Transfinite Composition

	Transfinite Composition Transfinite may refer to: * Transfinite number, a number larger than all finite numbers, yet not absolutely infinite * Transfinite induction, an extension of mathematical induction to well-ordered sets ** Transfinite recursion Transfinite induction is an extension of mathematical induction to well-ordered sets, for example to sets of ordinal numbers or cardinal numbers. Its correctness is a theorem of ZFC. Induction by cases Let P(\alpha) be a property defined for a ... * Transfinite arithmetic, the generalization of elementary arithmetic to infinite quantities * Transfinite interpolation, a method in numerical analysis to construct functions over a planar domain so that they match a given function on the boundary {{disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Transfinite Number In mathematics, transfinite numbers or infinite numbers are numbers that are " infinite" in the sense that they are larger than all finite numbers. These include the transfinite cardinals, which are cardinal numbers used to quantify the size of infinite sets, and the transfinite ordinals, which are ordinal numbers used to provide an ordering of infinite sets. The term ''transfinite'' was coined in 1895 by Georg Cantor, who wished to avoid some of the implications of the word ''infinite'' in connection with these objects, which were, nevertheless, not ''finite''. Few contemporary writers share these qualms; it is now accepted usage to refer to transfinite cardinals and ordinals as ''infinite numbers''. Nevertheless, the term ''transfinite'' also remains in use. Notable work on transfinite numbers was done by Wacław Sierpiński: ''Leçons sur les nombres transfinis'' (1928 book) much expanded into '' Cardinal and Ordinal Numbers'' (1958, 2nd ed. 1965). Definition Any finite natu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Transfinite Induction Transfinite induction is an extension of mathematical induction to well-ordered sets, for example to sets of ordinal numbers or cardinal numbers. Its correctness is a theorem of ZFC. Induction by cases Let P(\alpha) be a property defined for all ordinals \alpha. Suppose that whenever P(\beta) is true for all \beta < \alpha, then $P(\alpha)$ is also true. Then transfinite induction tells us that $P$ is true for all ordinals. Usually the proof is broken down into three cases: * Zero case: Prove that $P(0)$ is true. * Successor case: Prove that for any successor ordinal $\alpha+1$ , $P(\alpha+1)$ follows from $P(\alpha)$ (and, if necessary, $P(\beta)$ for all $\beta < \alpha$ ). * Limit case: Prove that for any [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Transfinite Arithmetic In the mathematical field of set theory, ordinal arithmetic describes the three usual operations on ordinal numbers: addition, multiplication, and exponentiation. Each can be defined in essentially two different ways: either by constructing an explicit well-ordered set that represents the result of the operation or by using transfinite recursion. Cantor normal form provides a standardized way of writing ordinals. In addition to these usual ordinal operations, there are also the "natural" arithmetic of ordinals and the nimber operations. Addition The sum of two well-ordered sets and is the ordinal representing the variant of lexicographical order with least significant position first, on the union of the Cartesian products and . This way, every element of is smaller than every element of , comparisons within keep the order they already have, and likewise for comparisons within . The definition of addition can also be given by transfinite recursion on . When the right ad ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]