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Buchholz Hydra
In mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal ..., specifically in graph theory and number theory, the Buchholz hydra game is a type of hydra game, which is a single-player game based on the idea of chopping pieces off a mathematical tree. The hydra game can be used to generate a rapidly growing function; BH(n), which eventually dominates all recursive functions that are provably total in "\mathsf", and is itself provably total in "\mathsf "transfinite induction with respect to TFB". Rules The game is played on a ''hydra'', a finite, rooted connected mathematical tree A with the following properties: * The root of A has a special label, usually denoted +. * Any other node of A has a label \nu \leq \omega. * All nodes directly above the root of A have ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Logic
Mathematical logic is the study of logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics. Since its inception, mathematical logic has both contributed to and been motivated by the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and Mathematical analysis, analysis. In the early 20th century it was shaped by David Hilbert's Hilbert's program, program to prove the consistency of foundational theories. Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to the program, and clarified the issues involved in pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordinal Number
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the least natural number that has not been previously used. To extend this process to various infinite sets, ordinal numbers are defined more generally as linearly ordered labels that include the natural numbers and have the property that every set of ordinals has a least element (this is needed for giving a meaning to "the least unused element"). This more general definition allows us to define an ordinal number \omega that is greater than every natural number, along with ordinal numbers \omega + 1, \omega + 2, etc., which are even greater than \omega. A linear order such that every subset has a least element is called a well-order. The axiom of choice implies that every set can be well-ordered, and given two well-ordered sets, one is isomorphic to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Large Numbers
Large numbers are numbers significantly larger than those typically used in everyday life (for instance in simple counting or in monetary transactions), appearing frequently in fields such as mathematics, cosmology, cryptography, and statistical mechanics. They are typically large positive integers, or more generally, large positive real numbers, but may also be other numbers in other contexts. Googology is the study of nomenclature and properties of large numbers. In the everyday world Scientific notation was created to handle the wide range of values that occur in scientific study. 1.0 × 109, for example, means one billion, or a 1 followed by nine zeros: 1 000 000 000. The reciprocal, 1.0 × 10−9, means one billionth, or 0.000 000 001. Writing 109 instead of nine zeros saves readers the effort and hazard of counting a long series of zeros to see how large the number is. Examples of large numbers describing everyday real-world objects include: * The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Games
A mathematical game is a game whose rules, strategies, and outcomes are defined by clear mathematical parameters. Often, such games have simple rules and match procedures, such as Tic-tac-toe and Dots and Boxes. Generally, mathematical games need not be conceptually intricate to involve deeper computational underpinnings. For example, even though the rules of Mancala are relatively basic, the game can be rigorously analyzed through the lens of combinatorial game theory. Mathematical games differ sharply from mathematical puzzles in that mathematical puzzles require specific mathematical expertise to complete, whereas mathematical games do not require a deep knowledge of mathematics to play. Often, the arithmetic core of mathematical games is not readily apparent to players untrained to note the statistical or mathematical aspects. Some mathematical games are of deep interest in the field of recreational mathematics. When studying a game's core mathematics, arithmetic theory i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trees (graph Theory)
In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, including only woody plants with secondary growth, plants that are usable as lumber or plants above a specified height. In wider definitions, the taller palms, tree ferns, bananas, and bamboos are also trees. Trees are not a taxonomic group but include a variety of plant species that have independently evolved a trunk and branches as a way to tower above other plants to compete for sunlight. The majority of tree species are angiosperms or hardwoods; of the rest, many are gymnosperms or softwoods. Trees tend to be long-lived, some reaching several thousand years old. Trees have been in existence for 370 million years. It is estimated that there are some three trillion mature trees in the world. A tree typically has many secondary branches supported clear of the ground by the trunk. This trunk typically co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Buchholz's Ordinal
In mathematics, ψ0(Ωω), widely known as Buchholz's ordinal, is a large countable ordinal that is used to measure the proof-theoretic strength of some mathematical systems. In particular, it is the proof theoretic ordinal of the subsystem \Pi_1^1-CA0 of second-order arithmetic; this is one of the "big five" subsystems studied in reverse mathematics (Simpson 1999). It is also the proof-theoretic ordinal of \mathsf, the theory of finitely iterated inductive definitions, and of KP\ell_0,T. CarlsonElementary Patterns of Resemblance(1999). Accessed 12 August 2022. a fragment of Kripke-Platek set theory extended by an axiom stating every set is contained in an admissible set In set theory, a discipline within mathematics, an admissible set is a transitive set A\, such that \langle A,\in \rangle is a model of Kripke–Platek set theory (Barwise 1975). The smallest example of an admissible set is the set of hereditaril .... Buchholz's ordinal is also the order type of the segment bou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bachmann–Howard Ordinal
In mathematics, the Bachmann–Howard ordinal (also known as the Howard ordinal, or Howard-Bachmann ordinal) is a large countable ordinal. It is the ordinal analysis, proof-theoretic ordinal of several mathematical theory (logic), theories, such as Kripke–Platek set theory (with the axiom of infinity) and the system CZF of constructive set theory. It was introduced by and . Definition The Bachmann–Howard ordinal is defined using an ordinal collapsing function: *''ε''''α'' enumerates the Epsilon numbers (mathematics), epsilon numbers, the ordinals ''ε'' such that ω''ε'' = ''ε''. *Ω = ω1 is the first uncountable ordinal. *''ε''Ω+1 is the first epsilon number after Ω = ''ε''Ω. *''ψ''(''α'') is defined to be the smallest ordinal that cannot be constructed by starting with 0, 1, ω and Ω, and repeatedly applying ordinal addition, multiplication and exponentiation, and ''ψ'' to previously constructed ordinals (except that ''ψ'' can only be applied to arguments les ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Large Veblen Ordinal
In mathematics, the large Veblen ordinal is a certain large countable ordinal, named after Oswald Veblen. There is no standard notation for ordinals beyond the Feferman–Schütte ordinal Γ0. Most systems of notation use symbols such as ψ(α), θ(α), ψα(β), some of which are modifications of the Veblen function In mathematics, the Veblen functions are a hierarchy of normal functions ( continuous strictly increasing functions from ordinals to ordinals), introduced by Oswald Veblen in . If φ0 is any normal function, then for any non-zero ordinal α, φ ...s to produce countable ordinals even for uncountable arguments, and some of which are ordinal collapsing functions. The large Veblen ordinal is sometimes denoted by \phi_(0) or \theta(\Omega^\Omega) or \psi(\Omega^). It was constructed by Veblen using an extension of Veblen functions allowing infinitely many arguments. References * * {{Number-stub Ordinal numbers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Small Veblen Ordinal
In mathematics, the small Veblen ordinal is a certain large countable ordinal, named after Oswald Veblen. It is occasionally called the Ackermann ordinal, though the Ackermann ordinal described by is somewhat smaller than the small Veblen ordinal. There is no standard notation for ordinals beyond the Feferman–Schütte ordinal \Gamma_0. Most systems of notation use symbols such as \psi(\alpha), \theta(\alpha), \psi_\alpha(\beta), some of which are modifications of the Veblen functions to produce countable ordinals even for uncountable arguments, and some of which are "collapsing functions". The small Veblen ordinal \theta_(0) or \psi(\Omega^) is the limit of ordinals that can be described using a version of Veblen functions with finitely many arguments. It is the ordinal that measures the strength of Kruskal's theorem. It is also the ordinal type of a certain ordering of rooted tree In graph theory, a tree is an undirected graph in which any two vertices are connected by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Friedman's SSCG Function
In mathematics, a simple subcubic graph (SSCG) is a finite simple Graph (discrete mathematics), graph in which each vertex (graph theory), vertex has vertex degree, degree at most three. Suppose we have a sequence of simple subcubic graphs ''G''1, ''G''2, ... such that each graph ''G''''i'' has at most ''i'' + ''k'' vertices (for some integer ''k'') and for no ''i'' < ''j'' is ''G''''i'' Homeomorphism (graph theory), homeomorphically embeddable into (i.e. is a graph minor of) ''G''''j''. The Robertson–Seymour theorem proves that subcubic graphs (simple or not) are well-founded by homeomorphic embeddability, implying such a sequence cannot be infinite. So, for each value of ''k'', there is a sequence with maximal length. The function SSCG(''k'') denotes that length for simple subcubic graphs. The function SCG(''k'') denotes that length for (general) subcubic graphs. The ''SCG'' sequence begins SCG(0) = 6, but then explodes to a value equivalent to fε ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hydra Game
In mathematics, specifically in graph theory and number theory, a hydra game is a single-player iterative mathematical game played on a tree (graph theory), mathematical tree called a ''hydra'' where, usually, the goal is to cut off the hydra's "heads" while the hydra simultaneously expands itself. Hydra games can be used to generate large numbers or infinite ordinals or prove the strength of certain mathematical theories. Unlike their combinatorial counterparts like Kruskal's tree theorem, TREE and Friedman's SSCG function, SCG, no search is required to compute these fast-growing function values – one must simply keep applying the transformation rule to the tree until the game says to stop. Introduction A simple hydra game can be defined as follows: * A hydra is a finite rooted tree, which is a network of points and lines with no loops and a single starting point known as the root. The root is referred to as R. * The player selects a leaf node x and a natural number n at eac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tree Function
In mathematics, Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding. History The theorem was conjectured by Andrew Vázsonyi and proved by ; a short proof was given by . It has since become a prominent example in reverse mathematics as a statement that cannot be proved within ATR0 (a form of arithmetical transfinite recursion), and a finitary application of the theorem gives the existence of the fast-growing TREE function. In 2004, the result was generalized from trees to graphs as the Robertson–Seymour theorem, a result that has also proved important in reverse mathematics and leads to the even-faster-growing SSCG function. Statement The version given here is that proven by Nash-Williams; Kruskal's formulation is somewhat stronger. All trees we consider are finite. Given a tree with a root, and given vertices , , call a successor of if the unique path from the r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
