Transfinite Recursion
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 picture info 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 all ordinals \alpha. Suppose that whenever P(\beta) is true for all \beta < \alpha, then $P\left(\alpha\right)$ 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 $P\left(0\right)$ is true. * Successor case: Prove that for any successor ordinal $\alpha+1$, $P\left(\alpha+1\right)$ follows from $P\left(\alpha\right)$ (and, if necessary, $P\left(\beta\right)$ for all $\beta < \alpha$). * Limit case: Prove that for any [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] Proper Class Proper may refer to: Mathematics * Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact * Proper morphism, in algebraic geometry, an analogue of a proper map for algebraic varieties * Proper transfer function, a transfer function in control theory in which the degree of the numerator does not exceed the degree of the denominator * Proper equilibrium, in game theory, a refinement of the Nash equilibrium * Proper subset * Proper space * Proper complex random variable Other uses * Proper (liturgy), the part of a Christian liturgy that is specific to the date within the Liturgical Year * Proper frame, such system of reference in which object is stationary (non moving), sometimes also called a co-moving frame * Proper (heraldry), in heraldry, means depicted in natural colors * Proper Records, a UK record label * Proper (album), an album by Into It. Over It. released in 2011 * Proper (often capi ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] Transfinite Number In mathematics, transfinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers, yet not necessarily absolutely infinite. 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 by Georg Cantor in 1895, 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. Definition Any finite natural number can be used in at least two ways: as an ordinal and as a cardinal. Cardinal numbers specify the size of sets (e.g., a bag of five marbles), whereas ordinal numbers specify t ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] Epsilon-induction In set theory, \in-induction, also called epsilon-induction or set-induction, is a principle that can be used to prove that all sets satisfy a given property. Considered as an axiomatic principle, it is called the axiom schema of set induction. The principle implies transfinite induction and recursion. It may also be studied in a general context of induction on well-founded relations. Statement The schema is for any given property \psi of sets and states that, if for every set x, the truth of \psi(x) follows from the truth of \psi for all elements of x, then this property \psi holds for all sets. In symbols: :\forall x. \Big(\big(\forall (y \in x). \psi(y)\big)\,\to\,\psi(x)\Big)\,\to\,\forall z. \psi(z) Note that for the "bottom case" where x denotes the empty set \, the subexpression \forall(y\in x).\psi(y) is vacuously true for all propositions and so that implication is proven by just proving \psi(\). In words, if a property is persistent when collecting any sets with t ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] picture info Mathematical Induction Mathematical induction is a method for proving that a statement ''P''(''n'') is true for every natural number ''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ...  all hold. Informal metaphors help to explain this technique, such as falling dominoes or climbing a ladder: A proof by induction consists of two cases. The first, the base case, proves the statement for ''n'' = 0 without assuming any knowledge of other cases. The second case, the induction step, proves that ''if'' the statement holds for any given case ''n'' = ''k'', ''then'' it must also hold for the next case ''n'' = ''k'' + 1. These two steps establish that the statement holds for every natural number ''n''. The base case does not necessarily begin with ''n'' = 0, but often with ''n'' = 1, and possibly with any fixed natural number ''n'' = ''N'', establishing the truth of the statement for all natu ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] Axiom Of Dependent Choice In mathematics, the axiom of dependent choice, denoted by \mathsf , is a weak form of the axiom of choice ( \mathsf ) that is still sufficient to develop most of real analysis. It was introduced by Paul Bernays in a 1942 article that explores which set-theoretic axioms are needed to develop analysis."The foundation of analysis does not require the full generality of set theory but can be accomplished within a more restricted frame." The axiom of dependent choice is stated on p. 86. Formal statement A homogeneous relation R on X is called a total relation if for every a \in X, there exists some b \in X such that a\,R~b is true. The axiom of dependent choice can be stated as follows: For every nonempty set X and every total relation R on X, there exists a sequence (x_n)_ in X such that :x_n\, R~x_ for all n \in \N. ''x''0 may be taken to be any desired element of ''X''. If the set X above is restricted to be the set of all real numbers, then the resulting axiom is den ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] picture info Countable Set In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; this means that each element in the set may be associated to a unique natural number, or that the elements of the set can be counted one at a time, although the counting may never finish due to an infinite number of elements. In more technical terms, assuming the axiom of countable choice, a set is ''countable'' if its cardinality (its number of elements) is not greater than that of the natural numbers. A countable set that is not finite is said countably infinite. The concept is attributed to Georg Cantor, who proved the existence of uncountable sets, that is, sets that are not countable; for example the set of the real numbers. A note on terminology Although the terms "countable" and "countably infinite" as defined here are quite c ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] picture info Rational Number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rational numbers, also referred to as "the rationals", the field of rationals or the field of rational numbers is usually denoted by boldface , or blackboard bold \mathbb. A rational number is a real number. The real numbers that are rational are those whose decimal expansion either terminates after a finite number of digits (example: ), or eventually begins to repeat the same finite sequence of digits over and over (example: ). This statement is true not only in base 10, but also in every other integer base, such as the binary and hexadecimal ones (see ). A real number that is not rational is called irrational. Irrational numbers include , , , and . Since the set of rational numbers is countable, and the set of real numbers is uncounta ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] Cardinality Of The Continuum In set theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbers \mathbb R, sometimes called the continuum. It is an infinite cardinal number and is denoted by \mathfrak c (lowercase fraktur "c") or , \mathbb R, . The real numbers \mathbb R are more numerous than the natural numbers \mathbb N. Moreover, \mathbb R has the same number of elements as the power set of \mathbb N. Symbolically, if the cardinality of \mathbb N is denoted as \aleph_0, the cardinality of the continuum is This was proven by Georg Cantor in his uncountability proof of 1874, part of his groundbreaking study of different infinities. The inequality was later stated more simply in his diagonal argument in 1891. Cantor defined cardinality in terms of bijective functions: two sets have the same cardinality if, and only if, there exists a bijective function between them. Between any two real numbers ''a''  \mathfrak c . Alternative explanation for 𝔠 = 2ℵ0 E ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] Well-ordering Theorem In mathematics, the well-ordering theorem, also known as Zermelo's theorem, states that every set can be well-ordered. A set ''X'' is ''well-ordered'' by a strict total order if every non-empty subset of ''X'' has a least element under the ordering. The well-ordering theorem together with Zorn's lemma are the most important mathematical statements that are equivalent to the axiom of choice (often called AC, see also ). Ernst Zermelo introduced the axiom of choice as an "unobjectionable logical principle" to prove the well-ordering theorem. One can conclude from the well-ordering theorem that every set is susceptible to transfinite induction, which is considered by mathematicians to be a powerful technique. One famous consequence of the theorem is the Banach–Tarski paradox. History Georg Cantor considered the well-ordering theorem to be a "fundamental principle of thought". However, it is considered difficult or even impossible to visualize a well-ordering of \mathbb; such a ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] picture info Real Number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and more generally in all mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives. The set of real numbers is denoted or \mathbb and is sometimes called "the reals". The adjective ''real'' in this context was introduced in the 17th century by René Descartes to distinguish real numbers, associated with physical reality, from imaginary numbers (such as the square roots of ), which seemed like a theoretical contrivance unrelated to physical reality. The real numbers include the rational numbers, such as the integer and the fraction . The rest of the real num ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] picture info Vitali Set In mathematics, a Vitali set is an elementary example of a set of real numbers that is not Lebesgue measurable, found by Giuseppe Vitali in 1905. The Vitali theorem is the existence theorem that there are such sets. There are uncountably many Vitali sets, and their existence depends on the axiom of choice. In 1970, Robert Solovay constructed a model of Zermelo–Fraenkel set theory without the axiom of choice where all sets of real numbers are Lebesgue measurable, assuming the existence of an inaccessible cardinal (see Solovay model). Measurable sets Certain sets have a definite 'length' or 'mass'. For instance, the interval , 1is deemed to have length 1; more generally, an interval 'a'', ''b'' ''a'' ≤ ''b'', is deemed to have length ''b'' − ''a''. If we think of such intervals as metal rods with uniform density, they likewise have well-defined masses. The set , 1∪ , 3is composed of two intervals of length one, so we take its total length to be 2. In terms o ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]