Contributions to the Founding of the Theory of Transfinite Numbers (el. knyga) (skaityta knyga) | knygos.lt

Aprašymas

In two memoirs of 1895 and 1897, Georg Cantor handed mathematics the actual infinite as something it could count with. He measured the sizes of infinite sets, built the transfinite cardinals and ordinals, and posed the continuum hypothesis that would outlast him by a century. Reproduced here is Philip Jourdain's 1915 Open Court translation - the standard English text, with Jourdain's historical introduction - in which Russell and a generation of readers first met set theory. When Cantor began his work in the early 1870s, the infinite was something mathematicians used but did not examine. Calculus depended on limits and infinite sums, yet the actual infinite - a completed collection with infinitely many members, treated as a single object - was regarded with suspicion inherited from Aristotle and Gauss alike. The respectable position was that infinity is only potential: a process that never ends, not a totality that exists. Cantor's achievement was to make the completed infinite a legitimate and calculable object, and in doing so to found set theory, the discipline that would become the common ground of modern mathematics. The break came in 1874. Cantor showed that the set of algebraic numbers can be put into one-to-one correspondence with the natural numbers - it is countable - while the set of real numbers cannot be. There are, in other words, at least two distinct sizes of infinity. The cardinality of the natural numbers he eventually named _0 (aleph-null), the smallest infinite cardinal; the cardinality of the real continuum is strictly larger. In 1891 he supplied the proof that has become emblematic of the whole enterprise: the diagonal argument. Suppose the real numbers between 0 and 1 could be listed; construct a new number differing from the first in its first digit, from the second in its second digit, and so on down the diagonal; the result cannot appear anywhere in the list. No enumeration can capture the continuum. The argument is a few lines long and has been reused ever since - in G"odel's incompleteness theorems, in Turing's analysis of computability - as the canonical way to prove that one infinity outruns another. From this beginning Cantor built two parallel hierarchies. The cardinal numbers measure size: how many elements a set has, whether finite or infinite. The ordinal numbers measure order: the position-structure of a well-ordered set, one in which every non-empty subset has a least element. For finite collections the two notions coincide, but in the transfinite they part company spectacularly. There is a smallest infinite ordinal, written, followed by + 1, + 2, and onward through an endless landscape of order types, all of them sharing the single cardinality _0. The memoirs reproduced in this volume are the mature exposition of both hierarchies: the arithmetic of cardinals, the theory of ordinal types, and the elaborate machinery of well-ordering that connects them. At the center stood a question Cantor could pose precisely but never settle. The continuum - the real line - has cardinality 2^ _0, the number of all subsets of a countable set, which Cantor's own theorem proves to be strictly greater than _0. Is it the very next cardinal, _1, with nothing in between? This is the continuum hypothesis, the assertion that 2^ _0 = _1. Cantor believed it and tried for years to prove it. He could not. The question proved deeper than anyone imagined: G"odel showed in 1940 that the hypothesis cannot be disproved from the standard axioms, and Paul Cohen showed in 1963 that it cannot be proved from them either. It is independent - a fact about the limits of axiomatic mathematics that Cantor's simple-seeming question forced into the open. The new theory was not welcomed quietly.