Non-standard Model Of Arithmetic

	Non-standard Model Of Arithmetic In mathematical logic, a non-standard model of arithmetic is a model of first-order Peano arithmetic that contains non-standard numbers. The term standard model of arithmetic refers to the standard natural numbers 0, 1, 2, …. The elements of any model of Peano arithmetic are linearly ordered and possess an initial segment isomorphic to the standard natural numbers. A non-standard model is one that has additional elements outside this initial segment. The construction of such models is due to Thoralf Skolem (1934). Non-standard models of arithmetic exist only for the first-order formulation of the Peano axioms; for the original second-order formulation, there is, up to isomorphism, only one model: the natural numbers themselves. Existence There are several methods that can be used to prove the existence of non-standard models of arithmetic. From the compactness theorem The existence of non-standard models of arithmetic can be demonstrated by an application of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Order Type In mathematics, especially in set theory, two ordered sets and are said to have the same order type if they are order isomorphic, that is, if there exists a bijection (each element pairs with exactly one in the other set) f\colon X \to Y such that both and its inverse are monotonic (preserving orders of elements). In the special case when is totally ordered, monotonicity of already implies monotonicity of its inverse. One and the same set may be equipped with different orders. Since order-equivalence is an equivalence relation, it partitions the class of all ordered sets into equivalence classes. Notation If a set X has order type denoted \sigma, the order type of the reversed order, the dual of X, is denoted \sigma^. The order type of a well-ordered set is sometimes expressed as . Examples The order type of the integers and rationals is usually denoted \pi and \eta, respectively. The set of integers and the set of even integers have the same order type, becaus ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Mathematical Logic Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability 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 th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Hypernatural In nonstandard analysis, a hyperinteger ''n'' is a hyperreal number that is equal to its own integer part. A hyperinteger may be either finite or infinite. A finite hyperinteger is an ordinary integer. An example of an infinite hyperinteger is given by the class of the sequence in the ultrapower construction of the hyperreals. Discussion The standard integer part function: :\lfloor x \rfloor is defined for all real ''x'' and equals the greatest integer not exceeding ''x''. By the transfer principle of nonstandard analysis, there exists a natural extension: :^\! \lfloor \,\cdot\, \rfloor defined for all hyperreal ''x'', and we say that ''x'' is a hyperinteger if x = ^\! \lfloor x \rfloor. Thus, the hyperintegers are the image of the integer part function on the hyperreals. Internal sets The set ^\mathbb of all hyperintegers is an internal subset of the hyperreal line ^\mathbb. The set of all finite hyperintegers (i.e. \mathbb itself) is not an internal subset. Ele ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Stanley Tennenbaum Stanley Tennenbaum (April 11, 1927 – May 4, 2005) was an American mathematician who contributed to the field of logic. In 1959, he published Tennenbaum's theorem, which states that no countable nonstandard model of Peano arithmetic (PA) can be recursive, i.e. the operations + and × of a nonstandard model of PA are not recursively definable in the + and × operations of the standard model. He was a professor at Yeshiva University in the 1960s. References External links Historical Remarks on Suslin's ProblemArticle by Akihiro Kanamori is a Japanese-born American mathematician. He specializes in set theory and is the author of the monograph on large cardinals, '' The Higher Infinite''. He has written several essays on the history of mathematics, especially set theory. Kanamor ... describing some of Tennenbaum's work, with some biographical info. 1927 births 2005 deaths 20th-century American mathematicians 21st-century American mathematicians {{US-mathemat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Recursion Theory Computability theory, also known as recursion theory, is a branch of mathematical logic, computer science, and the theory of computation that originated in the 1930s with the study of computable functions and Turing degrees. The field has since expanded to include the study of generalized computability and definable set, definability. In these areas, computability theory overlaps with proof theory and effective descriptive set theory. Basic questions addressed by computability theory include: * What does it mean for a function (mathematics), function on the natural numbers to be computable? * How can noncomputable functions be classified into a hierarchy based on their level of noncomputability? Although there is considerable overlap in terms of knowledge and methods, mathematical computability theorists study the theory of relative computability, reducibility notions, and degree structures; those in the computer science field focus on the theory of computational complexity theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Tennenbaum's Theorem Tennenbaum's theorem, named for Stanley Tennenbaum who presented the theorem in 1959, is a result in mathematical logic that states that no countable nonstandard model of first-order Peano arithmetic (PA) can be recursive (Kaye 1991:153ff). Recursive structures for PA A structure M in the language of PA is recursive if there are recursive functions \oplus and \otimes from \mathbb \times \mathbb to \mathbb, a recursive two-place relation <_''M'' on $\mathbb$ , and distinguished constants $n_0,n_1$ such that : $(\mathbb, \oplus,\otimes,<_M,n_,n_) \cong M,$ where $\cong$ indicates isomorphism and $\mathbb$ is the set of (standard) natural numbers . Because the isomorphism ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Approximations Of π Approximation#Mathematics, Approximations for the mathematical constant pi () in the history of mathematics reached an accuracy within 0.04% of the true value before the beginning of the Common Era. In Chinese mathematics, this was improved to approximations correct to what corresponds to about seven decimal digits by the 5th century. Further progress was not made until the 14th century, when Madhava of Sangamagrama developed approximations correct to eleven and then thirteen digits. Jamshīd al-Kāshī achieved sixteen digits next. Early modern mathematicians reached an accuracy of 35 digits by the beginning of the 17th century (Ludolph van Ceulen), and 126 digits by the 19th century (Jurij Vega). The record of manual approximation of is held by William Shanks, who calculated 527 decimals correctly in 1853. Since the middle of the 20th century, the approximation of has been the task of electronic digital computers (for a comprehensive account, see chronology of computation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Non-standard Analysis The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using (ε, δ)-definition of limit, limits rather than infinitesimals. Nonstandard analysis instead reformulates the calculus using a logically rigorous notion of infinitesimal numbers. Nonstandard analysis originated in the early 1960s by the mathematician Abraham Robinson. He wrote: ... the idea of infinitely small or ''infinitesimal'' quantities seems to appeal naturally to our intuition. At any rate, the use of infinitesimals was widespread during the formative stages of the Differential and Integral Calculus. As for the objection ... that the distance between two distinct real numbers cannot be infinitely small, Gottfried Wilhelm Leibniz argued that the theory of infinitesimals implies the introduction of ideal numbers which might be infinitely small or inf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Fred Landman Fred (Alfred) Landman (; born October 28, 1956) is a Dutch-born Israeli professor of semantics. He teaches at Tel Aviv University has written a number of books about linguistics. Biography Fred Landman was born in Holland. He immigrated to Israel in 1993. He was married to London-born linguist Susan Rothstein until her death in 2019. The couple had one daughter and resided in Tel Aviv. Academic career Landman is known for his work on progressives, polarity phenomena, groups, and other topics in semantics and pragmatics. He taught at Brown University and Cornell University Cornell University is a Private university, private Ivy League research university based in Ithaca, New York, United States. The university was co-founded by American philanthropist Ezra Cornell and historian and educator Andrew Dickson W ... before moving to Israel. Published works * ''Indefinites and the Type of Sets'' (2004) * ''Events and Plurality: The Jerusalem Lectures'' (2000) * ''Structures ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Cantor's Isomorphism Theorem In order theory and model theory, branches of mathematics, Cantor's isomorphism theorem states that every two countable dense unbounded linear orders are order-isomorphic. For instance, Minkowski's question-mark function produces an isomorphism (a one-to-one order-preserving correspondence) between the numerical ordering of the rational numbers and the numerical ordering of the dyadic rationals. The theorem is named after Georg Cantor, who first published it in 1895, using it to characterize the (uncountable) ordering on the real numbers. It can be proved by a back-and-forth method that is also sometimes attributed to Cantor but was actually published later, by Felix Hausdorff. The same back-and-forth method also proves that countable dense unbounded orders are highly symmetric, and can be applied to other kinds of structures. However, Cantor's original proof only used the "going forth" half of this method. In terms of model theory, the isomorphism theorem can be expressed by s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Dense Order In mathematics, a partial order or total order < on a set $X$ is said to be dense if, for all $x$ and $y$ in $X$ for which $x < y$ , there is a $z$ in $X$ such that $x < z < y$ . That is, for any two elements, one less than the other, there is another element between them. For total orders this can be simplified to "for any two distinct elements, there is another element between them", since all elements of a total order are comparable. Example The rational number s as a linearly ordered set are a densely ordered set in this sense, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Rational Numbers In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example, The set (mathematics), set of all rational numbers is often referred to as "the rationals", and is closure (mathematics), closed under addition, subtraction, multiplication, and division (mathematics), division by a nonzero rational number. It is a field (mathematics), field under these operations and therefore also called the field of rationals or the field of rational numbers. It is usually denoted by boldface , or blackboard bold 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 numerical digit, digits (example: ), or eventually begins to repeating decimal, repeat the same finite sequence of digits over and over (example: ). This st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]

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