In spite of the work of Gödel and Cohen, which showed the undecidability of the Generalized Continuum Hypothesis from the axioms of set theory, the problem still remains to decide GCH on the basis of new axioms. It is almost 100 years since Cantor first conjectured the Continuum Hypothesis, yet we seem to be no closer to determining its truth. Nevertheless, it is a sound methodological principle that given any undecidable set-theoretical statement, we should search (...) for “other axioms of set theory which a more profound understanding of the concepts underlying logic and mathematics would enable us to recognize as implied by these concepts”. (shrink)

This paper explores how the Generalized Continuum Hypothesis (GCH) arose from Cantor's Continuum Hypothesis in the work of Peirce, Jourdain, Hausdorff, Tarski, and how GCH was used up to Gödel's relative consistency result.

Kurt Gödel, mathematician and logician, was one of the most influential thinkers of the twentieth century. Gödel fled Nazi Germany, fearing for his Jewish wife and fed up with Nazi interference in the affairs of the mathematics institute at the University of Göttingen. In 1933 he settled at the Institute for Advanced Study in Princeton, where he joined the group of world-famous mathematicians who made up its original faculty. His 1940 book, better known by its short title, The Consistency of (...) the Continuum Hypothesis, is a classic of modern mathematics. The continuum hypothesis, introduced by mathematician George Cantor in 1877, states that there is no set of numbers between the integers and real numbers. It was later included as the first of mathematician David Hilbert's twenty-three unsolved math problems, famously delivered as a manifesto to the field of mathematics at the International Congress of Mathematicians in Paris in 1900. In The Consistency of the Continuum Hypothesis Gödel set forth his proof for this problem. In 1999, Time magazine ranked him higher than fellow scientists Edwin Hubble, Enrico Fermi, John Maynard Keynes, James Watson, Francis Crick, and Jonas Salk. He is most renowned for his proof in 1931 of the 'incompleteness theorem,' in which he demonstrated that there are problems that cannot be solved by any set of rules or procedures. His proof wrought fruitful havoc in mathematics, logic, and beyond. (shrink)

In this paper we study some statements similar to the Partition Principle and the Trichotomy. We prove some relationships between these statements, the Axiom of Choice, and the Generalized Continuum Hypothesis. We also prove some independence results. MSC: 03E25, 03E50, 04A25, 04A50.

In this paper we show how the assumption of the generalized continuum hypothesis (GCH) can be removed or partially removed from proofs in Zermelo-Frankel set theory (ZF) of statements expressible in the simple theory of types. We assume the reader is familiar with the latter language, especially with the classification of formulas and sentences of that language into Σκη and Πκη form (cf. [1]) and with how that language can be relatively interpreted into the language of ZF.

In this paper we consider whether L(R) has “enough information” to contain a counterexample to the continuum hypothesis. We believe this question provides deep insight into the difficulties surrounding the continuum hypothesis. We show sufficient conditions for L(R) not to contain such a counterexample. Along the way we establish many results about nonstationary towers, non-reflecting stationary sets, generalizations of proper and semiproper forcing and Chang's conjecture.

The material contained in this book is based on lectures given by Cohen at Harvard in 1965. It consists of a presentation of logic, set theory and other material, culminating in Cohen's ingenious proof of the independence of the continuum hypothesis and the axiom of choice. Since this proof is certainly one of the major developments in modern mathematics, Cohen's book is something of a necessity for every serious student of the foundations of set theory and mathematics. In (...) addition to Cohen's proof, the book also contains a chapter on Gödel's proof of the consistency of the continuum hypothesis and the axiom of choice, and a good deal of other material of general interest.—H. P. K. (shrink)

In this note, I will list instances where in the literature on subcomplete forcing and its forcing principles (mostly in articles of my own), the assumption of the continuum hypothesis, or that we are working above the continuum, was omitted. I state the correct statements and provide or point to correct proofs. There are also some new results, most of which revolve around showing the necessity of the extra assumption.

This article attempts to link the notion of absolute ego as the ultimate subjectivity of consciousness in continental tradition with a phenomenology of Mathematical Continuum as this term is generally established following Cantor’s pioneering ideas on the properties and cardinalities of sets. My motivation stems mainly from the inherent ambiguities underlying the nature and properties of this fundamental mathematical notion which, in my view, cannot be resolved in principle by the analytical means of any formal language not even by (...) the addition of any new axioms to a consistent first-order axiomatical system such as the Zermelo-Fraenkel Set Theory. In this phenomenologically motivated approach I deal, to some extent, with the undecidability of a fundamental statement about the cardinality of Continuum inside ZFC theory, namely the Continuum Hypothesis, and also with the underlying root of Gödel’s first incompleteness result.Este artigo procura conectar a noção de ego absoluto enquanto subjetividade última da consciência na tradição continental com a fenomenologia da matemática do contínuo tal como este termo foi estabelecido seguindo as idéias pioneiras de Cantor sobre as propriedades e cardinalidade de conjuntos. Minha motivação deriva basicamente das ambigüidades subjacentes à natureza e propriedades desta noção matemática fundamental as quais, no meu entender, não podem, em princípio, ser resolvidas pelos meios analíticos de qualquer linguagem formal e nem mesmo pela adição de novos axiomas a um sistema axiomático consistente de primeira ordem como a teoria de conjuntos de Zermelo Fraenkel . Neste tratamento fenomenológica-mente motivado eu lido em certa medida com a indecidibilidade de uma tese fundamental sobre a cardinalidade do contínuo em ZFC, a saber, a Hipótese do Contínuo, e também com aquilo que está na base do primeiro resultado de incompletude de Gödel. (shrink)

We construct models for the level-by-level equivalence between strong compactness and supercompactness containing failures of the Generalized Continuum Hypothesis at inaccessible cardinals. In one of these models, no cardinal is supercompact up to an inaccessible cardinal, and for every inaccessible cardinal $\delta $, $2^{\delta }\gt \delta ^{++}$. In another of these models, no cardinal is supercompact up to an inaccessible cardinal, and the only inaccessible cardinals at which GCH holds are also measurable. These results extend and generalize (...) earlier work of the author. (shrink)

In this article, I argue for the redundancy of Whitehead’s Platonic notion of the extensive continuum, counterposing it to his related notion of an atomic “ether of events.” I argue that Whitehead’s atomic ether is more compatible with orthodox general relativity than generally supposed and remarkably close to the contemporary idea of a discrete manifold in the causal set theory of quantum gravity. I argue that the method of extensive abstraction complements Whitehead’s atomic hypothesis by demonstrating the ultimately (...) fictive nature of any continuum. (shrink)

This paper presents and defends an argument that the continuum hypothesis is false, based on considerations about objective chance and an old theorem due to Banach and Kuratowski. More specifically, I argue that the probabilistic inductive methods standardly used in science presuppose that every proposition about the outcome of a chancy process has a certain chance between 0 and 1. I also argue in favour of the standard view that chances are countably additive. Since it is possible to (...) randomly pick out a point on a continuum, for instance using a roulette wheel or by flipping a countable infinity of fair coins, it follows, given the axioms of ZFC, that there are many different cardinalities between countable infinity and the cardinality of the continuum. (shrink)

Thanks to all the people who responded to my enquiry about the status of the Continuum Hypothesis. This is a really fascinating subject, which I could waste far too much time on. The following is a summary of some aspects of the feeling I got for the problems. This will be old hat to set theorists, and no doubt there are a couple of embarrassing misunderstandings, but it might be of some interest to non professionals.

This exploration of a notorious mathematical problem is the work of the man who discovered the solution. Written by an award-winning professor at Stanford University, it employs intuitive explanations as well as detailed mathematical proofs in a self-contained treatment. This unique text and reference is suitable for students and professionals. 1966 edition. Copyright renewed 1994.

CH is approached as a problem about the cardinality of the second number class Γ. For the purpose, the theory of constituents is extended to the countably infinite case where the nodes of a constituent tree are sequences of finite constituents. Certain branches (‘perfect’ ones) specify the structures of which a model of a countably infinite constituent consists. In the case of Γ, these branches keep on splitting indefinitely and hence have the cardinality of the continuum. Since Γ is (...) maximal, they are all satisfied in it. (shrink)

The major point of contention among the philosophers and mathematicians who have written about the independence results for the continuum hypothesis (CH) and related questions in set theory has been the question of whether these results give reason to doubt that the independent statements have definite truth values. This paper concerns the views of G. Kreisel, who gives arguments based on second order logic that the CH does have a truth value. The view defended here is that although (...) Kreisel's conclusion is correct, his arguments are unsatisfactory. Later sections of the paper advance a different argument that the independence results do not show lack of truth values. (shrink)

The purpose of this article is to explain why I believe that the Continuum Hypothesis (CH) is not a definite mathematical problem. My reason for that is that the concept of arbitrary set essential to its formulation is vague or underdetermined and there is no way to sharpen it without violating what it is supposed to be about. In addition, there is considerable circumstantial evidence to support the view that CH is not definite.

We consider various quotients of the C*-algebra of bounded operators on a nonseparable Hilbert space, and prove in some cases that, assuming some restriction of the Generalized Continuum Hypothesis, there are many outer automorphisms.