Large cardinals beyond choice bulletin of symbolic logic. Hugh woodin berkeley yes, false tony martin ucla i have no idea. Cantor and was on top of hilberts millenium list of open problems in mathematics in 1900. With infinite sets such as the set of integers or rational numbers, this becomes more complicated to demonstrate. Conjecture is true, then every theory obtained by adding to zfc an axiom which is compatible with the existence of large cardinals 2 1 1 1. Jan 24, 2018 the continuum hypothesis and the search for mathematical infinity, w. Continuum hypothesis wikipedia, the free encyclopedia. Woodin s results come very close to this conjecture, establishing it for a substantial part of the hierarchy of large cardinals. He has made many notable contributions to the theory of inner models and determinacy. The continuum hypothesis under one formulation is simply the statement that there is no such set of real numbers. There is no set whose cardinality is strictly between that of the integers and that of the real numbers. William hugh woodin is an american mathematician and set theorist at harvard university.
Theorem cantor the set n of all natural numbers and the set r of all real numbers do not have the same cardinality. Hugh woodin does the continuum hypothesis have an answer. Introduction in 1874, cantor cl showed that every set has cardinality strictly smaller than the cardinality of its power set. This paper illustrates woodins solutions to the problem, starting in sect. Suppose there are infinitely many woodin cardinals. The continuum hypothesis ch the hypothesis or conjecture that 2. Hugh woodin, 58, is the leading proponent of an axiom called.
There is no set whose cardinality is strictly between that of the integers and the real numbers the continuum hypothesis was advanced by georg cantor in 1878, and establishing its truth or falsehood is the first of hilberts 23 problems presented in 1900. While this might appear like an innocentseeming statement, its been a central question in mathematics since it was first stated by georg cantor in 1878. The problem of the continuum hypothesis quickly came to be widely regarded as one of the most important problems in all of modern mathematics. In mathematics, the continuum hypothesis abbreviated ch is a hypothesis about the possible sizes of infinite sets. Unsolvable problems, the continuum hypothesis, and the. William hugh woodin born april 23, 1955 is an american mathematician and set theorist at harvard university. To settle infinity dispute, a new law of logic by natalie wolchover as.
Hugh woodin has attracted considerable attention since the year 2000 woodin 2001a, 2001b. Hugh woodin has devoted his life to the study of infinity, attempting to solve the unsolvable. Please practice handwashing and social distancing, and check out our resources for adapting to these times. The powerset of x is the set px fy y is a subset of xg. The continuum hypothesis and the search for mathematical. In mathematics, the continuum hypothesis abbreviated ch is a hypothesis, advanced by georg cantor in 1878, about the possible sizes of infinite sets. Introduction the question of how many real numbers there are traces all the. The continuum hypothesis stanford encyclopedia of philosophy. A type of large cardinal, the woodin cardinal, bears his name biography. Is the continuum hypothesis a definite mathematical problem. About hugh woodin the modern mathematical study of in. The 20th century choice for the axioms 1 of set theory are the zermelofrankel axioms together with the axiom of choice, these are the zfc axioms. Continuum hypothesis wikimili, the best wikipedia reader. The continuum hypothesis the intrepid mathematician.
The generalized continuum hypothesis can fail everywhere. Doing so does require some mental gymnastics and willingness to change your mind if. This claim has been well documented in the 50 years since paul cohen established that the problem of the continuum hypothesis cannot be solved on the. Professor of mathematics department of mathematics university of california, berkeley berkeley, ca usa. To settle infinity question, a new law of mathematics quanta. Primarily consisting of talks presented at a workshop at the msri during its logic year 198990, this volume is intended to reflect the whole spectrum of activities in set theory. The formal independence of cantors continuum hypothesis from the axioms of set theory zfc is an immediate corollary of the following two theorems where the state ment of the cohens theorem is recast in. The formal independence of cantors continuum hypothesis from the axioms of set theory zfc is an immediate corollary of the following two theorems where the state ment of the cohens theorem is recast in the more modern formulation of the boolean valued universe. The generalized continuum hypothesis can fail everywhere by matthew foreman and w. For the assumption in fluid mechanics, see fluid mechanics. The continuum hypothesis is a famous problem of set theory concerning the cardinality of the real numbers the continuum.
Ive read about his work on ultimate l, which would allow for finally coming to terms with questions like the continuum hypothesis. He admits that 15 years ago he thought it was false. A type of large cardinal, the woodin cardinal, bears his name. I many tried to solve the problem of the continuum hypothesis and failed. In 1940, g odel showed that it is consistent with the axioms of set theory that the continuum hypothesis be true. Arguably the most famous formally unsolvable problem of mathematics is hilberts first prob lem. Unsolvable problems, the continuum hypothesis, and the nature of in nity w. For the latest developments, see hugh woodin, the continuum hypothesis, ams notices 48 2001, pp. The continuum hypothesis was advanced by georg cantor in 1878, andestablishing its truth or falsehood is the first of hilberts 23 problems presented in 1900. This book is the first detailed treatment of his method of the stationary tower that is generally accessible to graduate students in mathematical logic.
The universe of sets the power set suppose x is a set. Assuming the axiom of choice, the continuum hypothesis is. The continuum hypothesis and the search for mathematical infinity, w. Woodin is a brilliant mathematician and i think the work hes done in set theory has led to some interesting philosophical questions. In mathematics, the continuum hypothesis abbreviated ch is a hypothesis about the possible. The continuum hypothesis, the genericmultiverse of sets, and the. Hugh woodin, a leading set theorist, has not only taken it. Hugh woodin introduction arguably the most famous formally unsolvable problem of mathematics is hilberts first problem. This is the first example of such a model and moreover the model can be characterized in terms of.
The continuum hypothesis, part i, volume 48, number 6. Every set theory that is compatible with the existence of large cardinalsandmakesthepropertiesofsetswithhereditarycardinalityatmost. In mathematics, the continuum hypothesis abbreviated ch is a hypothesis, advanced by georg cantor in 1877, about the possible sizes of infinite sets. Independence, the continuum hypothesis, and the nature of infinity hugh woodin department of mathematics, university of california, berkeley, usa. Cantor asked c2 whether for infinite sets x there is a set y of cardinality strictly between cardinality x and. The continuum hypothesis or ch answers this question in the negative. His proposed solution has become known as the continuum hypothesis, c h. An overview and some objections ilarly, woodin s conjecture see above can be rephrased in the following theorem dehornoy 2003, 14. The continuum hypothesis, part ii, volume 48, number 7. William hugh woodin is a set theorist at university of california, berkeley. It was through his attempt to prove this hypothesis that led cantor do develop set theory into a sophisticated branch of mathematics. In the first part of this article, i identified the cor.
Hugh woodin is a leading figure in modern set theory, having made many deep and lasting contributions to the field, in particular to descriptive set theory and large cardinals. The continuum hypothesis, the genericmultiverse of sets. The independence of the continuum hypothesis from the zfc axioms. Woodin defended various forms of settheoretic realism, on which the con tinuum hypothesis. The rational numbers seemingly form a counterexample to the continuum hypothesis. Chief among the holes is the continuum hypothesis, a 140yearold statement about the possible sizes of infinity. Chapter 1 the trans nite universe harvard university. The starting point for this monograph is the previously unknown connection between the continuum hypothesis and the saturation of the nonstationary ideal on. Continuum hypothesis project gutenberg selfpublishing. Bagaria, matthew foreman, alexander kechris, john steel, hugh woodin, and, especially, stevo. 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. As incomprehensible as it may seem, endlessness comes in many measures.
The hypothesis in its classical form goes back to g. Notices of the american mathematical society junejuly and august. In 1963 paul cohen established that the continuum hypothesis is not provable under the zermelofraenkel set theory axioms with choice zfc enderton 1977. Available formats pdf please select a format to send. The continuum concept in search of happiness lost jean. The modern mathematical story of infinity began in the period 187984 with a series of papers by cantor that defined the fundamental framework of. Update sept 19 here are videos of the three 2016 bernays lectures by hugh woodin on the continuum hypothesis and also the videos of the three 2012 bernays lectures on the continuum hypothesis and related topics by solomon feferman.
The continuum hypothesis asserts that there is no infinity between the smallest. In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. Hugh woodins research works harvard university, ma. I the problem of the continuum hypothesis is the rst problem hilberts list of 23 problems from his icm address of 1900. Establishing the truth or falsehood of the continuum hypothesis is the first of hilberts 23 problems presented in the year 1900.