Jaakko Hintikka 1. How to Study Set Theory The continuum hypothesis (CH) is crucial in the core area of set theory, viz. in the theory of the hierarchies of infinite cardinal and infinite ordinal numbers. It is crucial in that it would, if true, help to relate the two hierarchies to each other. It says that the second infinite cardinal number, which is known to be the cardinality of the first uncountable ordinal, equals the cardinality 2 o of (...) the continuum. (Here o is the smallest infinite cardinal.). (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)

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.

Previously published: Princeton University Press, 1940.

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.

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)

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.

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)

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)

I argue that contemporary set theory, as depicted in the 2011–2012 EFI lecture series, lacks a program that promises to decide, in a genuinely realist fashion, the continuum hypothesis and related questions about the “width” of the universe. We can distinguish three possible objectives for a realist completion of set theory: maximizing structures, maximizing sets, and maximizing interpretive power. However, none of these is allied to a program that can plausibly decide CH. I discuss the implications of this (...) for set theory and other foundational programs. (shrink)

This is a survey paper which discusses the impact of large cardinals on provability of the Continuum Hypothesis. It was Gödel who first suggested that perhaps “strong axioms of infinity” could decide interesting set-theoretical statements independent over ZFC, such as CH. This hope proved largely unfounded for CH—one can show that virtually all large cardinals defined so far do not affect the status of CH. It seems to be an inherent feature of large cardinals that they do not (...) determine properties of sets low in the cumulative hierarchy if such properties can be forced to hold or fail by small forcings.The paper can also be used as an introductory text on large cardinals as it defines all relevant concepts. (shrink)

Mathematics and ethics are surprisingly similar. To some extent this is obvious, since neither looks to laboratory experiments nor sensory experience of any kind as a source of evidence. Both are based on reason and something commonly call “intuition.” This is not all. Interestingly, mathematics and ethics both possess similar distinctions between pure and applied. I explore some of the similarities and draw methodological lessons from them. We can use these lessons to explore how and why Freiling’s refutation of the (...) continuum hypothesis might be justified. (shrink)

The dream solution of the continuum hypothesis would be a solution by which we settle the continuum hypothesis on the basis of a newly discovered fundamental principle of set theory, a missing axiom, widely regarded as true. Such a dream solution would indeed be a solution, since we would all accept the new axiom along with its consequences. In this article, however, I argue that such a dream solution to $\mathrm {CH}$ is unattainable.

From the hypothesis that all Turing closed games are determined we prove: (1) the Continuum Hypothesis and (2) every subset of ℵ1 is constructible from a real.

Many results concerning the equivalence between a syntactic form of formulas and a model theoretic conditions are proven directly without using any form of a continuum hypothesis. In particular, it is demonstrated that any reduced product sentence is equivalent to a Horn sentence. Moreover, in any first order language without equality one now has that a reduced product sentence is equivalent to a Horn sentence and any sentence is equivalent to a Boolean combination of Horn sentences.

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)

The target article, in stressing the balance between neurobiological and psychological factors, makes a compelling argument in support of a continuum of perceptual and hallucinatory experience. Nevertheless, two points need to be addressed. First, the authors are probably underestimating the incidence of hallucinations in the normal population. Second, one should consider the role of absorption as a predisposing factor for hallucinations.

The nature of of Infinite Number is discussed in a rigorous but easy-to-follow manner. Special attention is paid to Cantor's proof that any given set has more subsets than members, and it is discussed how this fact bears on the question: How many infinite numbers are there? This work is ideal for people with little or no background in set theory who would like an introduction to the mathematics of the infinite.

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 Continuum Problem has inspired set theorists and philosophers since the days of Cantorian set theory. In the last 15 years, W. Hugh Woodin, a leading set theorist, has not only taken it upon himself to engage in this question, he has also changed his mind about the answer. This paper illustrates Woodin’s solutions to the problem, starting in Sect. 3 with his 1999–2004 argument that Cantor’s hypothesis about the continuum was incorrect. From 2010 onwards, Woodin presents (...) a very different argument, an argument that Cantor’s hypothesis is in fact true. This argument is still incomplete, but according to Woodin, some of the philosophical issues surrounding the Continuum Problem have been reduced to precise mathematical questions, questions that are, unlike Cantor’s hypothesis, solvable from our current theory of sets. (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.

Gitik, M., On measurable cardinals violating the continuum hypothesis, Annals of Pure and Applied Logic 63 227-240. It is shown that an extender used uncountably many times in an iteration is reconstructible. This together with the Weak Covering Lemma is used to show that the assumption o=κ+α is necessary for a measurable κ with 2κ=κ+α.

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.

Einsteinian gravity, of which Newtonian gravity is a part, is fraught with the problem of singularity that has been established as a theorem by Hawking and Penrose. The _hypothesis_ that founds the basis of both Einsteinian and Newtonian theories of gravity is that bodies with unequal magnitudes of masses fall with the same acceleration under the gravity of a source object. Since, the Einstein’s equations is one of the assumptions that underlies the proof of the singularity theorem, therefore, the above (...) hypothesis is implicitly one of the founding pillars of the same. In this work, I demonstrate how one can possibly write a non-singular theory of gravity which manifests that the above mentioned hypothesis is only valid in an approximate sense in the “large distance” scenario. To mention a specific instance, under the gravity of the earth, a 5 kg and a 500 kg fall with accelerations which differ by approximately \(113.148\times 10^{-32}\) meter/sec \(^2\) and the more massive object falls with less acceleration. Further, I demonstrate why the concept of gravitational field is not definable in the “small distance” regime which automatically justifies why the Einstein’s and Newton’s theories fail to provide any “small distance” analysis. In course of writing down this theory, I demonstrate why the continuum hypothesis as spelled out by Goedel, is undecidable. The theory has several aspects which provide the following realizations: (i) Descartes’ self-skepticism concerning exact representation of numbers by drawing lines (ii) Born’s wish of taking into account “natural uncertainty in all observations” while describing “a physical situation” by means of “real numbers” (iii) Klein’s vision of having “a fusion of arithmetic and geometry” where “a point is replaced by a small spot” (iv) Goedel’s assertion about “non-standard analysis, in some version” being “the analysis of the future”. (shrink)

We present here an approach to the fine structure of L based solely on elementary model theoretic ideas, and illustrate its use in a proof of Global Square in L. We thereby avoid the Lévy hierarchy of formulas and the subtleties of master codes and projecta, introduced by Jensen [3] in the original form of the theory. Our theory could appropriately be called ”Hyperfine Structure Theory”, as we make use of a hierarchy of structures and hull operations which refines the (...) traditional Lα -or Jα-sequences with their Σn-hull operations.§1. Introduction. In 1938, K. Gödel defined the model L of set theory to show the relative consistency of Cantor's Continuum Hypothesis. L is defined as a unionof initial segments which satisfy: L0 = ∅, Lλ = ∪α<λLα for limit ordinals λ, and, crucially, Lα + 1 = the collection of 1st order definable subsets of Lα. Since every transitive model of set theory must be closed under 1st order definability, L turns out to be the smallest inner model of set theory. Thus it occupies the central place in the set theoretic spectrum of models.The proof of the continuum hypothesis in L is based on the very uniform hierarchical definition of the L-hierarchy. The Condensation Lemma states that if π : M → Lα is an elementary embedding, M transitive, then some ; the lemma can be proved by induction on α. If a real, i.e., a subset of ω, is definable over some Lα,then by a Löwenheim-Skolem argument it is definable over some countable M as above, and hence over some, < ω1. This allows one to list the reals in L in length ω1 and therefore proves the Continuum Hypothesis in L. (shrink)

We present here an approach to the fine structure of L based solely on elementary model theoretic ideas, and illustrate its use in a proof of Global Square in L. We thereby avoid the Lévy hierarchy of formulas and the subtleties of master codes and projecta, introduced by Jensen [3] in the original form of the theory. Our theory could appropriately be called ”Hyperfine Structure Theory”, as we make use of a hierarchy of structures and hull operations which refines the (...) traditional Lα -or Jα-sequences with their Σn-hull operations.§1. Introduction. In 1938, K. Gödel defined the model L of set theory to show the relative consistency of Cantor's Continuum Hypothesis. L is defined as a unionof initial segments which satisfy: L0 = ∅, Lλ = ∪α<λLα for limit ordinals λ, and, crucially, Lα + 1 = the collection of 1st order definable subsets of Lα. Since every transitive model of set theory must be closed under 1st order definability, L turns out to be the smallest inner model of set theory. Thus it occupies the central place in the set theoretic spectrum of models.The proof of the continuum hypothesis in L is based on the very uniform hierarchical definition of the L-hierarchy. The Condensation Lemma states that if π : M → Lα is an elementary embedding, M transitive, then some ; the lemma can be proved by induction on α. If a real, i.e., a subset of ω, is definable over some Lα,then by a Löwenheim-Skolem argument it is definable over some countable M as above, and hence over some, < ω1. This allows one to list the reals in L in length ω1 and therefore proves the Continuum Hypothesis in L. (shrink)