|
Tarski's Exponential Function Problem
In model theory, Tarski's exponential function problem asks whether the theory of the real numbers together with the exponential function is decidable. Alfred Tarski had previously shown that the theory of the real numbers (without the exponential function) is decidable. The problem The ordered real field R is a structure over the language of ordered rings ''L''or = (+,·,−, ''η''−1. Workaround Recently there are attempts at handling the theory of the real numbers with functions such as the exponential function or sine by relaxing decidability to the weaker notion of quasi-decidability. A theory is quasi-decidable if there is a procedure that decides satisfiability but that may run forever for inputs that are not robust in a certain, well-defined sense. Such a procedure exists for systems of equations in variables . References * * *{{citation, first1=Peter, last1=Franek, first2=Stefan, last2=Ratschan, first3=Piotr, last3=Zgliczynski, chapter=S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Model Theory
In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the statements of the theory hold). The aspects investigated include the number and size of models of a theory, the relationship of different models to each other, and their interaction with the formal language itself. In particular, model theorists also investigate the sets that can be defined in a model of a theory, and the relationship of such definable sets to each other. As a separate discipline, model theory goes back to Alfred Tarski, who first used the term "Theory of Models" in publication in 1954. Since the 1970s, the subject has been shaped decisively by Saharon Shelah's stability theory. Compared to other areas of mathematical logic such as proof theory, model theory is often less concerned with formal rigour and closer in spirit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theory (mathematical Logic)
In mathematical logic, a theory (also called a formal theory) is a set of sentences in a formal language. In most scenarios, a deductive system is first understood from context, after which an element \phi\in T of a deductively closed theory T is then called a theorem of the theory. In many deductive systems there is usually a subset \Sigma \subseteq T that is called "the set of axioms" of the theory T, in which case the deductive system is also called an "axiomatic system". By definition, every axiom is automatically a theorem. A first-order theory is a set of first-order sentences (theorems) recursively obtained by the inference rules of the system applied to the set of axioms. General theories (as expressed in formal language) When defining theories for foundational purposes, additional care must be taken, as normal set-theoretic language may not be appropriate. The construction of a theory begins by specifying a definite non-empty ''conceptual class'' \mathcal, the element ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponential Function
The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, although it can be extended to the complex numbers or generalized to other mathematical objects like matrices or Lie algebras. The exponential function originated from the notion of exponentiation (repeated multiplication), but modern definitions (there are several equivalent characterizations) allow it to be rigorously extended to all real arguments, including irrational numbers. Its ubiquitous occurrence in pure and applied mathematics led mathematician Walter Rudin to opine that the exponential function is "the most important function in mathematics". The exponential function satisfies the exponentiation identity e^ = e^x e^y \text x,y\in\mathbb, which, along with the definition e = \exp(1), shows that e^n=\underbrace_ for positive i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Decidability (logic)
In logic, a true/false decision problem is decidable if there exists an effective method for deriving the correct answer. Zeroth-order logic (propositional logic) is decidable, whereas first-order and higher-order logic are not. Logical systems are decidable if membership in their set of logically valid formulas (or theorems) can be effectively determined. A theory (set of sentences closed under logical consequence) in a fixed logical system is decidable if there is an effective method for determining whether arbitrary formulas are included in the theory. Many important problems are undecidable, that is, it has been proven that no effective method for determining membership (returning a correct answer after finite, though possibly very long, time in all cases) can exist for them. Decidability of a logical system Each logical system comes with both a syntactic component, which among other things determines the notion of provability, and a semantic component, which determines ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alfred Tarski
Alfred Tarski (, born Alfred Teitelbaum;School of Mathematics and Statistics, University of St Andrews ''School of Mathematics and Statistics, University of St Andrews''. January 14, 1901 – October 26, 1983) was a Polish-American logician and mathematician. A prolific author best known for his work on model theory, metamathematics, and algebraic logic, he also contributed to abstract algebra, topology, geometry, measure theory, mathematical logic, set theory, and analytic philosophy. Educated in Poland at the University of Warsaw, and a member of the Lwów–Warsaw school of logic and the Warsaw school of mathematics, he immigrated to the United States in 1939 where he became a naturalized citizen in 1945. Tarski taught and carried out research in mathematics at the University of California, Berkeley, from 1942 until his death in 1983. Feferman A. His biographers Anita Burdman Feferman and Solomon Feferman state that, "Along with his contemporary, Kurt Gödel, he cha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Decidability Of First-order Theories Of The Real Numbers
In mathematical logic, a first-order language of the real numbers is the set of all well-formed sentences of first-order logic that involve universal and existential quantifiers and logical combinations of equalities and inequalities of expressions over real variables. The corresponding first-order theory is the set of sentences that are actually true of the real numbers. There are several different such theories, with different expressive power, depending on the primitive operations that are allowed to be used in the expression. A fundamental question in the study of these theories is whether they are decidable: that is, is there an algorithm that can take a sentence as input and produce as output an answer "yes" or "no" to the question of whether the sentence is true in the theory. The theory of real closed fields is the theory in which the primitive operations are multiplication and addition; this implies that, in this theory, the only numbers that can be defined are the real ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordered Ring
In abstract algebra, an ordered ring is a (usually commutative) ring ''R'' with a total order ≤ such that for all ''a'', ''b'', and ''c'' in ''R'': * if ''a'' ≤ ''b'' then ''a'' + ''c'' ≤ ''b'' + ''c''. * if 0 ≤ ''a'' and 0 ≤ ''b'' then 0 ≤ ''ab''. Examples Ordered rings are familiar from arithmetic. Examples include the integers, the rationals and the real numbers. (The rationals and reals in fact form ordered fields.) The complex numbers, in contrast, do not form an ordered ring or field, because there is no inherent order relationship between the elements 1 and ''i''. Positive elements In analogy with the real numbers, we call an element ''c'' of an ordered ring ''R'' positive if 0 < ''c'', and negative if ''c'' < 0. 0 is considered to be neither positive nor negative. The set of positive elements of an ordered ring ''R'' is often denoted by ''R''+. An alternative notation, favored in some disciplines, is to use ''R''+ for the set of non ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Closed Field
In mathematics, a real closed field is a field ''F'' that has the same first-order properties as the field of real numbers. Some examples are the field of real numbers, the field of real algebraic numbers, and the field of hyperreal numbers. Definitions A real closed field is a field ''F'' in which any of the following equivalent conditions is true: #''F'' is elementarily equivalent to the real numbers. In other words, it has the same first-order properties as the reals: any sentence in the first-order language of fields is true in ''F'' if and only if it is true in the reals. #There is a total order on ''F'' making it an ordered field such that, in this ordering, every positive element of ''F'' has a square root in ''F'' and any polynomial of odd degree with coefficients in ''F'' has at least one root in ''F''. #''F'' is a formally real field such that every polynomial of odd degree with coefficients in ''F'' has at least one root in ''F'', and for every element ''a'' of ''F'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponential Polynomial
In mathematics, exponential polynomials are functions on fields, rings, or abelian groups that take the form of polynomials in a variable and an exponential function. Definition In fields An exponential polynomial generally has both a variable ''x'' and some kind of exponential function ''E''(''x''). In the complex numbers there is already a canonical exponential function, the function that maps ''x'' to '' e''''x''. In this setting the term exponential polynomial is often used to mean polynomials of the form ''P''(''x'', ''e''''x'') where ''P'' ∈ C 'x'', ''y''is a polynomial in two variables. There is nothing particularly special about C here; exponential polynomials may also refer to such a polynomial on any exponential field or exponential ring with its exponential function taking the place of ''e''''x'' above. Similarly, there is no reason to have one variable, and an exponential polynomial in ''n'' variables would be of the form ''P''(''x''1, ..., ''x''''n'', ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schanuel's Conjecture
In mathematics, specifically transcendental number theory, Schanuel's conjecture is a conjecture made by Stephen Schanuel in the 1960s concerning the transcendence degree of certain field extensions of the rational numbers. Statement The conjecture is as follows: :Given any complex numbers that are linearly independent over the rational numbers \mathbb, the field extension \mathbb(''z''1, ..., ''z''''n'', ''e''''z''1, ..., ''e''''z''''n'') has transcendence degree at least over \mathbb. The conjecture can be found in Lang (1966). Consequences The conjecture, if proven, would generalize most known results in transcendental number theory. The special case where the numbers ''z''1,...,''z''''n'' are all algebraic is the Lindemann–Weierstrass theorem. If, on the other hand, the numbers are chosen so as to make exp(''z''1),...,exp(''z''''n'') all algebraic then one would prove that linearly independent logarithms of algebraic numbers are algebraically independent, a streng ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Necessary And Sufficient Condition
In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth of is guaranteed by the truth of (equivalently, it is impossible to have without ). Similarly, is sufficient for , because being true always implies that is true, but not being true does not always imply that is not true. In general, a necessary condition is one that must be present in order for another condition to occur, while a sufficient condition is one that produces the said condition. The assertion that a statement is a "necessary ''and'' sufficient" condition of another means that the former statement is true if and only if the latter is true. That is, the two statements must be either simultaneously true, or simultaneously false. In ordinary English (also natural language) "necessary" and "sufficient" indicate relations betw ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
