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classical involution theorem | In mathematical finite group theory, the classical involution theorem of Aschbacher (1977a, 1977b, 1980) classifies simple groups with a classical involution and satisfying some other conditions, showing that they are mostly groups of Lie type over a field of odd characteristic. Berkman (2001) extended the classical involution theorem to groups of finite Morley rank. A classical involution t of a finite group G is an involution whose centralizer has a subnormal subgroup containing t with quaternion Sylow 2-subgroups. | wikipedia |
no free lunch theorem | In mathematical folklore, the "no free lunch" (NFL) theorem (sometimes pluralized) of David Wolpert and William Macready, alludes to the saying "no such thing as a free lunch", that is, there are no easy shortcuts to success. It appeared in the 1997 "No Free Lunch Theorems for Optimization". Wolpert had previously derived no free lunch theorems for machine learning (statistical inference).In 2005, Wolpert and Macready themselves indicated that the first theorem in their paper "state that any two optimization algorithms are equivalent when their performance is averaged across all possible problems".The "no free lunch" (NFL) theorem is an easily stated and easily understood consequence of theorems Wolpert and Macready actually prove. It is weaker than the proven theorems, and thus does not encapsulate them. Various investigators have extended the work of Wolpert and Macready substantively. In terms of how the NFL theorem is used in the context of the research area, the no free lunch in search and optimization is a field that is dedicated for purposes of mathematically analyzing data for statistical identity, particularly search and optimization.While some scholars argue that NFL conveys important insight, others argue that NFL is of little relevance to machine learning research. | wikipedia |
kimeme | In mathematical folklore, the no free lunch theorem (sometimes pluralized) of David Wolpert and William G. Macready appears in the 1997 "No Free Lunch Theorems for Optimization. "This mathematical result states the need for a specific effort in the design of a new algorithm, tailored to the specific problem to be optimized. Kimeme allows the design and experimentation of new optimization algorithms through the new paradigm of memetic computing, a subject of computational intelligence which studies algorithmic structures composed of multiple interacting and evolving modules (memes). | wikipedia |
minus-plus sign | In mathematical formulas, the ± symbol may be used to indicate a symbol that may be replaced by either the plus and minus signs, + or −, allowing the formula to represent two values or two equations.If x2 = 9, one may give the solution as x = ±3. This indicates that the equation has two solutions: x = +3 and x = −3. A common use of this notation is found in the quadratic formula x = − b ± b 2 − 4 a c 2 a , {\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}},} which describes the two solutions to the quadratic equation ax2+bx+c = 0. Similarly, the trigonometric identity sin ( A ± B ) = sin ( A ) cos ( B ) ± cos ( A ) sin ( B ) {\displaystyle \sin(A\pm B)=\sin(A)\cos(B)\pm \cos(A)\sin(B)} can be interpreted as a shorthand for two equations: one with + on both sides of the equation, and one with − on both sides. | wikipedia |
minus-plus sign | + x 5 5 ! − x 7 7 ! | wikipedia |
minus-plus sign | + ⋯ ± 1 ( 2 n + 1 ) ! x 2 n + 1 + ⋯ . {\displaystyle \sin \left(x\right)=x-{\frac {x^{3}}{3! | wikipedia |
minus-plus sign | }}+{\frac {x^{5}}{5! }}-{\frac {x^{7}}{7! }}+\cdots \pm {\frac {1}{(2n+1)! | wikipedia |
minus-plus sign | }}x^{2n+1}+\cdots ~.} Here, the plus-or-minus sign indicates that the term may be added or subtracted depending on whether n is odd or even; a rule which can be deduced from the first few terms. A more rigorous presentation would multiply each term by a factor of (−1)n, which gives +1 when n is even, and −1 when n is odd. | wikipedia |
minus-plus sign | In older texts one occasionally finds (−)n, which means the same. When the standard presumption that the plus-or-minus signs all take on the same value of +1 or all −1 is not true, then the line of text that immediately follows the equation must contain a brief description of the actual connection, if any, most often of the form "where the ‘±’ signs are independent" or similar. If a brief, simple description is not possible, the equation must be re-written to provide clarity; e.g. by introducing variables such as s1, s2, ... and specifying a value of +1 or −1 separately for each, or some appropriate relation, like s 3 = s 1 ⋅ ( s 2 ) n , {\displaystyle s_{3}=s_{1}\cdot (s_{2})^{n}\,,} or similar. | wikipedia |
bernstein algebra | In mathematical genetics, a genetic algebra is a (possibly non-associative) algebra used to model inheritance in genetics. Some variations of these algebras are called train algebras, special train algebras, gametic algebras, Bernstein algebras, copular algebras, zygotic algebras, and baric algebras (also called weighted algebra). The study of these algebras was started by Ivor Etherington (1939). In applications to genetics, these algebras often have a basis corresponding to the genetically different gametes, and the structure constant of the algebra encode the probabilities of producing offspring of various types. The laws of inheritance are then encoded as algebraic properties of the algebra. For surveys of genetic algebras see Bertrand (1966), Wörz-Busekros (1980) and Reed (1997). | wikipedia |
hall–higman theorem | In mathematical group theory, the Hall–Higman theorem, due to Philip Hall and Graham Higman (1956, Theorem B), describes the possibilities for the minimal polynomial of an element of prime power order for a representation of a p-solvable group. | wikipedia |
balance theorem | In mathematical group theory, the balance theorem states that if G is a group with no core then G either has disconnected Sylow 2-subgroups or it is of characteristic 2 type or it is of component type (Gorenstein 1983, p. 7). The significance of this theorem is that it splits the classification of finite simple groups into three major subcases. | wikipedia |
root data | In mathematical group theory, the root datum of a connected split reductive algebraic group over a field is a generalization of a root system that determines the group up to isomorphism. They were introduced by Michel Demazure in SGA III, published in 1970. | wikipedia |
green's function number | In mathematical heat conduction, the Green's function number is used to uniquely categorize certain fundamental solutions of the heat equation to make existing solutions easier to identify, store, and retrieve. | wikipedia |
link (knot theory) | In mathematical knot theory, a link is a collection of knots which do not intersect, but which may be linked (or knotted) together. A knot can be described as a link with one component. Links and knots are studied in a branch of mathematics called knot theory. Implicit in this definition is that there is a trivial reference link, usually called the unlink, but the word is also sometimes used in context where there is no notion of a trivial link. | wikipedia |
link (knot theory) | For example, a co-dimension 2 link in 3-dimensional space is a subspace of 3-dimensional Euclidean space (or often the 3-sphere) whose connected components are homeomorphic to circles. The simplest nontrivial example of a link with more than one component is called the Hopf link, which consists of two circles (or unknots) linked together once. The circles in the Borromean rings are collectively linked despite the fact that no two of them are directly linked. The Borromean rings thus form a Brunnian link and in fact constitute the simplest such link. | wikipedia |
valuation (logic) | In mathematical logic (especially model theory), a valuation is an assignment of truth values to formal sentences that follows a truth schema. Valuations are also called truth assignments. In propositional logic, there are no quantifiers, and formulas are built from propositional variables using logical connectives. | wikipedia |
valuation (logic) | In this context, a valuation begins with an assignment of a truth value to each propositional variable. This assignment can be uniquely extended to an assignment of truth values to all propositional formulas. In first-order logic, a language consists of a collection of constant symbols, a collection of function symbols, and a collection of relation symbols. | wikipedia |
valuation (logic) | Formulas are built out of atomic formulas using logical connectives and quantifiers. A structure consists of a set (domain of discourse) that determines the range of the quantifiers, along with interpretations of the constant, function, and relation symbols in the language. Corresponding to each structure is a unique truth assignment for all sentences (formulas with no free variables) in the language. | wikipedia |
gabbay's separation theorem | In mathematical logic and computer science, Gabbay's separation theorem, named after Dov Gabbay, states that any arbitrary temporal logic formula can be rewritten in a logically equivalent "past → future" form. I.e. the future becomes what must be satisfied. This form can be used as execution rules; a MetateM program is a set of such rules. == References == | wikipedia |
mu-recursive function | In mathematical logic and computer science, a general recursive function, partial recursive function, or μ-recursive function is a partial function from natural numbers to natural numbers that is "computable" in an intuitive sense – as well as in a formal one. If the function is total, it is also called a total recursive function (sometimes shortened to recursive function). In computability theory, it is shown that the μ-recursive functions are precisely the functions that can be computed by Turing machines (this is one of the theorems that supports the Church–Turing thesis). The μ-recursive functions are closely related to primitive recursive functions, and their inductive definition (below) builds upon that of the primitive recursive functions. | wikipedia |
mu-recursive function | However, not every total recursive function is a primitive recursive function—the most famous example is the Ackermann function. Other equivalent classes of functions are the functions of lambda calculus and the functions that can be computed by Markov algorithms. The subset of all total recursive functions with values in {0,1} is known in computational complexity theory as the complexity class R. | wikipedia |
calculus of constructions | In mathematical logic and computer science, the calculus of constructions (CoC) is a type theory created by Thierry Coquand. It can serve as both a typed programming language and as constructive foundation for mathematics. For this second reason, the CoC and its variants have been the basis for Coq and other proof assistants. Some of its variants include the calculus of inductive constructions (which adds inductive types), the calculus of (co)inductive constructions (which adds coinduction), and the predicative calculus of inductive constructions (which removes some impredicativity). | wikipedia |
lambda-mu calculus | In mathematical logic and computer science, the lambda-mu calculus is an extension of the lambda calculus introduced by M. Parigot. It introduces two new operators: the μ operator (which is completely different both from the μ operator found in computability theory and from the μ operator of modal μ-calculus) and the bracket operator. Proof-theoretically, it provides a well-behaved formulation of classical natural deduction. One of the main goals of this extended calculus is to be able to describe expressions corresponding to theorems in classical logic. | wikipedia |
lambda-mu calculus | According to the Curry–Howard isomorphism, lambda calculus on its own can express theorems in intuitionistic logic only, and several classical logical theorems can't be written at all. However with these new operators one is able to write terms that have the type of, for example, Peirce's law. Semantically these operators correspond to continuations, found in some functional programming languages. | wikipedia |
universal horn theory | In mathematical logic and logic programming, a Horn clause is a logical formula of a particular rule-like form which gives it useful properties for use in logic programming, formal specification, and model theory. Horn clauses are named for the logician Alfred Horn, who first pointed out their significance in 1951. | wikipedia |
collapsing function | In mathematical logic and set theory, an ordinal collapsing function (or projection function) is a technique for defining (notations for) certain recursive large countable ordinals, whose principle is to give names to certain ordinals much larger than the one being defined, perhaps even large cardinals (though they can be replaced with recursively large ordinals at the cost of extra technical difficulty), and then "collapse" them down to a system of notations for the sought-after ordinal. For this reason, ordinal collapsing functions are described as an impredicative manner of naming ordinals. The details of the definition of ordinal collapsing functions vary, and get more complicated as greater ordinals are being defined, but the typical idea is that whenever the notation system "runs out of fuel" and cannot name a certain ordinal, a much larger ordinal is brought "from above" to give a name to that critical point. | wikipedia |
collapsing function | An example of how this works will be detailed below, for an ordinal collapsing function defining the Bachmann–Howard ordinal (i.e., defining a system of notations up to the Bachmann–Howard ordinal). The use and definition of ordinal collapsing functions is inextricably intertwined with the theory of ordinal analysis, since the large countable ordinals defined and denoted by a given collapse are used to describe the ordinal-theoretic strength of certain formal systems, typically subsystems of analysis (such as those seen in the light of reverse mathematics), extensions of Kripke–Platek set theory, Bishop-style systems of constructive mathematics or Martin-Löf-style systems of intuitionistic type theory. Ordinal collapsing functions are typically denoted using some variation of either the Greek letter ψ {\displaystyle \psi } (psi) or θ {\displaystyle \theta } (theta). | wikipedia |
theory of pure equality | In mathematical logic the theory of pure equality is a first-order theory. It has a signature consisting of only the equality relation symbol, and includes no non-logical axioms at all.This theory is consistent but incomplete, as a non-empty set with the usual equality relation provides an interpretation making certain sentences true. It is an example of a decidable theory and is a fragment of more expressive decidable theories, including monadic class of first-order logic (which also admits unary predicates and is, via Skolem normal form, related to set constraints in program analysis) and monadic second-order theory of a pure set (which additionally permits quantification over predicates and whose signature extends to monadic second-order logic of k successors). | wikipedia |
craig's theorem | In mathematical logic, Craig's theorem (also known as Craig's trick) states that any recursively enumerable set of well-formed formulas of a first-order language is (primitively) recursively axiomatizable. This result is not related to the well-known Craig interpolation theorem, although both results are named after the same logician, William Craig. | wikipedia |
diaconescu theorem | In mathematical logic, Diaconescu's theorem, or the Goodman–Myhill theorem, states that the full axiom of choice is sufficient to derive the law of the excluded middle or restricted forms of it. The theorem was discovered in 1975 by Radu Diaconescu and later by Goodman and Myhill. Already in 1967, Errett Bishop posed the theorem as an exercise (Problem 2 on page 58 in Foundations of constructive analysis). | wikipedia |
frege's propositional calculus | In mathematical logic, Frege's propositional calculus was the first axiomatization of propositional calculus. It was invented by Gottlob Frege, who also invented predicate calculus, in 1879 as part of his second-order predicate calculus (although Charles Peirce was the first to use the term "second-order" and developed his own version of the predicate calculus independently of Frege). It makes use of just two logical operators: implication and negation, and it is constituted by six axioms and one inference rule: modus ponens. Frege's propositional calculus is equivalent to any other classical propositional calculus, such as the "standard PC" with 11 axioms. | wikipedia |
frege's propositional calculus | Frege's PC and standard PC share two common axioms: THEN-1 and THEN-2. Notice that axioms THEN-1 through THEN-3 only make use of (and define) the implication operator, whereas axioms FRG-1 through FRG-3 define the negation operator. | wikipedia |
frege's propositional calculus | The following theorems will aim to find the remaining nine axioms of standard PC within the "theorem-space" of Frege's PC, showing that the theory of standard PC is contained within the theory of Frege's PC. (A theory, also called here, for figurative purposes, a "theorem-space", is a set of theorems that are a subset of a universal set of well-formed formulas. The theorems are linked to each other in a directed manner by inference rules, forming a sort of dendritic network. At the roots of the theorem-space are found the axioms, which "generate" the theorem-space much like a generating set generates a group.) | wikipedia |
kirby–paris theorem | In mathematical logic, Goodstein's theorem is a statement about the natural numbers, proved by Reuben Goodstein in 1944, which states that every Goodstein sequence eventually terminates at 0. Laurence Kirby and Jeff Paris showed that it is unprovable in Peano arithmetic (but it can be proven in stronger systems, such as second-order arithmetic). This was the third example of a true statement that is unprovable in Peano arithmetic, after the examples provided by Gödel's incompleteness theorem and Gerhard Gentzen's 1943 direct proof of the unprovability of ε0-induction in Peano arithmetic. The Paris–Harrington theorem gave another example. | wikipedia |
kirby–paris theorem | Kirby and Paris introduced a graph-theoretic hydra game with behavior similar to that of Goodstein sequences: the "Hydra" (named for the mythological multi-headed Hydra of Lerna) is a rooted tree, and a move consists of cutting off one of its "heads" (a branch of the tree), to which the hydra responds by growing a finite number of new heads according to certain rules. Kirby and Paris proved that the Hydra will eventually be killed, regardless of the strategy that Hercules uses to chop off its heads, though this may take a very long time. Just like for Goodstein sequences, Kirby and Paris showed that it cannot be proven in Peano arithmetic alone. | wikipedia |
gödel's β function | In mathematical logic, Gödel's β function is a function used to permit quantification over finite sequences of natural numbers in formal theories of arithmetic. The β function is used, in particular, in showing that the class of arithmetically definable functions is closed under primitive recursion, and therefore includes all primitive recursive functions. The β function was introduced without the name in the proof of the first of Gödel's incompleteness theorems (Gödel 1931). The β function lemma given below is an essential step of that proof. Gödel gave the β function its name in (Gödel 1934). | wikipedia |
lindstrom's theorem | In mathematical logic, Lindström's theorem (named after Swedish logician Per Lindström, who published it in 1969) states that first-order logic is the strongest logic (satisfying certain conditions, e.g. closure under classical negation) having both the (countable) compactness property and the (downward) Löwenheim–Skolem property.Lindström's theorem is perhaps the best known result of what later became known as abstract model theory, the basic notion of which is an abstract logic; the more general notion of an institution was later introduced, which advances from a set-theoretical notion of model to a category-theoretical one. Lindström had previously obtained a similar result in studying first-order logics extended with Lindström quantifiers.Lindström's theorem has been extended to various other systems of logic, in particular modal logics by Johan van Benthem and Sebastian Enqvist. | wikipedia |
löb's theorem | In mathematical logic, Löb's theorem states that in Peano arithmetic (PA) (or any formal system including PA), for any formula P, if it is provable in PA that "if P is provable in PA then P is true", then P is provable in PA. If Prov(P) means that the formula P is provable, we may express this more formally as If P A ⊢ P r o v ( P ) → P {\displaystyle {\mathit {PA}}\vdash {\mathrm {Prov} (P)\rightarrow P}} then P A ⊢ P {\displaystyle {\mathit {PA}}\vdash P} .An immediate corollary (the contrapositive) of Löb's theorem is that, if P is not provable in PA, then "if P is provable in PA, then P is true" is not provable in PA. For example, "If 1 + 1 = 3 {\displaystyle 1+1=3} is provable in PA, then 1 + 1 = 3 {\displaystyle 1+1=3} " is not provable in PA.Löb's theorem is named for Martin Hugo Löb, who formulated it in 1955. It is related to Curry's paradox. | wikipedia |
typed set theory | In mathematical logic, New Foundations (NF) is an axiomatic set theory, conceived by Willard Van Orman Quine as a simplification of the theory of types of Principia Mathematica. Quine first proposed NF in a 1937 article titled "New Foundations for Mathematical Logic"; hence the name. Much of this entry discusses NF with urelements (NFU), an important variant of NF due to Jensen and clarified by Holmes. In 1940 and in a revision in 1951, Quine introduced an extension of NF sometimes called "Mathematical Logic" or "ML", that included proper classes as well as sets. | wikipedia |
typed set theory | New Foundations has a universal set, so it is a non-well-founded set theory. That is to say, it is an axiomatic set theory that allows infinite descending chains of membership, such as ... xn ∈ xn-1 ∈ ... ∈ x2 ∈ x1. It avoids Russell's paradox by permitting only stratifiable formulas to be defined using the axiom schema of comprehension. For instance, x ∈ y is a stratifiable formula, but x ∈ x is not. New Foundations is closely related to Russellian unramified typed set theory (TST), a streamlined version of the theory of types of Principia Mathematica with a linear hierarchy of types. | wikipedia |
tarski's high school algebra problem | In mathematical logic, Tarski's high school algebra problem was a question posed by Alfred Tarski. It asks whether there are identities involving addition, multiplication, and exponentiation over the positive integers that cannot be proved using eleven axioms about these operations that are taught in high-school-level mathematics. The question was solved in 1980 by Alex Wilkie, who showed that such unprovable identities do exist. | wikipedia |
boolean-valued model | In mathematical logic, a Boolean-valued model is a generalization of the ordinary Tarskian notion of structure from model theory. In a Boolean-valued model, the truth values of propositions are not limited to "true" and "false", but instead take values in some fixed complete Boolean algebra. Boolean-valued models were introduced by Dana Scott, Robert M. Solovay, and Petr Vopěnka in the 1960s in order to help understand Paul Cohen's method of forcing. They are also related to Heyting algebra semantics in intuitionistic logic. | wikipedia |
gödel encoding | In mathematical logic, a Gödel numbering is a function that assigns to each symbol and well-formed formula of some formal language a unique natural number, called its Gödel number. The concept was developed by Kurt Gödel for the proof of his incompleteness theorems. (Gödel 1931) A Gödel numbering can be interpreted as an encoding in which a number is assigned to each symbol of a mathematical notation, after which a sequence of natural numbers can then represent a sequence of symbols. These sequences of natural numbers can again be represented by single natural numbers, facilitating their manipulation in formal theories of arithmetic. Since the publishing of Gödel's paper in 1931, the term "Gödel numbering" or "Gödel code" has been used to refer to more general assignments of natural numbers to mathematical objects. | wikipedia |
deduction theorem | In mathematical logic, a deduction theorem is a metatheorem that justifies doing conditional proofs from a hypothesis in systems that do not explicitly axiomatize that hypothesis, i.e. to prove an implication A → B, it is sufficient to assume A as an hypothesis and then proceed to derive B. Deduction theorems exist for both propositional logic and first-order logic. The deduction theorem is an important tool in Hilbert-style deduction systems because it permits one to write more comprehensible and usually much shorter proofs than would be possible without it. In certain other formal proof systems the same conveniency is provided by an explicit inference rule; for example natural deduction calls it implication introduction. In more detail, the propositional logic deduction theorem states that if a formula B {\displaystyle B} is deducible from a set of assumptions Δ ∪ { A } {\displaystyle \Delta \cup \{A\}} then the implication A → B {\displaystyle A\to B} is deducible from Δ {\displaystyle \Delta } ; in symbols, Δ ∪ { A } ⊢ B {\displaystyle \Delta \cup \{A\}\vdash B} implies Δ ⊢ A → B {\displaystyle \Delta \vdash A\to B} . | wikipedia |
deduction theorem | In the special case where Δ {\displaystyle \Delta } is the empty set, the deduction theorem claim can be more compactly written as: A ⊢ B {\displaystyle A\vdash B} implies ⊢ A → B {\displaystyle \vdash A\to B} . The deduction theorem for predicate logic is similar, but comes with some extra constraints (that would for example be satisfied if A {\displaystyle A} is a closed formula). In general a deduction theorem needs to take into account all logical details of the theory under consideration, so each logical system technically needs its own deduction theorem, although the differences are usually minor. | wikipedia |
deduction theorem | The deduction theorem holds for all first-order theories with the usual deductive systems for first-order logic. However, there are first-order systems in which new inference rules are added for which the deduction theorem fails. Most notably, the deduction theorem fails to hold in Birkhoff–von Neumann quantum logic, because the linear subspaces of a Hilbert space form a non-distributive lattice. | wikipedia |
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. | wikipedia |
decidability of first-order theories of the real numbers | 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 algebraic numbers. As proven by Tarski, this theory is decidable; see Tarski–Seidenberg theorem and Quantifier elimination. Current implementations of decision procedures for the theory of real closed fields are often based on quantifier elimination by cylindrical algebraic decomposition. | wikipedia |
decidability of first-order theories of the real numbers | Tarski's exponential function problem concerns the extension of this theory to another primitive operation, the exponential function. It is an open problem whether this theory is decidable, but if Schanuel's conjecture holds then the decidability of this theory would follow. In contrast, the extension of the theory of real closed fields with the sine function is undecidable since this allows encoding of the undecidable theory of integers (see Richardson's theorem). | wikipedia |
decidability of first-order theories of the real numbers | Still, one can handle the undecidable case with functions such as sine by using algorithms that do not necessarily terminate always. In particular, one can design algorithms that are only required to terminate for input formulas that are robust, that is, formulas whose satisfiability does not change if the formula is slightly perturbed. Alternatively, it is also possible to use purely heuristic approaches. | wikipedia |
formal language theory | In mathematical logic, a formal theory is a set of sentences expressed in a formal language. A formal system (also called a logical calculus, or a logical system) consists of a formal language together with a deductive apparatus (also called a deductive system). The deductive apparatus may consist of a set of transformation rules, which may be interpreted as valid rules of inference, or a set of axioms, or have both. | wikipedia |
formal language theory | A formal system is used to derive one expression from one or more other expressions. Although a formal language can be identified with its formulas, a formal system cannot be likewise identified by its theorems. Two formal systems F S {\displaystyle {\mathcal {FS}}} and F S ′ {\displaystyle {\mathcal {FS'}}} may have all the same theorems and yet differ in some significant proof-theoretic way (a formula A may be a syntactic consequence of a formula B in one but not another for instance). | wikipedia |
formal language theory | A formal proof or derivation is a finite sequence of well-formed formulas (which may be interpreted as sentences, or propositions) each of which is an axiom or follows from the preceding formulas in the sequence by a rule of inference. The last sentence in the sequence is a theorem of a formal system. Formal proofs are useful because their theorems can be interpreted as true propositions. | wikipedia |
mathematical theorem | In mathematical logic, a formal theory is a set of sentences within a formal language. A sentence is a well-formed formula with no free variables. A sentence that is a member of a theory is one of its theorems, and the theory is the set of its theorems. Usually a theory is understood to be closed under the relation of logical consequence. | wikipedia |
mathematical theorem | Some accounts define a theory to be closed under the semantic consequence relation ( ⊨ {\displaystyle \models } ), while others define it to be closed under the syntactic consequence, or derivability relation ( ⊢ {\displaystyle \vdash } ). For a theory to be closed under a derivability relation, it must be associated with a deductive system that specifies how the theorems are derived. The deductive system may be stated explicitly, or it may be clear from the context. | wikipedia |
mathematical theorem | The closure of the empty set under the relation of logical consequence yields the set that contains just those sentences that are the theorems of the deductive system. In the broad sense in which the term is used within logic, a theorem does not have to be true, since the theory that contains it may be unsound relative to a given semantics, or relative to the standard interpretation of the underlying language. A theory that is inconsistent has all sentences as theorems. | wikipedia |
mathematical theorem | The definition of theorems as sentences of a formal language is useful within proof theory, which is a branch of mathematics that studies the structure of formal proofs and the structure of provable formulas. It is also important in model theory, which is concerned with the relationship between formal theories and structures that are able to provide a semantics for them through interpretation. Although theorems may be uninterpreted sentences, in practice mathematicians are more interested in the meanings of the sentences, i.e. in the propositions they express. | wikipedia |
mathematical theorem | What makes formal theorems useful and interesting is that they may be interpreted as true propositions and their derivations may be interpreted as a proof of their truth. A theorem whose interpretation is a true statement about a formal system (as opposed to within a formal system) is called a metatheorem. Some important theorems in mathematical logic are: Compactness of first-order logic Completeness of first-order logic Gödel's incompleteness theorems of first-order arithmetic Consistency of first-order arithmetic Tarski's undefinability theorem Church-Turing theorem of undecidability Löb's theorem Löwenheim–Skolem theorem Lindström's theorem Craig's theorem Cut-elimination theorem | wikipedia |
formula | In mathematical logic, a formula (often referred to as a well-formed formula) is an entity constructed using the symbols and formation rules of a given logical language. For example, in first-order logic, ∀ x ∀ y ( P ( f ( x ) ) → ¬ ( P ( x ) → Q ( f ( y ) , x , z ) ) ) {\displaystyle \forall x\forall y(P(f(x))\rightarrow \neg (P(x)\rightarrow Q(f(y),x,z)))} is a formula, provided that f {\displaystyle f} is a unary function symbol, P {\displaystyle P} a unary predicate symbol, and Q {\displaystyle Q} a ternary predicate symbol. | wikipedia |
shoenfield absoluteness theorem | In mathematical logic, a formula is said to be absolute to some class of structures (also called models), if it has the same truth value in each of the members of that class. One can also speak of absoluteness of a formula between two structures, if it is absolute to some class which contains both of them.. Theorems about absoluteness typically establish relationships between the absoluteness of formulas and their syntactic form. There are two weaker forms of partial absoluteness. If the truth of a formula in each substructure N of a structure M follows from its truth in M, the formula is downward absolute. | wikipedia |
shoenfield absoluteness theorem | If the truth of a formula in a structure N implies its truth in each structure M extending N, the formula is upward absolute. Issues of absoluteness are particularly important in set theory and model theory, fields where multiple structures are considered simultaneously. In model theory, several basic results and definitions are motivated by absoluteness. | wikipedia |
shoenfield absoluteness theorem | In set theory, the issue of which properties of sets are absolute is well studied. The Shoenfield absoluteness theorem, due to Joseph Shoenfield (1961), establishes the absoluteness of a large class of formulas between a model of set theory and its constructible universe, with important methodological consequences. The absoluteness of large cardinal axioms is also studied, with positive and negative results known. | wikipedia |
skolem function | In mathematical logic, a formula of first-order logic is in Skolem normal form if it is in prenex normal form with only universal first-order quantifiers. Every first-order formula may be converted into Skolem normal form while not changing its satisfiability via a process called Skolemization (sometimes spelled Skolemnization). The resulting formula is not necessarily equivalent to the original one, but is equisatisfiable with it: it is satisfiable if and only if the original one is satisfiable.Reduction to Skolem normal form is a method for removing existential quantifiers from formal logic statements, often performed as the first step in an automated theorem prover. | wikipedia |
judgment (mathematical logic) | In mathematical logic, a judgment (or judgement) or assertion is a statement or enunciation in a metalanguage. For example, typical judgments in first-order logic would be that a string is a well-formed formula, or that a proposition is true. Similarly, a judgment may assert the occurrence of a free variable in an expression of the object language, or the provability of a proposition. In general, a judgment may be any inductively definable assertion in the metatheory. | wikipedia |
judgment (mathematical logic) | Judgments are used in formalizing deduction systems: a logical axiom expresses a judgment, premises of a rule of inference are formed as a sequence of judgments, and their conclusion is a judgment as well (thus, hypotheses and conclusions of proofs are judgments). A characteristic feature of the variants of Hilbert-style deduction systems is that the context is not changed in any of their rules of inference, while both natural deduction and sequent calculus contain some context-changing rules. Thus, if we are interested only in the derivability of tautologies, not hypothetical judgments, then we can formalize the Hilbert-style deduction system in such a way that its rules of inference contain only judgments of a rather simple form. | wikipedia |
judgment (mathematical logic) | The same cannot be done with the other two deductions systems: as context is changed in some of their rules of inferences, they cannot be formalized so that hypothetical judgments could be avoided—not even if we want to use them just for proving derivability of tautologies. This basic diversity among the various calculi allows such difference, that the same basic thought (e.g. deduction theorem) must be proven as a metatheorem in Hilbert-style deduction system, while it can be declared explicitly as a rule of inference in natural deduction. In type theory, some analogous notions are used as in mathematical logic (giving rise to connections between the two fields, e.g. Curry–Howard correspondence). The abstraction in the notion of judgment in mathematical logic can be exploited also in foundation of type theory as well. | wikipedia |
boolean literal | In mathematical logic, a literal is an atomic formula (also known as an atom or prime formula) or its negation. The definition mostly appears in proof theory (of classical logic), e.g. in conjunctive normal form and the method of resolution. Literals can be divided into two types: A positive literal is just an atom (e.g., x {\displaystyle x} ). A negative literal is the negation of an atom (e.g., ¬ x {\displaystyle \lnot x} ).The polarity of a literal is positive or negative depending on whether it is a positive or negative literal. | wikipedia |
boolean literal | In logics with double negation elimination (where ¬ ¬ x ≡ x {\displaystyle \lnot \lnot x\equiv x} ) the complementary literal or complement of a literal l {\displaystyle l} can be defined as the literal corresponding to the negation of l {\displaystyle l} . We can write l ¯ {\displaystyle {\bar {l}}} to denote the complementary literal of l {\displaystyle l} . More precisely, if l ≡ x {\displaystyle l\equiv x} then l ¯ {\displaystyle {\bar {l}}} is ¬ x {\displaystyle \lnot x} and if l ≡ ¬ x {\displaystyle l\equiv \lnot x} then l ¯ {\displaystyle {\bar {l}}} is x {\displaystyle x} . | wikipedia |
boolean literal | Double negation elimination occurs in classical logics but not in intuitionistic logic. In the context of a formula in the conjunctive normal form, a literal is pure if the literal's complement does not appear in the formula. In Boolean functions, each separate occurrence of a variable, either in inverse or uncomplemented form, is a literal. For example, if A {\displaystyle A} , B {\displaystyle B} and C {\displaystyle C} are variables then the expression A ¯ B C {\displaystyle {\bar {A}}BC} contains three literals and the expression A ¯ C + B ¯ C ¯ {\displaystyle {\bar {A}}C+{\bar {B}}{\bar {C}}} contains four literals. However, the expression A ¯ C + B ¯ C {\displaystyle {\bar {A}}C+{\bar {B}}C} would also be said to contain four literals, because although two of the literals are identical ( C {\displaystyle C} appears twice) these qualify as two separate occurrences. | wikipedia |
finite model property | In mathematical logic, a logic L has the finite model property (fmp for short) if any non-theorem of L is falsified by some finite model of L. Another way of putting this is to say that L has the fmp if for every formula A of L, A is an L-theorem if and only if A is a theorem of the theory of finite models of L. If L is finitely axiomatizable (and has a recursive set of inference rules) and has the fmp, then it is decidable. However, the result does not hold if L is merely recursively axiomatizable. Even if there are only finitely many finite models to choose from (up to isomorphism) there is still the problem of checking whether the underlying frames of such models validate the logic, and this may not be decidable when the logic is not finitely axiomatizable, even when it is recursively axiomatizable. (Note that a logic is recursively enumerable if and only if it is recursively axiomatizable, a result known as Craig's theorem.) | wikipedia |
soundness | In mathematical logic, a logical system has the soundness property if every formula that can be proved in the system is logically valid with respect to the semantics of the system. In most cases, this comes down to its rules having the property of preserving truth. The converse of soundness is known as completeness. A logical system with syntactic entailment ⊢ {\displaystyle \vdash } and semantic entailment ⊨ {\displaystyle \models } is sound if for any sequence A 1 , A 2 , . | wikipedia |
soundness | , A n {\displaystyle A_{1},A_{2},...,A_{n}} of sentences in its language, if A 1 , A 2 , . . . | wikipedia |
soundness | , A n ⊢ C {\displaystyle A_{1},A_{2},...,A_{n}\vdash C} , then A 1 , A 2 , . . . | wikipedia |
soundness | , A n ⊨ C {\displaystyle A_{1},A_{2},...,A_{n}\models C} . In other words, a system is sound when all of its theorems are tautologies. Soundness is among the most fundamental properties of mathematical logic. | wikipedia |
soundness | The soundness property provides the initial reason for counting a logical system as desirable. The completeness property means that every validity (truth) is provable. Together they imply that all and only validities are provable. | wikipedia |
soundness | Most proofs of soundness are trivial. For example, in an axiomatic system, proof of soundness amounts to verifying the validity of the axioms and that the rules of inference preserve validity (or the weaker property, truth). If the system allows Hilbert-style deduction, it requires only verifying the validity of the axioms and one rule of inference, namely modus ponens. (and sometimes substitution) Soundness properties come in two main varieties: weak and strong soundness, of which the former is a restricted form of the latter. | wikipedia |
epistemic structural realism | In mathematical logic, a mathematical structure is a standard concept. A mathematical structure is a set of abstract entities with relations between them. The natural numbers under arithmetic constitute a structure, with relations such as "is evenly divisible by" and "is greater than". Here the relation "is greater than" includes the element (3, 4), but not the element (4, 3). | wikipedia |
epistemic structural realism | Points in space and the real numbers under Euclidean geometry are another structure, with relations such as "the distance between point P1 and point P2 is real number R1"; equivalently, the "distance" relation includes the element (P1, P2, R1). Other structures include the Riemann space of general relativity and the Hilbert space of quantum mechanics. The entities in a mathematical structure do not have any independent existence outside their participation in relations. Two descriptions of a structure are considered equivalent, and to be describing the same underlying structure, if there is a correspondence between the descriptions that preserves all relations.Many proponents of structural realism formally or informally ascribe "properties" to the abstract objects; some argue that such properties, while they can perhaps be "shoehorned" into the formalism of relations, should instead be considered distinct from relations. | wikipedia |
non-standard number | 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). | wikipedia |
proof system | In mathematical logic, a proof calculus or a proof system is built to prove statements. | wikipedia |
redundant proof | In mathematical logic, a redundant proof is a proof that has a subset that is a shorter proof of the same result. In other words, a proof is redundant if it has more proof steps than are actually necessary to prove the result. Formally, a proof ψ {\displaystyle \psi } of κ {\displaystyle \kappa } is considered redundant if there exists another proof ψ ′ {\displaystyle \psi ^{\prime }} of κ ′ {\displaystyle \kappa ^{\prime }} such that κ ′ ⊆ κ {\displaystyle \kappa ^{\prime }\subseteq \kappa } (i.e. κ ′ subsumes κ {\displaystyle \kappa ^{\prime }\;{\text{subsumes}}\;\kappa } ) and | ψ ′ | < | ψ | {\displaystyle |\psi ^{\prime }|<|\psi |} where | φ | {\displaystyle |\varphi |} is the number of nodes in φ {\displaystyle \varphi } . | wikipedia |
sentence (mathematical logic) | In mathematical logic, a sentence (or closed formula) of a predicate logic is a Boolean-valued well-formed formula with no free variables. A sentence can be viewed as expressing a proposition, something that must be true or false. The restriction of having no free variables is needed to make sure that sentences can have concrete, fixed truth values: as the free variables of a (general) formula can range over several values, the truth value of such a formula may vary. Sentences without any logical connectives or quantifiers in them are known as atomic sentences; by analogy to atomic formula. | wikipedia |
sentence (mathematical logic) | Sentences are then built up out of atomic formulas by applying connectives and quantifiers. A set of sentences is called a theory; thus, individual sentences may be called theorems. | wikipedia |
sentence (mathematical logic) | To properly evaluate the truth (or falsehood) of a sentence, one must make reference to an interpretation of the theory. For first-order theories, interpretations are commonly called structures. Given a structure or interpretation, a sentence will have a fixed truth value. A theory is satisfiable when it is possible to present an interpretation in which all of its sentences are true. The study of algorithms to automatically discover interpretations of theories that render all sentences as being true is known as the satisfiability modulo theories problem. | wikipedia |
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 ϕ ∈ T {\displaystyle \phi \in T} of a deductively closed theory T {\displaystyle T} is then called a theorem of the theory. In many deductive systems there is usually a subset Σ ⊆ T {\displaystyle \Sigma \subseteq T} that is called "the set of axioms" of the theory T {\displaystyle 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. | wikipedia |
extension by new constant and function names | In mathematical logic, a theory can be extended with new constants or function names under certain conditions with assurance that the extension will introduce no contradiction. Extension by definitions is perhaps the best-known approach, but it requires unique existence of an object with the desired property. Addition of new names can also be done safely without uniqueness. Suppose that a closed formula ∃ x 1 … ∃ x m φ ( x 1 , … , x m ) {\displaystyle \exists x_{1}\ldots \exists x_{m}\,\varphi (x_{1},\ldots ,x_{m})} is a theorem of a first-order theory T {\displaystyle T} . | wikipedia |
extension by new constant and function names | Let T 1 {\displaystyle T_{1}} be a theory obtained from T {\displaystyle T} by extending its language with new constants a 1 , … , a m {\displaystyle a_{1},\ldots ,a_{m}} and adding a new axiom φ ( a 1 , … , a m ) {\displaystyle \varphi (a_{1},\ldots ,a_{m})} .Then T 1 {\displaystyle T_{1}} is a conservative extension of T {\displaystyle T} , which means that the theory T 1 {\displaystyle T_{1}} has the same set of theorems in the original language (i.e., without constants a i {\displaystyle a_{i}} ) as the theory T {\displaystyle T} . Such a theory can also be conservatively extended by introducing a new functional symbol:Suppose that a closed formula ∀ x → ∃ y φ ( y , x → ) {\displaystyle \forall {\vec {x}}\,\exists y\,\!\,\varphi (y,{\vec {x}})} is a theorem of a first-order theory T {\displaystyle T} , where we denote x → := ( x 1 , … , x n ) {\displaystyle {\vec {x}}:=(x_{1},\ldots ,x_{n})} . Let T 1 {\displaystyle T_{1}} be a theory obtained from T {\displaystyle T} by extending its language with a new functional symbol f {\displaystyle f} (of arity n {\displaystyle n} ) and adding a new axiom ∀ x → φ ( f ( x → ) , x → ) {\displaystyle \forall {\vec {x}}\,\varphi (f({\vec {x}}),{\vec {x}})} . | wikipedia |
extension by new constant and function names | Then T 1 {\displaystyle T_{1}} is a conservative extension of T {\displaystyle T} , i.e. the theories T {\displaystyle T} and T 1 {\displaystyle T_{1}} prove the same theorems not involving the functional symbol f {\displaystyle f} ). Shoenfield states the theorem in the form for a new function name, and constants are the same as functions of zero arguments. In formal systems that admit ordered tuples, extension by multiple constants as shown here can be accomplished by addition of a new constant tuple and the new constant names having the values of elements of the tuple. | wikipedia |
existential theory of the reals | In mathematical logic, a theory is a formal language consisting of a set of sentences written using a fixed set of symbols. The first-order theory of real closed fields has the following symbols: the constants 0 and 1, a countable collection of variables X i {\displaystyle X_{i}} , the addition, subtraction, multiplication, and (optionally) division operations, symbols <, ≤, =, ≥, >, and ≠ for comparisons of real values, the logical connectives ∧, ∨, ¬, and ⇔, parentheses, and the universal quantifier ∀ and the existential quantifier ∃A sequence of these symbols forms a sentence that belongs to the first-order theory of the reals if it is grammatically well formed, all its variables are properly quantified, and (when interpreted as a mathematical statement about the real numbers) it is a true statement. As Tarski showed, this theory can be described by an axiom schema and a decision procedure that is complete and effective: for every fully quantified and grammatical sentence, either the sentence or its negation (but not both) can be derived from the axioms. The same theory describes every real closed field, not just the real numbers. | wikipedia |
existential theory of the reals | However, there are other number systems that are not accurately described by these axioms; in particular, the theory defined in the same way for integers instead of real numbers is undecidable, even for existential sentences (Diophantine equations) by Matiyasevich's theorem.The existential theory of the reals is the fragment of the first-order theory consisting of sentences in which all the quantifiers are existential and they appear before any of the other symbols. That is, it is the set of all true sentences of the form where F ( X 1 , … X n ) {\displaystyle F(X_{1},\dots X_{n})} is a quantifier-free formula involving equalities and inequalities of real polynomials. The decision problem for the existential theory of the reals is the algorithmic problem of testing whether a given sentence belongs to this theory; equivalently, for strings that pass the basic syntactical checks (they use the correct symbols with the correct syntax, and have no unquantified variables) it is the problem of testing whether the sentence is a true statement about the real numbers. | wikipedia |
existential theory of the reals | The set of n {\displaystyle n} -tuples of real numbers ( X 1 , … X n ) {\displaystyle (X_{1},\dots X_{n})} for which F ( X 1 , … X n ) {\displaystyle F(X_{1},\dots X_{n})} is true is called a semialgebraic set, so the decision problem for the existential theory of the reals can equivalently be rephrased as testing whether a given semialgebraic set is nonempty.In determining the time complexity of algorithms for the decision problem for the existential theory of the reals, it is important to have a measure of the size of the input. The simplest measure of this type is the length of a sentence: that is, the number of symbols it contains. However, in order to achieve a more precise analysis of the behavior of algorithms for this problem, it is convenient to break down the input size into several variables, separating out the number of variables to be quantified, the number of polynomials within the sentence, and the degree of these polynomials. | wikipedia |
categorical (model theory) | In mathematical logic, a theory is categorical if it has exactly one model (up to isomorphism). Such a theory can be viewed as defining its model, uniquely characterizing the model's structure. In first-order logic, only theories with a finite model can be categorical. | wikipedia |
categorical (model theory) | Higher-order logic contains categorical theories with an infinite model. For example, the second-order Peano axioms are categorical, having a unique model whose domain is the set of natural numbers N . {\displaystyle \mathbb {N} .} | wikipedia |
categorical (model theory) | In model theory, the notion of a categorical theory is refined with respect to cardinality. A theory is κ-categorical (or categorical in κ) if it has exactly one model of cardinality κ up to isomorphism. Morley's categoricity theorem is a theorem of Michael D. Morley (1965) stating that if a first-order theory in a countable language is categorical in some uncountable cardinality, then it is categorical in all uncountable cardinalities. Saharon Shelah (1974) extended Morley's theorem to uncountable languages: if the language has cardinality κ and a theory is categorical in some uncountable cardinal greater than or equal to κ then it is categorical in all cardinalities greater than κ. | wikipedia |
complete theory | In mathematical logic, a theory is complete if it is consistent and for every closed formula in the theory's language, either that formula or its negation is provable. That is, for every sentence φ , {\displaystyle \varphi ,} the theory T {\displaystyle T} contains the sentence or its negation but not both (that is, either T ⊢ φ {\displaystyle T\vdash \varphi } or T ⊢ ¬ φ {\displaystyle T\vdash \neg \varphi } ). Recursively axiomatizable first-order theories that are consistent and rich enough to allow general mathematical reasoning to be formulated cannot be complete, as demonstrated by Gödel's first incompleteness theorem. This sense of complete is distinct from the notion of a complete logic, which asserts that for every theory that can be formulated in the logic, all semantically valid statements are provable theorems (for an appropriate sense of "semantically valid"). | wikipedia |
complete theory | Gödel's completeness theorem is about this latter kind of completeness. Complete theories are closed under a number of conditions internally modelling the T-schema: For a set of formulas S {\displaystyle S}: A ∧ B ∈ S {\displaystyle A\land B\in S} if and only if A ∈ S {\displaystyle A\in S} and B ∈ S {\displaystyle B\in S} , For a set of formulas S {\displaystyle S}: A ∨ B ∈ S {\displaystyle A\lor B\in S} if and only if A ∈ S {\displaystyle A\in S} or B ∈ S {\displaystyle B\in S} .Maximal consistent sets are a fundamental tool in the model theory of classical logic and modal logic. Their existence in a given case is usually a straightforward consequence of Zorn's lemma, based on the idea that a contradiction involves use of only finitely many premises. In the case of modal logics, the collection of maximal consistent sets extending a theory T (closed under the necessitation rule) can be given the structure of a model of T, called the canonical model. | wikipedia |
witness (mathematics) | In mathematical logic, a witness is a specific value t to be substituted for variable x of an existential statement of the form ∃x φ(x) such that φ(t) is true. | wikipedia |
abstract algebraic logic | In mathematical logic, abstract algebraic logic is the study of the algebraization of deductive systems arising as an abstraction of the well-known Lindenbaum–Tarski algebra, and how the resulting algebras are related to logical systems. | wikipedia |
abstract model theory | In mathematical logic, abstract model theory is a generalization of model theory that studies the general properties of extensions of first-order logic and their models.Abstract model theory provides an approach that allows us to step back and study a wide range of logics and their relationships. The starting point for the study of abstract models, which resulted in good examples was Lindström's theorem.In 1974 Jon Barwise provided an axiomatization of abstract model theory. | wikipedia |
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