set theory continuum hypothesis von cohen paul (10 Ergebnisse)

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Taschenbuch. Zustand: Neu. Neuware - This exploration of a notorious mathematical problem is the work of the man who discovered the solution. The award-winning author employs intuitive explanations and detailed proofs in this self-contained treatment. 1966 edition. Copyright renewed 1994.

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First edition. "THE GREATEST ADVANCE IN THE FOUNDATIONS OF SET THEORY SINCE ITS AXIOMATIZATION" (KURT GÖDEL). First edition, extremely rare offprints, of Cohen's proof that the continuum hypothesis (CH) and the axiom of choice (AC) cannot be proved from the generally accepted Zermelo-Fraenkel axioms (ZF) of set theory; the method of 'forcing' which Cohen devised for the purposes of his proof revolutionized the subsequent development of set theory. Kurt Gödel wrote in 1964 that this work "no doubt is the greatest advance in the foundations of set theory since its axiomatization" (Gödel, Works 2, p. 270). Gödel had proved in 1938 that CH and AC cannot be disproved from ZF Cohen's and Gödel's results together showed that CH and AC are independent of ZF. Gödel had, in fact, also proved that AC could not be proved from ZF, although he never published this result, but his methods had failed to establish this for CH. In 1878, German mathematician Georg Cantor put forward the 'continuum hypothesis': any infinite subset of the set of all real numbers can be put into one-to-one correspondence either with the set of integers or with the set of real numbers ("There is no set whose cardinality is strictly between that of the integers and that of the real numbers"). This was the first in Hilbert's famous list of mathematical problems, presented in an address to the International Congress of Mathematicians at Paris in 1900. The 'Axiom of Choice', proposed by the German logician Ernst Zermelo in 1904, states that, given any collection of sets (even an infinite collection), each containing at least one element, it is possible to construct a new set by arbitrarily choosing one element from each set. "The principle of set theory known as the 'Axiom of Choice' has been hailed as 'probably the most interesting and, in spite of its late appearance, the most discussed axiom of mathematics, second only to Euclid's axiom of parallels which was introduced more than two thousand years ago" (Stanford Encyclopedia of Philosophy). All attempts to prove or disprove CH or AC failed until the work of Gödel and Cohen. For this work, Cohen was in 1966 awarded a Fields Medal (the equivalent for mathematics of a Nobel Prize). Just as Gödel followed up his 1938 announcement with a monograph based on a series of lectures, published in 1940 by the Princeton University Press, so Cohen followed up his announcement with a monograph entitled Set Theory and the Continuum Hypothesis, which acted as a very readable introduction to both set theory and his remarkable results. Cohen's offprints and book are accompanied here by an autograph letter from Cohen to the mathematician Martin Davis, whom Cohen writes was "directly responsible for my looking once more at set theory". RBH lists one other copy of each offprint. Not on OCLC. Provenance:MartinDavis (1928-2023), American logician and computer scientist (ownership signature and his notes on Set Theory and the Continuum Hypothesis; ALS from Cohen to Davis). Two problems had preoccupied workers in the field of set theory since its creation by Cantor beginning in the 1870s: the well-ordering principle and the cardinality of the continuum. An 'ordering' of a set X is a rule for deciding, given any two different elements of X, which one precedes the other (such as the usual ordering on the set of integers , 2, 1, 0, 1, 2, ). A 'well-ordering' of X is an ordering with the property that every non-empty subset Y of X has a least element (an element that precedes all the other elements of Y). So the usual ordering of the integers is not a well-ordering (there is no least integer), but the same ordering on the set of positive integers is a well-ordering. "Cantor had conjectured the proposition, now called the 'well-ordering theorem,' that every set can be well-ordered. In 1904 Zermelo gave a proof of this conjecture, using in an essential way the following mathematical principle: for every set X there is a 'choice function,' f, which is defined on the collection of non-empty subsets of X, such that for every subset A of X,f(A) is an element of A [so f 'chooses' an element of each subset of X]. Subsequently, in 1908, Zermelo presented an axiomatic version of set theory in which his proof of the well-ordering theorem could be carried out. One of the axioms was the principle just stated, which Zermelo referred to as the 'Axiom der Auswahl' (the Axiom of Choice, abbreviated AC). "Zermelo's proof was the subject of considerable controversy. The well-ordering theorem is quite remarkable, since, for example, there is no obvious way to define a well-ordering of the set of real numbers. Nor is such an explicit well-ordering provided by Zermelo's proof. Thus, many people who thought Zermelo's result implausible cast doubt upon the validity of AC. The other set-existence axioms all have the form that some collection of sets, explicitly definable from certain given parameters, is itself a set. The axiom of choice, on the other hand, asserts the existence of a choice function but does not provide an explicit definition of such a choice function. Zerrnelo was well aware that his axiom had this purely existential character, but many other mathematicians were uncomfortable with existence proofs that did not provide the construction of specific examples of what was asserted to exist" (Gödel, Works 2, pp. 1-2). Beginning in 1874, Cantor introduced the concept of the 'cardinality' of a set. For a set with a finite number of elements, its cardinality is just the number of elements in the set. But the cardinality is also defined for infinite sets. Two sets have the same cardinality if and only if they can be put into one-to-one correspondence. If ? is the cardinal number of a set X, the cardinal number of the set of all subsets of X is denoted by 2?(because if a finite set has n elements, the set of all its subsets has 2n elements). A set X has cardinality less than or equal to that of another set Y if X can be put into one.

Washington, 1963-64. 8vo. The two complete volumes 50 and 51 of the 'Proceedings of the National Academy of Sciences' in contemporary half cloth bindings + original printed wrappers. First editions. In this work Cohen established the independence of the Axiom of Choice (AC) from ZF and the independence of the Continuum Hypothesis (CH) from ZFC. This result is among the most outstanding achievements in 20th century mathematics. For this work Cohen was awarded the Fields Medal in 1966 and the National Medal of Science in 1967 the 1966 Fields Medal continues to be the only Fields Medal to have been awarded for a work in mathematical logic.