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c_yyf8a91sb8zd | Uniqueness case | Summary | Uniqueness_case | In mathematical finite group theory, the uniqueness case is one of the three possibilities for groups of characteristic 2 type given by the trichotomy theorem. The uniqueness case covers groups G of characteristic 2 type with e(G) ≥ 3 that have an almost strongly p-embedded maximal 2-local subgroup for all primes p who... |
c_wiuqw5s7m1eg | Vertex of a representation | Summary | Vertex_of_a_representation | In mathematical finite group theory, the vertex of a representation of a finite group is a subgroup associated to it, that has a special representation called a source. Vertices and sources were introduced by Green (1958–1959). |
c_ak0r69vi6z84 | No free lunch theorem | Summary | 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 deri... |
c_zmcv9slvsx6y | Kimeme | Algorithm design | Kimeme > Features > Algorithm design | 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 optim... |
c_y178lgcbs02q | Minus-plus sign | In mathematics | Plus–minus_sign > Usage > In mathematics | 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 ... |
c_et1ak5c099j0 | Minus-plus sign | In mathematics | Plus–minus_sign > Usage > In mathematics | + x 5 5 ! − x 7 7 ! |
c_5nsft66jssul | Minus-plus sign | In mathematics | Plus–minus_sign > Usage > In mathematics | + ⋯ ± 1 ( 2 n + 1 ) ! x 2 n + 1 + ⋯ . {\displaystyle \sin \left(x\right)=x-{\frac {x^{3}}{3! |
c_v396ql0ee8fw | Minus-plus sign | In mathematics | Plus–minus_sign > Usage > In mathematics | }}+{\frac {x^{5}}{5! }}-{\frac {x^{7}}{7! }}+\cdots \pm {\frac {1}{(2n+1)! |
c_nnkx2ii5jj9r | Minus-plus sign | In mathematics | Plus–minus_sign > Usage > In mathematics | }}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 o... |
c_t5rhdpv0dla3 | Minus-plus sign | In mathematics | Plus–minus_sign > Usage > In mathematics | 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 o... |
c_93zq8lrn6u68 | Penrose inequality | Summary | Penrose_inequality | In mathematical general relativity, the Penrose inequality, first conjectured by Sir Roger Penrose, estimates the mass of a spacetime in terms of the total area of its black holes and is a generalization of the positive mass theorem. The Riemannian Penrose inequality is an important special case. Specifically, if (M, g... |
c_f1n0z6ykaaxg | Penrose inequality | Summary | Penrose_inequality | This is purely a geometrical fact, and it corresponds to the case of a complete three-dimensional, space-like, totally geodesic submanifold of a (3 + 1)-dimensional spacetime. Such a submanifold is often called a time-symmetric initial data set for a spacetime. The condition of (M, g) having nonnegative scalar curvatur... |
c_lsirgjald43t | Penrose inequality | Summary | Penrose_inequality | This inequality was first proved by Gerhard Huisken and Tom Ilmanen in 1997 in the case where A is the area of the largest component of the outermost minimal surface. Their proof relied on the machinery of weakly defined inverse mean curvature flow, which they developed. In 1999, Hubert Bray gave the first complete pro... |
c_q789s4bz4c5i | Bernstein algebra | Summary | Bernstein_problem_in_mathematical_genetics | 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 weight... |
c_qztcsv96vslb | Higman-Sims graph | Summary | Higman-Sims_graph | In mathematical graph theory, the Higman–Sims graph is a 22-regular undirected graph with 100 vertices and 1100 edges. It is the unique strongly regular graph srg(100,22,0,6), where no neighboring pair of vertices share a common neighbor and each non-neighboring pair of vertices share six common neighbors. It was first... |
c_n1nvd2wbi5o0 | 3-transposition group | Summary | 3-transposition_group | In mathematical group theory, a 3-transposition group is a group generated by a conjugacy class of involutions, called the 3-transpositions, such that the product of any two involutions from the conjugacy class has order at most 3. They were first studied by Bernd Fischer (1964, 1970, 1971) who discovered the three Fis... |
c_vevaozxmv6ym | C-group | Summary | C-group | In mathematical group theory, a C-group is a group such that the centralizer of any involution has a normal Sylow 2-subgroup. They include as special cases CIT-groups where the centralizer of any involution is a 2-group, and TI-groups where any Sylow 2-subgroups have trivial intersection. The simple C-groups were deter... |
c_c87kg8n4g9vi | Demushkin group | Summary | Demushkin_group | In mathematical group theory, a Demushkin group (also written as Demuškin or Demuskin) is a pro-p group G having a certain properties relating to duality in group cohomology. More precisely, G must be such that the first cohomology group with coefficients in Fp = Z/p Z has finite rank, the second cohomology group has r... |
c_exnt22hj9qiu | Normal p-complement | Summary | P-nilpotent_group | In mathematical group theory, a normal p-complement of a finite group for a prime p is a normal subgroup of order coprime to p and index a power of p. In other words the group is a semidirect product of the normal p-complement and any Sylow p-subgroup. A group is called p-nilpotent if it has a normal p-complement. |
c_ppjvpsbonm83 | Special abelian subgroup | Summary | Special_abelian_subgroup | In mathematical group theory, a subgroup of a group is termed a special abelian subgroup or SA-subgroup if the centralizer of any nonidentity element in the subgroup is precisely the subgroup(Curtis & Reiner 1981, p.354). Equivalently, an SA subgroup is a centrally closed abelian subgroup. Any SA subgroup is a maximal ... |
c_k9515ivgpvy5 | Tame group | Summary | Tame_group | In mathematical group theory, a tame group is a certain kind of group defined in model theory. Formally, we define a bad field as a structure of the form (K, T), where K is an algebraically closed field and T is an infinite, proper, distinguished subgroup of K, such that (K, T) is of finite Morley rank in its full lang... |
c_rvclinxpsdcw | Hall–Higman theorem | Summary | 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. |
c_fis02p4wqglx | Schur cover | Summary | Schur_multiplier | In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group H 2 ( G , Z ) {\displaystyle H_{2}(G,\mathbb {Z} )} of a group G. It was introduced by Issai Schur (1904) in his work on projective representations. |
c_pjehhdga13xl | Automorphism group of a free group | Summary | Automorphism_group_of_a_free_group | In mathematical group theory, the automorphism group of a free group is a discrete group of automorphisms of a free group. The quotient by the inner automorphisms is the outer automorphism group of a free group, which is similar in some ways to the mapping class group of a surface. |
c_gfo6ooudq94p | Balance theorem | Summary | 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 ... |
c_5nd6rbtrxlby | Root data | Summary | Root_datum | 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. |
c_nkwvo4tkbocr | Green's function number | Summary | 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. |
c_gx60uskl35uh | Perpetuant | Summary | Perpetuant | In mathematical invariant theory, a perpetuant is informally an irreducible covariant of a form or infinite degree. More precisely, the dimension of the space of irreducible covariants of given degree and weight for a binary form stabilizes provided the degree of the form is larger than the weight of the covariant, and... |
c_mq3j531oid1j | Perpetuant | Summary | Perpetuant | Elliott (1907) describes the early history of perpetuants and gives an annotated bibliography. MacMahon conjectured and Stroh proved that the dimension of the space of perpetuants of degree n>2 and weight w is the coefficient of xw of x 2 n − 1 − 1 ( 1 − x 2 ) ( 1 − x 3 ) ⋯ ( 1 − x n ) {\displaystyle {\frac {x^{2^{n-1}... |
c_a2ltk9yo1rrm | Transvectant | Summary | Transvectant | In mathematical invariant theory, a transvectant is an invariant formed from n invariants in n variables using Cayley's Ω process. |
c_7kdx9ojzebg7 | Evectant | Summary | Evectant | In mathematical invariant theory, an evectant is a contravariant constructed from an invariant by acting on it with a differential operator called an evector. Evectants and evectors were introduced by Sylvester (1854, p.95). |
c_tjsfo7g8ztzm | Invariant of a binary form | Summary | Invariants_of_binary_form | In mathematical invariant theory, an invariant of a binary form is a polynomial in the coefficients of a binary form in two variables x and y that remains invariant under the special linear group acting on the variables x and y. |
c_8uaognnplxym | Canonizant | Summary | Canonizant | In mathematical invariant theory, the canonizant or canonisant is a covariant of forms related to a canonical form for them. |
c_mw1d7w456sqx | Catalecticant | Summary | Catalecticant | In mathematical invariant theory, the catalecticant of a form of even degree is a polynomial in its coefficients that vanishes when the form is a sum of an unusually small number of powers of linear forms. It was introduced by Sylvester (1852); see Miller (2010). The word catalectic refers to an incomplete line of vers... |
c_pry42kdmesja | Osculant | Summary | Osculant | In mathematical invariant theory, the osculant or tacinvariant or tact invariant is an invariant of a hypersurface that vanishes if the hypersurface touches itself, or an invariant of several hypersurfaces that osculate, meaning that they have a common point where they meet to unusually high order. |
c_hdn6e4ykd1f2 | 74 knot | Summary | 74_knot | In mathematical knot theory, 74 is the name of a 7-crossing knot which can be visually depicted in a highly-symmetric form, and so appears in the symbolism and/or artistic ornamentation of various cultures. |
c_y4yzmru03fp6 | Conway sphere | Summary | Conway_sphere | In mathematical knot theory, a Conway sphere, named after John Horton Conway, is a 2-sphere intersecting a given knot or link in a 3-manifold transversely in four points. In a knot diagram, a Conway sphere can be represented by a simple closed curve crossing four points of the knot, the cross-section of the sphere; suc... |
c_l5bcz1hcs78g | Link (knot theory) | Summary | 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 refe... |
c_53y99gt7o86j | Link (knot theory) | Summary | 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 unkno... |
c_ogips1fqd5mb | Hopf link | Summary | Hopf_link | In mathematical knot theory, the Hopf link is the simplest nontrivial link with more than one component. It consists of two circles linked together exactly once, and is named after Heinz Hopf. |
c_q71ykl3gofzh | Equisatisfiability | Summary | Equisatisfiability | In mathematical logic (a subtopic within the field of formal logic), two formulae are equisatisfiable if the first formula is satisfiable whenever the second is and vice versa; in other words, either both formulae are satisfiable or both are not. Equisatisfiable formulae may disagree, however, for a particular choice o... |
c_vrkl2kut5feg | Equisatisfiability | Summary | Equisatisfiability | Whereas within equisatisfiable formulae, only the primitive proposition the formula imposes is valued. Equisatisfiability is generally used in the context of translating formulae, so that one can define a translation to be correct if the original and resulting formulae are equisatisfiable. Examples of translations invo... |
c_ernahugexvqf | Valuation (logic) | Mathematical logic | Valuation_(logic) > Mathematical 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. |
c_2vni3jgt2pdu | Valuation (logic) | Mathematical logic | Valuation_(logic) > Mathematical 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... |
c_i2mz1cwtjw96 | Valuation (logic) | Mathematical logic | Valuation_(logic) > Mathematical 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 t... |
c_fzrmhkzdb9nu | Resolution inference | Summary | First-order_resolution | In mathematical logic and automated theorem proving, resolution is a rule of inference leading to a refutation complete theorem-proving technique for sentences in propositional logic and first-order logic. For propositional logic, systematically applying the resolution rule acts as a decision procedure for formula unsa... |
c_kw6bfezbgqeq | ⊢ | Summary | Turnstile_(symbol) | In mathematical logic and computer science the symbol ⊢ ( ⊢ {\displaystyle \vdash } ) has taken the name turnstile because of its resemblance to a typical turnstile if viewed from above. It is also referred to as tee and is often read as "yields", "proves", "satisfies" or "entails". |
c_elyf3hdf76bu | Gabbay's separation theorem | Summary | 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 i... |
c_118525g6llis | Mu-recursive function | Summary | 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 recursi... |
c_yw3mbukrfglv | Mu-recursive function | Summary | 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... |
c_8cgw3qvm2pxp | Univalence axiom | Summary | Higher_inductive_type | In mathematical logic and computer science, homotopy type theory (HoTT ) refers to various lines of development of intuitionistic type theory, based on the interpretation of types as objects to which the intuition of (abstract) homotopy theory applies. This includes, among other lines of work, the construction of homot... |
c_kqnqpt1xgj7k | Univalence axiom | Summary | Higher_inductive_type | Although neither is precisely delineated, and the terms are sometimes used interchangeably, the choice of usage also sometimes corresponds to differences in viewpoint and emphasis. As such, this article may not represent the views of all researchers in the fields equally. This kind of variability is unavoidable when a ... |
c_pgwr14muxwi5 | Top type | Summary | Top_type | In mathematical logic and computer science, some type theories and type systems include a top type that is commonly denoted with top or the symbol ⊤. The top type is sometimes called also universal type, or universal supertype as all other types in the type system of interest are subtypes of it, and in most cases, it c... |
c_5xuxzguq6qba | Kleene star | Summary | Kleene_star | In mathematical logic and computer science, the Kleene star (or Kleene operator or Kleene closure) is a unary operation, either on sets of strings or on sets of symbols or characters. In mathematics, it is more commonly known as the free monoid construction. The application of the Kleene star to a set V {\displaystyle ... |
c_7v9pxc0djz0i | Kleene star | Summary | Kleene_star | If V {\displaystyle V} is a set of strings, then V ∗ {\displaystyle V^{*}} is defined as the smallest superset of V {\displaystyle V} that contains the empty string ε {\displaystyle \varepsilon } and is closed under the string concatenation operation. If V {\displaystyle V} is a set of symbols or characters, then V ∗ {... |
c_g7wdu4a6wexk | Calculus of Constructions | Summary | Calculus_of_Inductive_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 assi... |
c_c7m22j7nswnm | Lambda-mu calculus | Summary | 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 b... |
c_uui5zih03bff | Lambda-mu calculus | Summary | 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 operat... |
c_k3b44wxsv67t | Two-variable logic with counting | Summary | Two-variable_logic_with_counting | In mathematical logic and computer science, two-variable logic is the fragment of first-order logic where formulae can be written using only two different variables. This fragment is usually studied without function symbols. |
c_yrkc2q2o60to | Analytical hierarchy | Summary | Analytical_hierarchy | In mathematical logic and descriptive set theory, the analytical hierarchy is an extension of the arithmetical hierarchy. The analytical hierarchy of formulas includes formulas in the language of second-order arithmetic, which can have quantifiers over both the set of natural numbers, N {\displaystyle \mathbb {N} } , a... |
c_ws5ljpkjxpvp | Implication graph | Summary | Implication_graph | In mathematical logic and graph theory, an implication graph is a skew-symmetric, directed graph G = (V, E) composed of vertex set V and directed edge set E. Each vertex in V represents the truth status of a Boolean literal, and each directed edge from vertex u to vertex v represents the material implication "If the li... |
c_w8b47n4kr7nx | Potential isomorphism | Summary | Potential_isomorphism | In mathematical logic and in particular in model theory, a potential isomorphism is a collection of finite partial isomorphisms between two models which satisfies certain closure conditions. Existence of a partial isomorphism entails elementary equivalence, however the converse is not generally true, but it holds for ω... |
c_mz23ddu2plqu | Universal Horn theory | Summary | 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. |
c_96qtjxzbzzmi | Completeness (logic) | Summary | Completeness_(logic) | In mathematical logic and metalogic, a formal system is called complete with respect to a particular property if every formula having the property can be derived using that system, i.e. is one of its theorems; otherwise the system is said to be incomplete. The term "complete" is also used without qualification, with di... |
c_7uj5b2om3zdn | Skolem paradox | Summary | Skolem_paradox | In mathematical logic and philosophy, Skolem's paradox is a seeming contradiction that arises from the downward Löwenheim–Skolem theorem. Thoralf Skolem (1922) was the first to discuss the seemingly contradictory aspects of the theorem, and to discover the relativity of set-theoretic notions now known as non-absolutene... |
c_s33uvyqvvl6v | Skolem paradox | Summary | Skolem_paradox | 295). Skolem's paradox is that every countable axiomatisation of set theory in first-order logic, if it is consistent, has a model that is countable. This appears contradictory because it is possible to prove, from those same axioms, a sentence that intuitively says (or that precisely says in the standard model of the ... |
c_5a834ri7i3zu | Skolem paradox | Summary | Skolem_paradox | Thus the seeming contradiction is that a model that is itself countable, and which therefore contains only countable sets, satisfies the first-order sentence that intuitively states "there are uncountable sets". A mathematical explanation of the paradox, showing that it is not a contradiction in mathematics, was given ... |
c_wsi6kh0nb6hy | Skolem paradox | Summary | Skolem_paradox | The philosophical implications of Skolem's paradox have received much study. One line of inquiry questions whether it is accurate to claim that any first-order sentence actually states "there are uncountable sets". This line of thought can be extended to question whether any set is uncountable in an absolute sense. Mor... |
c_w3u21j6y48d8 | Collapsing function | Summary | 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... |
c_dd90lo24zrkl | Collapsing function | Summary | 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, s... |
c_szisawowvdly | Ordinal notation | Summary | Ordinal_notation | In mathematical logic and set theory, an ordinal notation is a partial function mapping the set of all finite sequences of symbols, themselves members of a finite alphabet, to a countable set of ordinals. A Gödel numbering is a function mapping the set of well-formed formulae (a finite sequence of symbols on which the ... |
c_hbsygajvmqvi | Ordinal notation | Summary | Ordinal_notation | A recursive ordinal notation must satisfy the following two additional properties: the subset of natural numbers is a recursive set the induced well-ordering on the subset of natural numbers is a recursive relationThere are many such schemes of ordinal notations, including schemes by Wilhelm Ackermann, Heinz Bachmann, ... |
c_yeznq96dv6pp | Ordinal notation | Summary | Ordinal_notation | In many systems, such as Veblen's well known system, the functions are normal functions, that is, they are strictly increasing and continuous in at least one of their arguments, and increasing in other arguments. Another desirable property for such functions is that the value of the function is greater than each of its... |
c_twsyd5l85fjp | Minsky machine | Summary | Minsky_machine | In mathematical logic and theoretical computer science, a register machine is a generic class of abstract machines used in a manner similar to a Turing machine. All the models are Turing equivalent. |
c_ksi9zdg0mk5x | Convergent term rewriting system | Summary | Abstract_rewriting | In mathematical logic and theoretical computer science, an abstract rewriting system (also (abstract) reduction system or abstract rewrite system; abbreviated ARS) is a formalism that captures the quintessential notion and properties of rewriting systems. In its simplest form, an ARS is simply a set (of "objects") toge... |
c_rxcru5fd1jnz | Convergent term rewriting system | Summary | Abstract_rewriting | Historically, there have been several formalizations of rewriting in an abstract setting, each with its idiosyncrasies. This is due in part to the fact that some notions are equivalent, see below in this article. The formalization that is most commonly encountered in monographs and textbooks, and which is generally fol... |
c_8uqb63h257ym | Lambda cube | Summary | Lambda_cube | In mathematical logic and type theory, the λ-cube (also written lambda cube) is a framework introduced by Henk Barendregt to investigate the different dimensions in which the calculus of constructions is a generalization of the simply typed λ-calculus. Each dimension of the cube corresponds to a new kind of dependency ... |
c_vznrpqueur2c | Lambda cube | Summary | Lambda_cube | y-axis ( ↑ {\displaystyle \uparrow } ): terms that can bind types, corresponding to polymorphism. z-axis ( ↗ {\displaystyle \nearrow } ): types that can bind types, corresponding to (binding) type operators.The different ways to combine these three dimensions yield the 8 vertices of the cube, each corresponding to a di... |
c_k1ic5vx9d3hn | Löwenheim number | Summary | Löwenheim_number | In mathematical logic the Löwenheim number of an abstract logic is the smallest cardinal number for which a weak downward Löwenheim–Skolem theorem holds. They are named after Leopold Löwenheim, who proved that these exist for a very broad class of logics. |
c_xkesrom7e758 | Theory of pure equality | Summary | 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... |
c_c8nw7ovtlxwy | Beth definability | Summary | Beth_definability | In mathematical logic, Beth definability is a result that connects implicit definability of a property to its explicit definability. Specifically Beth definability states that the two senses of definability are equivalent. First-order logic has the Beth definability property. |
c_qkbn0tq4ev3y | Craig interpolation | Summary | Craig_interpolation | In mathematical logic, Craig's interpolation theorem is a result about the relationship between different logical theories. Roughly stated, the theorem says that if a formula φ implies a formula ψ, and the two have at least one atomic variable symbol in common, then there is a formula ρ, called an interpolant, such tha... |
c_9cyi99ypd68l | Craig's theorem | Summary | 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 th... |
c_syi6v2dp3b99 | Diaconescu theorem | Summary | Diaconescu's_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 t... |
c_r6uc4nhqy2l3 | Frege's propositional calculus | Summary | 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... |
c_yvcc2ejrqn1k | Frege's propositional calculus | Summary | 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. |
c_whljmdm08z2j | Frege's propositional calculus | Summary | 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 ... |
c_p0yb2ywqyoer | Kirby–Paris theorem | Summary | Goodstein_sequence | 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 a... |
c_2wvmv08b4rl8 | Kirby–Paris theorem | Summary | Goodstein_sequence | 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 fini... |
c_6fyqk4i34ow1 | Gödel's β function | Summary | 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 inclu... |
c_hjcrcjvv2os4 | Heyting arithmetic | Summary | Heyting_arithmetic | In mathematical logic, Heyting arithmetic H A {\displaystyle {\mathsf {HA}}} is an axiomatization of arithmetic in accordance with the philosophy of intuitionism. It is named after Arend Heyting, who first proposed it. |
c_fkx473k7dlud | Lindenbaum's lemma | Summary | Lindenbaum's_lemma | In mathematical logic, Lindenbaum's lemma, named after Adolf Lindenbaum, states that any consistent theory of predicate logic can be extended to a complete consistent theory. The lemma is a special case of the ultrafilter lemma for Boolean algebras, applied to the Lindenbaum algebra of a theory. |
c_t2w0hjofc3g2 | Lindstrom's theorem | Summary | Lindström'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–Skol... |
c_m4vahwo0nazg | Löb's Theorem | Summary | 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 ⊢... |
c_z2iyebfqa7nx | Morley rank | Summary | Morley_rank | In mathematical logic, Morley rank, introduced by Michael D. Morley (1965), is a means of measuring the size of a subset of a model of a theory, generalizing the notion of dimension in algebraic geometry. |
c_ldthcj2lr6ii | Typed set theory | Summary | 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 wit... |
c_z0067ijyeq8i | Typed set theory | Summary | 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 schem... |
c_5ijy5ppmsw0q | Peano–Russell notation | Summary | Peano–Russell_notation | In mathematical logic, Peano–Russell notation was Bertrand Russell's application of Giuseppe Peano's logical notation to the logical notions of Frege and was used in the writing of Principia Mathematica in collaboration with Alfred North Whitehead: "The notation adopted in the present work is based upon that of Peano, ... |
c_pbe6ylg8n1bt | Rosser's trick | Summary | Rosser's_trick | In mathematical logic, Rosser's trick is a method for proving Gödel's incompleteness theorems without the assumption that the theory being considered is ω-consistent (Smorynski 1977, p. 840; Mendelson 1977, p. 160). This method was introduced by J. Barkley Rosser in 1936, as an improvement of Gödel's original proof of ... |
c_n8tpkd0xwk1v | Russell paradox | Summary | Russel's_paradox | In mathematical logic, Russell's paradox (also known as Russell's antinomy) is a set-theoretic paradox published by the British philosopher and mathematician Bertrand Russell in 1901. Russell's paradox shows that every set theory that contains an unrestricted comprehension principle leads to contradictions. The paradox... |
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