• The set of all alephs and certain of Cantor's beliefs regarding this set.
• The legitimacy of the concept of a supertask, as well as of the concept of so-called infinite minds. We discuss these ideas in relation to the "finite" minds of humans and the question of the extent to which the human mind is able to understand infinity.
• Non-well-founded sets, and their legitimacy. The paper demystifies the concept of non-well-founded sets and provides a way to integrate a proper understanding of such sets into a logically valid version of set theory.
• The usefulness and limitations of the concept of the universe of sets.
• Poincaré's comment on the contradictory nature of Cantor's theory of infinity.
• Comments made by at least one practitioner regarding the lack of intuitiveness in Cantor's diagonal argument.
• Potter's comments on equinumerosity, and the extent to which this concept is applicable in discussing infinite sets.
• A critical assessment of cardinal and ordinal arithmetic, which are widely-accepted and widely-used procedures in set theory for combining and manipulating infinite numbers.
• A critique of the standard proof of the uncountability of the real numbers. We show that the standard proof is flawed, and discuss implications of this finding.
• The concept of limitation of size.

    At various points in the paper we discuss the psychological and emotional underpinnings of the acceptance by the vast majority of modern set theorists, philosophers, and logicians of the concepts of uncountability and multiple levels of infinity. It is shown that these psychological and emotional underpinnings, and the religious overtones (both direct and indirect) that are often associated with them, are the primary reason CH has been an outstanding problem for almost 150 years. The paper provides commentary on the path to overcoming these intellectual limitations, both in the context of set theory and in general intellectual circumstances, and shows the connection between the near-universal acceptance of modern set theory's understanding of infinity and humanity's longing for immortality.

    Since our beginning, humanity has sought to know and to conquer infinity. One of the primary goals of set theory is to do precisely this. But the desire to conquer infinity, and thus, in a crude but powerful emotional way, avoid mortality, makes it extremely difficult to understand certain foundational aspects of the universe, and extremely difficult to accept certain unpleasant realities. Understanding how to correct the flaws in modern set theory entails the revisiting and reworking of certain foundational components of its framework, which in turn entails the direct and full recognition of the illusion and the logical illegitimacy of the concept of uncountable infinity. But this concept can be a very difficult one to relinquish, since it is foundational not only in a set theoretical and philosophical sense, but, most importantly, in an emotional sense. Our intellectual work is driven by emotional need, and we have no incentive to do this work, or any work, outside of this need. This does not mean that we cannot be objective or rational in our understanding of things or in our conclusions. But it does mean that when a conclusion that has an intellectual or scholarly format purports also to satisfy a deeply felt emotional need, we have a powerful incentive to latch onto this association between idea and emotion, and to reinforce this association in our minds; and an idea to which we are emotionally attached is one which is difficult to think clearly about, and difficult to reject even if it is untrue. But such clarity of thought, and such rejection, is an essential component of being able to perceive the world as it is, and thus of coming to a mature understanding of both ourselves and our surroundings. This mature understanding is more sober, and it is more limited in various ways than one built solely by desire; but it is an understanding which allows us to live a fuller and happier life, a life which accepts unpleasant truths instead of hiding from them, because it is an understanding which allows us to experience the world from more angles and perspectives, and in more layers, contexts, and scope than rigid emotionalism could ever accommodate.