 transfinite number

Math.an infinite cardinal or ordinal number.[190005]
* * *
denotation of the size of an infinite collection of objects. Comparison of certain infinite collections suggests that they have different sizes even though they are all infinite. For example, the sets of integers, rational numbers, and real numbers are all infinite; but each is a subset of the next. Ordering the size of sets according to the subset relation results in too many classifications and gives no way of comparing the size of sets involving different elements. Sets of different elements can be compared by pairing them off and seeing which set has leftover elements. If the fractions are listed in a special way, they can be paired off with the integers with no numbers left over from either set. Any infinite set that can be thus paired off with the integers is called countably, or denumerably, infinite. It has been demonstrated that the real numbers cannot be paired off in this way; and so they are called uncountable or nondenumerable and are considered as larger sets. There are still larger sets, such as the set of all functions involving real numbers. The size of infinite sets is indicated by the cardinal numbers symbolized by the Hebrew letter aleph (alef>) with subscript. Alephnull symbolizes the cardinality of any set that can be matched with the integers. The cardinality of the real numbers, or the continuum, is c. The continuum hypothesis asserts that c equals alephone, the next cardinal number; that is, no sets exist with cardinality between alephnull and alephone. The set of all subsets of a given set has a larger cardinal number than the set itself, resulting in an infinite succession of cardinal numbers of increasing size.* * *
Universalium. 2010.
Look at other dictionaries:
Transfinite number — Transfinite numbers are cardinal numbers or ordinal numbers that are larger than all finite numbers, yet not necessarily absolutely infinite. The term transfinite was coined by Georg Cantor, who wished to avoid some of the implications of the… … Wikipedia
transfinite number — noun any of several numbers which are infinite, but which are not equal to each other. They are represented by the Hebrew letter aleph (ℵ, א) with subscripts 0, 1, etc. At the beginning of Xanadus second decade of development, Nelson was pleased… … Wiktionary
transfinite number — Math. an infinite cardinal or ordinal number. [1900 05] … Useful english dictionary
number — The natural numbers are 0, 1, 2, 3… The integers are…–3, –2, –1, 0, 1, 2, 3…The rationals, as their name implies, measure ratios: any number that can be written as a/b, where a and b are integers, and b ? 0 (in other words, values of x that give… … Philosophy dictionary
Transfinite arithmetic — In mathematics, transfinite arithmetic is the generalization of elementary arithmetic to infinite quantities like infinite sets. It was originally invented by the German mathematician Georg Cantor. See also * transfinite number * cardinal… … Wikipedia
transfinite — /trans fuy nuyt/, adj. 1. going beyond or surpassing the finite. n. 2. See transfinite number. [1900 05; TRANS + FINITE] * * * … Universalium
Transfinite induction — is an extension of mathematical induction to well ordered sets, for instance to sets of ordinals or cardinals. Transfinite induction Let P(α) be a property defined for all ordinals α. Suppose that whenever P(β) is true for all β < α, then P(α) is … Wikipedia
transfinite — [transfī′nīt] adj. 1. extending beyond or surpassing the finite 2. Math. designating or of a cardinal or ordinal number that is larger than any positive integer … English World dictionary
Number — For other uses, see Numbers (disambiguation). A number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational… … Wikipedia
number game — Introduction any of various puzzles and games that involve aspects of mathematics. Mathematical recreations comprise puzzles and games that vary from naive amusements to sophisticated problems, some of which have never been solved.… … Universalium