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Additively Indecomposable Ordinal
In set theory, a branch of mathematics, an additively indecomposable ordinal ''α'' is any ordinal number that is not 0 such that for any \beta,\gamma<\alpha, we have Additively indecomposable ordinals were named the ''gamma numbers'' by Cantor,A. Rhea, The Ordinals as a Consummate Abstraction of Number Systems (2017), preprint.p.20 and are also called ''additive principal numbers''. The class (set theory), class of additively indecomposable ordinals may be denoted , from the German "Hauptzahl".W. Pohlers, "A short course in ordinal analysis", pp. 27–78. Appearing in Peter Aczel, Aczel, Simmons, ''Proof Theory: A selection of papers from the Leeds Proof Theory Programme 1990'' (1992). Cambridge University Press, [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set Theory
Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathematics – is mostly concerned with those that are relevant to mathematics as a whole. The modern study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s. In particular, Georg Cantor is commonly considered the founder of set theory. The non-formalized systems investigated during this early stage go under the name of ''naive set theory''. After the discovery of Paradoxes of set theory, paradoxes within naive set theory (such as Russell's paradox, Cantor's paradox and the Burali-Forti paradox), various axiomatic systems were proposed in the early twentieth century, of which Zermelo–Fraenkel set theory (with or without the axiom of choice) is still the best-known and most studied. Set the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Function
In axiomatic set theory, a function is called normal (or a normal function) if it is continuous (with respect to the order topology) and strictly monotonically increasing. This is equivalent to the following two conditions: # For every limit ordinal (i.e. is neither zero nor a successor), it is the case that . # For all ordinals , it is the case that . Examples A simple normal function is given by (see ordinal arithmetic). But is ''not'' normal because it is not continuous at any limit ordinal (for example, f(\omega) = \omega+1 \ne \omega = \sup \). If is a fixed ordinal, then the functions , (for ), and (for ) are all normal. More important examples of normal functions are given by the aleph numbers f(\alpha) = \aleph_\alpha, which connect ordinal and cardinal numbers, and by the beth numbers f(\alpha) = \beth_\alpha. Properties If is normal, then for any ordinal , :. Proof: If not, choose minimal such that . Since is strictly monotonically increasing, , contr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Knuth's Up-arrow Notation
In mathematics, Knuth's up-arrow notation is a method of notation for very large integers, introduced by Donald Knuth in 1976. In his 1947 paper, R. L. Goodstein introduced the specific sequence of operations that are now called ''hyperoperations''. Goodstein also suggested the Greek names tetration, pentation, etc., for the extended operations beyond exponentiation. The sequence starts with a unary operation (the successor function with ''n'' = 0), and continues with the binary operations of addition (''n'' = 1), multiplication (''n'' = 2), exponentiation (''n'' = 3), tetration (''n'' = 4), pentation (''n'' = 5), etc. Various notations have been used to represent hyperoperations. One such notation is H_n(a,b). Knuth's up-arrow notation \uparrow is another. For example: * the single arrow \uparrow represents exponentiation (iterated multiplication) 2 \uparrow 4 = H_3(2,4) = 2\times(2\times(2\times 2)) = 2^4 = 16 * the double arrow \uparrow\uparrow represents tetration (iterated ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Ordinal
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] |
Epsilon Numbers (mathematics)
In mathematics, the epsilon numbers are a collection of transfinite numbers whose defining property is that they are fixed points of an exponential map. Consequently, they are not reachable from 0 via a finite series of applications of the chosen exponential map and of "weaker" operations like addition and multiplication. The original epsilon numbers were introduced by Georg Cantor in the context of ordinal arithmetic; they are the ordinal numbers ''ε'' that satisfy the equation :\varepsilon = \omega^\varepsilon, \, in which ω is the smallest infinite ordinal. The least such ordinal is ''ε''0 (pronounced epsilon nought (chiefly British), epsilon naught (chiefly American), or epsilon zero), which can be viewed as the "limit" obtained by transfinite recursion from a sequence of smaller limit ordinals: :\varepsilon_0 = \omega^ = \sup \left\\,, where is the supremum, which is equivalent to set union in the case of the von Neumann representation of ordinals. Larger ordinal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called the ''length'' of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. Formally, a sequence can be defined as a function from natural numbers (the positions of elements in the sequence) to the elements at each position. The notion of a sequence can be generalized to an indexed family, defined as a function from an ''arbitrary'' index set. For example, (M, A, R, Y) is a sequence of letters with the letter "M" first and "Y" last. This sequence differs from (A, R, M, Y). Also, the sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a valid sequence. Sequences can be '' finite'', as in these examples, or '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Epsilon Numbers (mathematics)
In mathematics, the epsilon numbers are a collection of transfinite numbers whose defining property is that they are fixed points of an exponential map. Consequently, they are not reachable from 0 via a finite series of applications of the chosen exponential map and of "weaker" operations like addition and multiplication. The original epsilon numbers were introduced by Georg Cantor in the context of ordinal arithmetic; they are the ordinal numbers ''ε'' that satisfy the equation :\varepsilon = \omega^\varepsilon, \, in which ω is the smallest infinite ordinal. The least such ordinal is ''ε''0 (pronounced epsilon nought (chiefly British), epsilon naught (chiefly American), or epsilon zero), which can be viewed as the "limit" obtained by transfinite recursion from a sequence of smaller limit ordinals: :\varepsilon_0 = \omega^ = \sup \left\\,, where is the supremum, which is equivalent to set union in the case of the von Neumann representation of ordinals. Larger ordinal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fixed Point (mathematics)
In mathematics, a fixed point (sometimes shortened to fixpoint), also known as an invariant point, is a value that does not change under a given transformation (mathematics), transformation. Specifically, for function (mathematics), functions, a fixed point is an element that is mapped to itself by the function. Any set of fixed points of a transformation is also an invariant set. Fixed point of a function Formally, is a fixed point of a function if belongs to both the domain of a function, domain and the codomain of , and . In particular, cannot have any fixed point if its domain is disjoint from its codomain. If is defined on the real numbers, it corresponds, in graphical terms, to a curve in the Euclidean plane, and each fixed-point corresponds to an intersection of the curve with the line , cf. picture. For example, if is defined on the real numbers by f(x) = x^2 - 3 x + 4, then 2 is a fixed point of , because . Not all functions have fixed points: for example, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order Topology
In mathematics, an order topology is a specific topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets. If ''X'' is a totally ordered set, the order topology on ''X'' is generated by the subbase of "open rays" :\ :\ for all ''a, b'' in ''X''. Provided ''X'' has at least two elements, this is equivalent to saying that the open intervals :(a,b) = \ together with the above rays form a base for the order topology. The open sets in ''X'' are the sets that are a union of (possibly infinitely many) such open intervals and rays. A topological space ''X'' is called orderable or linearly orderable if there exists a total order on its elements such that the order topology induced by that order and the given topology on ''X'' coincide. The order topology makes ''X'' into a completely normal Hausdorff space. The standard topologies on R, Q, Z, and N are the order topologies. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cardinal Number
In mathematics, a cardinal number, or cardinal for short, is what is commonly called the number of elements of a set. In the case of a finite set, its cardinal number, or cardinality is therefore a natural number. For dealing with the case of infinite sets, the infinite cardinal numbers have been introduced, which are often denoted with the Hebrew letter \aleph (aleph) marked with subscript indicating their rank among the infinite cardinals. Cardinality is defined in terms of bijective functions. Two sets have the same cardinality if, and only if, there is a one-to-one correspondence (bijection) between the elements of the two sets. In the case of finite sets, this agrees with the intuitive notion of number of elements. In the case of infinite sets, the behavior is more complex. A fundamental theorem due to Georg Cantor shows that it is possible for two infinite sets to have different cardinalities, and in particular the cardinality of the set of real numbers is gre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Initial Ordinal
The von Neumann cardinal assignment is a cardinal assignment that uses ordinal numbers. For a well-orderable set ''U'', we define its cardinal number to be the smallest ordinal number equinumerous to ''U'', using the von Neumann definition of an ordinal number. More precisely: :, U, = \mathrm(U) = \inf \, where ON is the class of ordinals. This ordinal is also called the initial ordinal of the cardinal. That such an ordinal exists and is unique is guaranteed by the fact that ''U'' is well-orderable and that the class of ordinals is well-ordered, using the axiom of replacement. With the full axiom of choice, every set is well-orderable, so every set has a cardinal; we order the cardinals using the inherited ordering from the ordinal numbers. This is readily found to coincide with the ordering via ≤''c''. This is a well-ordering of cardinal numbers. Initial ordinal of a cardinal Each ordinal has an associated cardinal, its cardinality, obtained by simply forgetting the ord ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
