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c_8j0mw7pycj8m | Complete theory | Summary | Maximal_consistent_set | 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. |
c_eh0aojdvgk4r | Tolerant sequence | Summary | Tolerant_sequence | In mathematical logic, a tolerant sequence is a sequence T 1 {\displaystyle T_{1}} ,..., T n {\displaystyle T_{n}} of formal theories such that there are consistent extensions S 1 {\displaystyle S_{1}} ,..., S n {\displaystyle S_{n}} of these theories with each S i + 1 {\displaystyle S_{i+1}} interpretable in S i {\displaystyle S_{i}} . Tolerance naturally generalizes from sequences of theories to trees of theories. Weak interpretability can be shown to be a special, binary case of tolerance. This concept, together with its dual concept of cotolerance, was introduced by Japaridze in 1992, who also proved that, for Peano arithmetic and any stronger theories with effective axiomatizations, tolerance is equivalent to Π 1 {\displaystyle \Pi _{1}} -consistency. |
c_psv7eu26br6s | Universal quantifier | Summary | Universally_quantify | In mathematical logic, a universal quantification is a type of quantifier, a logical constant which is interpreted as "given any", "for all", or "for any". It expresses that a predicate can be satisfied by every member of a domain of discourse. In other words, it is the predication of a property or relation to every member of the domain. It asserts that a predicate within the scope of a universal quantifier is true of every value of a predicate variable. |
c_my06ol4g9yrp | Universal quantifier | Summary | Universally_quantify | It is usually denoted by the turned A (∀) logical operator symbol, which, when used together with a predicate variable, is called a universal quantifier ("∀x", "∀(x)", or sometimes by "(x)" alone). Universal quantification is distinct from existential quantification ("there exists"), which only asserts that the property or relation holds for at least one member of the domain. Quantification in general is covered in the article on quantification (logic). The universal quantifier is encoded as U+2200 ∀ FOR ALL in Unicode, and as \forall in LaTeX and related formula editors. |
c_91avl568pkxm | Witness (mathematics) | Summary | 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. |
c_4ohz0dwm0lg6 | Abstract algebraic logic | Summary | 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. |
c_im7gxxv9df1q | Abstract model theory | Summary | 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. |
c_ojfr45wiqh4h | Logic of relations | Summary | Calculus_of_relations | In mathematical logic, algebraic logic is the reasoning obtained by manipulating equations with free variables. What is now usually called classical algebraic logic focuses on the identification and algebraic description of models appropriate for the study of various logics (in the form of classes of algebras that constitute the algebraic semantics for these deductive systems) and connected problems like representation and duality. Well known results like the representation theorem for Boolean algebras and Stone duality fall under the umbrella of classical algebraic logic (Czelakowski 2003). Works in the more recent abstract algebraic logic (AAL) focus on the process of algebraization itself, like classifying various forms of algebraizability using the Leibniz operator (Czelakowski 2003). |
c_pw2jv1nkhgst | Algebraic semantics (mathematical logic) | Summary | Algebraic_semantics_(mathematical_logic) | In mathematical logic, algebraic semantics is a formal semantics based on algebras studied as part of algebraic logic. For example, the modal logic S4 is characterized by the class of topological boolean algebras—that is, boolean algebras with an interior operator. Other modal logics are characterized by various other algebras with operators. The class of boolean algebras characterizes classical propositional logic, and the class of Heyting algebras propositional intuitionistic logic. MV-algebras are the algebraic semantics of Łukasiewicz logic. |
c_87nwfiug865p | Modus ponens | Algebraic semantics | Modus_ponens > Correspondence to other mathematical frameworks > Algebraic semantics | In mathematical logic, algebraic semantics treats every sentence as a name for an element in an ordered set. Typically, the set can be visualized as a lattice-like structure with a single element (the "always-true") at the top and another single element (the "always-false") at the bottom. Logical equivalence becomes identity, so that when ¬ ( P ∧ Q ) {\displaystyle \neg {(P\wedge Q)}} and ¬ P ∨ ¬ Q {\displaystyle \neg {P}\vee \neg {Q}} , for instance, are equivalent (as is standard), then ¬ ( P ∧ Q ) = ¬ P ∨ ¬ Q {\displaystyle \neg {(P\wedge Q)}=\neg {P}\vee \neg {Q}} . Logical implication becomes a matter of relative position: P {\displaystyle P} logically implies Q {\displaystyle Q} just in case P ≤ Q {\displaystyle P\leq Q} , i.e., when either P = Q {\displaystyle P=Q} or else P {\displaystyle P} lies below Q {\displaystyle Q} and is connected to it by an upward path. |
c_cmf5gesdqlng | Modus ponens | Algebraic semantics | Modus_ponens > Correspondence to other mathematical frameworks > Algebraic semantics | In this context, to say that P {\textstyle P} and P → Q {\displaystyle P\rightarrow Q} together imply Q {\displaystyle Q} —that is, to affirm modus ponens as valid—is to say that P ∧ ( P → Q ) ≤ Q {\displaystyle P\wedge (P\rightarrow Q)\leq Q} . In the semantics for basic propositional logic, the algebra is Boolean, with → {\displaystyle \rightarrow } construed as the material conditional: P → Q = ¬ P ∨ Q {\displaystyle P\rightarrow Q=\neg {P}\vee Q} . Confirming that P ∧ ( P → Q ) ≤ Q {\displaystyle P\wedge (P\rightarrow Q)\leq Q} is then straightforward, because P ∧ ( P → Q ) = P ∧ Q {\displaystyle P\wedge (P\rightarrow Q)=P\wedge Q} . With other treatments of → {\displaystyle \rightarrow } , the semantics becomes more complex, the algebra may be non-Boolean, and the validity of modus ponens cannot be taken for granted. |
c_fxhfzrja45bo | Substructure (mathematics) | Summary | Extension_(model_theory) | In mathematical logic, an (induced) substructure or (induced) subalgebra is a structure whose domain is a subset of that of a bigger structure, and whose functions and relations are restricted to the substructure's domain. Some examples of subalgebras are subgroups, submonoids, subrings, subfields, subalgebras of algebras over a field, or induced subgraphs. Shifting the point of view, the larger structure is called an extension or a superstructure of its substructure. |
c_4gjrkm2okngo | Substructure (mathematics) | Summary | Extension_(model_theory) | In model theory, the term "submodel" is often used as a synonym for substructure, especially when the context suggests a theory of which both structures are models. In the presence of relations (i.e. for structures such as ordered groups or graphs, whose signature is not functional) it may make sense to relax the conditions on a subalgebra so that the relations on a weak substructure (or weak subalgebra) are at most those induced from the bigger structure. Subgraphs are an example where the distinction matters, and the term "subgraph" does indeed refer to weak substructures. Ordered groups, on the other hand, have the special property that every substructure of an ordered group which is itself an ordered group, is an induced substructure. |
c_h8sw1rh3us9m | Abstract logic | Summary | Abstract_logic | In mathematical logic, an abstract logic is a formal system consisting of a class of sentences and a satisfaction relation with specific properties related to occurrence, expansion, isomorphism, renaming and quantification.Based on Lindström's characterization, first-order logic is, up to equivalence, the only abstract logic that is countably compact and has Löwenheim number ω. |
c_fj7cmznzgv0k | Algebraic definition | Summary | Algebraic_definition | In mathematical logic, an algebraic definition is one that can be given using only equations between terms with free variables. Inequalities and quantifiers are specifically disallowed.Saying that a definition is algebraic is a stronger condition than saying it is elementary. |
c_musk3i5w5pi0 | Algebraic sentence | Summary | Algebraic_sentence | In mathematical logic, an algebraic sentence is one that can be stated using only equations between terms with free variables. Inequalities and quantifiers are specifically disallowed. Sentential logic is the subset of first-order logic involving only algebraic sentences. Saying that a sentence is algebraic is a stronger condition than saying it is elementary. |
c_my6yge2ptvc9 | Alternative Set Theory | Summary | Alternative_Set_Theory | In mathematical logic, an alternative set theory is any of the alternative mathematical approaches to the concept of set and any alternative to the de facto standard set theory described in axiomatic set theory by the axioms of Zermelo–Fraenkel set theory. |
c_qd01ph9l5yks | Arithmetical numbers | Summary | Arithmetical_numbers | In mathematical logic, an arithmetical set (or arithmetic set) is a set of natural numbers that can be defined by a formula of first-order Peano arithmetic. The arithmetical sets are classified by the arithmetical hierarchy. The definition can be extended to an arbitrary countable set A (e.g. the set of n-tuples of integers, the set of rational numbers, the set of formulas in some formal language, etc.) by using Gödel numbers to represent elements of the set and declaring a subset of A to be arithmetical if the set of corresponding Gödel numbers is arithmetical. |
c_dnffvd7zynk7 | Arithmetical numbers | Summary | Arithmetical_numbers | A function f :⊆ N k → N {\displaystyle f:\subseteq \mathbb {N} ^{k}\to \mathbb {N} } is called arithmetically definable if the graph of f {\displaystyle f} is an arithmetical set. A real number is called arithmetical if the set of all smaller rational numbers is arithmetical. A complex number is called arithmetical if its real and imaginary parts are both arithmetical. |
c_ids38uyjoo2f | Atomic formula | Summary | Atomic_formula | In mathematical logic, an atomic formula (also known as an atom or a prime formula) is a formula with no deeper propositional structure, that is, a formula that contains no logical connectives or equivalently a formula that has no strict subformulas. Atoms are thus the simplest well-formed formulas of the logic. Compound formulas are formed by combining the atomic formulas using the logical connectives. |
c_y3yukz0o4dtb | Atomic formula | Summary | Atomic_formula | The precise form of atomic formulas depends on the logic under consideration; for propositional logic, for example, a propositional variable is often more briefly referred to as an "atomic formula", but, more precisely, a propositional variable is not an atomic formula but a formal expression that denotes an atomic formula. For predicate logic, the atoms are predicate symbols together with their arguments, each argument being a term. In model theory, atomic formulas are merely strings of symbols with a given signature, which may or may not be satisfiable with respect to a given model. |
c_yc309uy0gvi4 | Finite axiomatization | Summary | Finite_axiomatization | In mathematical logic, an axiom schema (plural: axiom schemata or axiom schemas) generalizes the notion of axiom. |
c_0s86dwjpcexk | Effective Polish space | Summary | Effective_Polish_space | In mathematical logic, an effective Polish space is a complete separable metric space that has a computable presentation. Such spaces are studied in effective descriptive set theory and in constructive analysis. In particular, standard examples of Polish spaces such as the real line, the Cantor set and the Baire space are all effective Polish spaces. |
c_p26082gep7ic | Elementary definition | Summary | Elementary_definition | In mathematical logic, an elementary definition is a definition that can be made using only finitary first-order logic, and in particular without reference to set theory or using extensions such as plural quantification. Elementary definitions are of particular interest because they admit a complete proof apparatus while still being expressive enough to support most everyday mathematics (via the addition of elementarily-expressible axioms such as Zermelo–Fraenkel set theory (ZFC)). Saying that a definition is elementary is a weaker condition than saying it is algebraic. |
c_k3htmrvfcnng | Elementary sentence | Summary | Elementary_sentence | In mathematical logic, an elementary sentence is one that is stated using only finitary first-order logic, without reference to set theory or using any axioms which have consistency strength equal to set theory. Saying that a sentence is elementary is a weaker condition than saying it is algebraic. |
c_q7pubpldhpg2 | Countably categorical theory | Summary | Omega-categorical_theory | In mathematical logic, an omega-categorical theory is a theory that has exactly one countably infinite model up to isomorphism. Omega-categoricity is the special case κ = ℵ 0 {\displaystyle \aleph _{0}} = ω of κ-categoricity, and omega-categorical theories are also referred to as ω-categorical. The notion is most important for countable first-order theories. |
c_i8gskck5y54l | Function letter | Summary | Uninterpreted_function | In mathematical logic, an uninterpreted function or function symbol is one that has no other property than its name and n-ary form. Function symbols are used, together with constants and variables, to form terms. The theory of uninterpreted functions is also sometimes called the free theory, because it is freely generated, and thus a free object, or the empty theory, being the theory having an empty set of sentences (in analogy to an initial algebra). Theories with a non-empty set of equations are known as equational theories. The satisfiability problem for free theories is solved by syntactic unification; algorithms for the latter are used by interpreters for various computer languages, such as Prolog. Syntactic unification is also used in algorithms for the satisfiability problem for certain other equational theories, see Unification (computer science). |
c_6nmxumggn8km | Omega-consistent theory | Summary | Omega-consistent_theory | In mathematical logic, an ω-consistent (or omega-consistent, also called numerically segregative) theory is a theory (collection of sentences) that is not only (syntactically) consistent (that is, does not prove a contradiction), but also avoids proving certain infinite combinations of sentences that are intuitively contradictory. The name is due to Kurt Gödel, who introduced the concept in the course of proving the incompleteness theorem. |
c_q2wgh6wfsngc | O-minimal theory | Summary | O-minimal_structure | In mathematical logic, and more specifically in model theory, an infinite structure (M,<,...) that is totally ordered by < is called an o-minimal structure if and only if every definable subset X ⊆ M (with parameters taken from M) is a finite union of intervals and points. O-minimality can be regarded as a weak form of quantifier elimination. A structure M is o-minimal if and only if every formula with one free variable and parameters in M is equivalent to a quantifier-free formula involving only the ordering, also with parameters in M. This is analogous to the minimal structures, which are exactly the analogous property down to equality. A theory T is an o-minimal theory if every model of T is o-minimal. It is known that the complete theory T of an o-minimal structure is an o-minimal theory. This result is remarkable because, in contrast, the complete theory of a minimal structure need not be a strongly minimal theory, that is, there may be an elementarily equivalent structure that is not minimal. |
c_ysan8rgfmkcv | Saturated model | Summary | Saturated_model | In mathematical logic, and particularly in its subfield model theory, a saturated model M is one that realizes as many complete types as may be "reasonably expected" given its size. For example, an ultrapower model of the hyperreals is ℵ 1 {\displaystyle \aleph _{1}} -saturated, meaning that every descending nested sequence of internal sets has a nonempty intersection. |
c_2zh8j0avscji | Kappa calculus | Summary | Kappa_calculus | In mathematical logic, category theory, and computer science, kappa calculus is a formal system for defining first-order functions. Unlike lambda calculus, kappa calculus has no higher-order functions; its functions are not first class objects. Kappa-calculus can be regarded as "a reformulation of the first-order fragment of typed lambda calculus".Because its functions are not first-class objects, evaluation of kappa calculus expressions does not require closures. |
c_twwfkrvpa7sf | Cointerpretability | Summary | Cointerpretability | In mathematical logic, cointerpretability is a binary relation on formal theories: a formal theory T is cointerpretable in another such theory S, when the language of S can be translated into the language of T in such a way that S proves every formula whose translation is a theorem of T. The "translation" here is required to preserve the logical structure of formulas. This concept, in a sense dual to interpretability, was introduced by Japaridze (1993), who also proved that, for theories of Peano arithmetic and any stronger theories with effective axiomatizations, cointerpretability is equivalent to Σ 1 {\displaystyle \Sigma _{1}} -conservativity. |
c_0ml3828llw2o | Existential theory of the reals | Summary | Existential_theory_of_the_reals | In mathematical logic, computational complexity theory, and computer science, the existential theory of the reals is the set of all true sentences of the form where the variables X i {\displaystyle X_{i}} are interpreted as having real number values, and 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. A sentence of this form is true if it is possible to find values for all of the variables that, when substituted into formula F {\displaystyle F} , make it become true.The decision problem for the existential theory of the reals is the problem of finding an algorithm that decides, for each such sentence, whether it is true or false. Equivalently, it is the problem of testing whether a given semialgebraic set is non-empty. This decision problem is NP-hard and lies in PSPACE, giving it significantly lower complexity than Alfred Tarski's quantifier elimination procedure for deciding statements in the first-order theory of the reals without the restriction to existential quantifiers. |
c_p58en2h9rlqx | Existential theory of the reals | Summary | Existential_theory_of_the_reals | However, in practice, general methods for the first-order theory remain the preferred choice for solving these problems.The complexity class ∃ R {\displaystyle \exists \mathbb {R} } has been defined to describe the class of computational problems that may be translated into equivalent sentences of this form. In structural complexity theory, it lies between NP and PSPACE. Many natural problems in geometric graph theory, especially problems of recognizing geometric intersection graphs and straightening the edges of graph drawings with crossings, belong to ∃ R {\displaystyle \exists \mathbb {R} } , and are complete for this class. Here, completeness means that there exists a translation in the reverse direction, from an arbitrary sentence over the reals into an equivalent instance of the given problem. |
c_amlm24gzxw4f | Fuzzy concept | Clarifying methods | Fuzzy_concept > Clarifying methods | In mathematical logic, computer programming, philosophy and linguistics fuzzy concepts can be analyzed and defined more accurately or comprehensively, by describing or modelling the concepts using the terms of fuzzy logic or other substructural logics. More generally, clarification techniques can be used such as: 1. Contextualizing the concept by defining the setting or situation in which the concept is used, or how it is used appropriately (context). 2. |
c_bhqsn8g96ap0 | Fuzzy concept | Clarifying methods | Fuzzy_concept > Clarifying methods | Identifying the intention, purpose, aim or goal associated with the concept (teleology and design). 3. Comparing and contrasting the concept with related ideas in the present or the past (comparative and comparative research). |
c_rgu9la7exeum | Fuzzy concept | Clarifying methods | Fuzzy_concept > Clarifying methods | 4. Creating a model, likeness, analogy, metaphor, prototype or narrative which shows what the concept is about or how it is applied (isomorphism, simulation or successive approximation ). 5. |
c_r1rkfus3w9zl | Fuzzy concept | Clarifying methods | Fuzzy_concept > Clarifying methods | Probing the assumptions on which a concept is based, or which are associated with its use (critical thought, tacit assumption). 6. Mapping or graphing the applications of the concept using some basic parameters, or using some diagrams or flow charts to understand the relationships between elements involved (visualization and concept map). |
c_96x2giswap2y | Fuzzy concept | Clarifying methods | Fuzzy_concept > Clarifying methods | 7. Examining how likely it is that the concept applies, statistically or intuitively (probability theory). |
c_47vce6r97okp | Fuzzy concept | Clarifying methods | Fuzzy_concept > Clarifying methods | 8. Specifying relevant conditions to which the concept applies, as a procedure (computer programming, formal concept analysis). 9. |
c_xe4wskdsav1t | Fuzzy concept | Clarifying methods | Fuzzy_concept > Clarifying methods | Concretizing the concept – finding specific examples, illustrations, details or cases to which it applies (exemplar, exemplification). 10. Reducing or restating fuzzy concepts in terms which are simpler or similar, and which are not fuzzy or less fuzzy (simplification, dimensionality reduction, plain language, KISS principle or concision). |
c_agv2u2hcprg6 | Fuzzy concept | Clarifying methods | Fuzzy_concept > Clarifying methods | 11. Trying out a concept, by using it in interactions, practical work or in communication, and assessing the feedback to understand how the boundaries and distinctions of the concept are being drawn (trial and error or pilot experiment). 12. |
c_7psjsr3fjh1z | Fuzzy concept | Clarifying methods | Fuzzy_concept > Clarifying methods | Engaging in a structured dialogue or repeated discussion, to exchange ideas about how to get specific about what it means and how to clear it up (scrum method). 13. Allocating different applications of the concept to different but related sets (Boolean logic). |
c_bucbb3zecgzt | Fuzzy concept | Clarifying methods | Fuzzy_concept > Clarifying methods | 14. Identifying operational rules defining the use of the concept, which can be stated in a language and which cover all or most cases (material conditional). 15. |
c_3pplu4u2sw0e | Fuzzy concept | Clarifying methods | Fuzzy_concept > Clarifying methods | Classifying, categorizing, grouping, or inventorizing all or most cases or uses to which the concept applies (taxonomy, cluster analysis and typology).16. Applying a meta-language which includes fuzzy concepts in a more inclusive categorical system which is not fuzzy (meta). 17. |
c_e22rh9nq3dzb | Fuzzy concept | Clarifying methods | Fuzzy_concept > Clarifying methods | Creating a measure or scale of the degree to which the concept applies (metrology). 18. Examining the distribution patterns or distributional frequency of (possibly different) uses of the concept (statistics). |
c_xkxnp328pepk | Fuzzy concept | Clarifying methods | Fuzzy_concept > Clarifying methods | 19. Specifying a series of logical operators or inferential system which captures all or most cases to which the concept applies (algorithm). 20. |
c_vbmhqdm4zglc | Fuzzy concept | Clarifying methods | Fuzzy_concept > Clarifying methods | Relating the fuzzy concept to other concepts which are not fuzzy or less fuzzy, or simply by replacing the fuzzy concept altogether with another, alternative concept which is not fuzzy yet "works the same way" (proxy) 21. Engaging in meditation, or taking the proverbial "run around the block" to clarify the mind, and thus improve precision of thought about the definitional issue (self-care).In this way, we can obtain a more exact understanding of the meaning and use of a fuzzy concept, and possibly decrease the amount of fuzziness. It may not be possible to specify all the possible meanings or applications of a concept completely and exhaustively, but if it is possible to capture the majority of them, statistically or otherwise, this may be useful enough for practical purposes. |
c_zijsdxjtrt58 | Descriptive set theory | Summary | Descriptive_set_theory | In mathematical logic, descriptive set theory (DST) is the study of certain classes of "well-behaved" subsets of the real line and other Polish spaces. As well as being one of the primary areas of research in set theory, it has applications to other areas of mathematics such as functional analysis, ergodic theory, the study of operator algebras and group actions, and mathematical logic. |
c_q8qgk6ol7jse | Buchholz hydra | Summary | Buchholz_hydra | In mathematical logic, especially in graph theory and number theory, the Buchholz hydra game is a type of hydra game, which is a single-player game based on the idea of chopping pieces off of a mathematical tree. The hydra game can be used to generate a rapidly growing function, B H ( n ) {\displaystyle BH(n)} , which eventually dominates all recursive functions that are provably total in " ID ν {\displaystyle {\textrm {ID}}_{\nu }} ", and the termination of all hydra games is not provably total in ( Π 1 1 -CA)+BI {\displaystyle {\textrm {(}}\Pi _{1}^{1}{\textrm {-CA)+BI}}} . |
c_4sch13njdp3m | Back-and-forth method | Summary | Back-and-forth_method | In mathematical logic, especially set theory and model theory, the back-and-forth method is a method for showing isomorphism between countably infinite structures satisfying specified conditions. In particular it can be used to prove that any two countably infinite densely ordered sets (i.e., linearly ordered in such a way that between any two members there is another) without endpoints are isomorphic. An isomorphism between linear orders is simply a strictly increasing bijection. This result implies, for example, that there exists a strictly increasing bijection between the set of all rational numbers and the set of all real algebraic numbers. |
c_bbsyt53wduow | Back-and-forth method | Summary | Back-and-forth_method | any two countably infinite atomless Boolean algebras are isomorphic to each other. any two equivalent countable atomic models of a theory are isomorphic. the Erdős–Rényi model of random graphs, when applied to countably infinite graphs, almost surely produces a unique graph, the Rado graph. any two many-complete recursively enumerable sets are recursively isomorphic. |
c_d05u06nti5mg | Unstable fixed point | Fixed-point logics | Repulsive_fixed_point > Fixed-point logics | In mathematical logic, fixed-point logics are extensions of classical predicate logic that have been introduced to express recursion. Their development has been motivated by descriptive complexity theory and their relationship to database query languages, in particular to Datalog. |
c_n8idd2ucarpc | Least fixed-point logic | Summary | Partial_fixed-point_logic | In mathematical logic, fixed-point logics are extensions of classical predicate logic that have been introduced to express recursion. Their development has been motivated by descriptive complexity theory and their relationship to database query languages, in particular to Datalog. Least fixed-point logic was first studied systematically by Yiannis N. Moschovakis in 1974, and it was introduced to computer scientists in 1979, when Alfred Aho and Jeffrey Ullman suggested fixed-point logic as an expressive database query language. |
c_e6cpm9xa3bxs | Focused proof | Summary | Focused_proof | In mathematical logic, focused proofs are a family of analytic proofs that arise through goal-directed proof-search, and are a topic of study in structural proof theory and reductive logic. They form the most general definition of goal-directed proof-search—in which someone chooses a formula and performs hereditary reductions until the result meets some condition. The extremal case where reduction only terminates when axioms are reached forms the sub-family of uniform proofs.A sequent calculus is said to have the focusing property when focused proofs are complete for some terminating condition. For System LK, System LJ, and System LL, uniform proofs are focused proofs where all the atoms are assigned negative polarity. Many other sequent calculi has been shown to have the focusing property, notably the nested sequent calculi of both the classical and intuitionistic variants of the modal logics in the S5 cube. |
c_h0uxv8kixc6w | Equiconsistency | Consistency | Consistency_strength > Consistency | In mathematical logic, formal theories are studied as mathematical objects. Since some theories are powerful enough to model different mathematical objects, it is natural to wonder about their own consistency. Hilbert proposed a program at the beginning of the 20th century whose ultimate goal was to show, using mathematical methods, the consistency of mathematics. Since most mathematical disciplines can be reduced to arithmetic, the program quickly became the establishment of the consistency of arithmetic by methods formalizable within arithmetic itself. |
c_j7q4n4uemiyy | Equiconsistency | Consistency | Consistency_strength > Consistency | Gödel's incompleteness theorems show that Hilbert's program cannot be realized: if a consistent recursively enumerable theory is strong enough to formalize its own metamathematics (whether something is a proof or not), i.e. strong enough to model a weak fragment of arithmetic (Robinson arithmetic suffices), then the theory cannot prove its own consistency. There are some technical caveats as to what requirements the formal statement representing the metamathematical statement "The theory is consistent" needs to satisfy, but the outcome is that if a (sufficiently strong) theory can prove its own consistency then either there is no computable way of identifying whether a statement is even an axiom of the theory or not, or else the theory itself is inconsistent (in which case it can prove anything, including false statements such as its own consistency). Given this, instead of outright consistency, one usually considers relative consistency: Let S and T be formal theories. |
c_czmi742fauct | Equiconsistency | Consistency | Consistency_strength > Consistency | Assume that S is a consistent theory. Does it follow that T is consistent? If so, then T is consistent relative to S. Two theories are equiconsistent if each one is consistent relative to the other. |
c_kb17rf2ypojs | Formation rule | Summary | Formation_rules | In mathematical logic, formation rules are rules for describing which strings of symbols formed from the alphabet of a formal language are syntactically valid within the language. These rules only address the location and manipulation of the strings of the language. It does not describe anything else about a language, such as its semantics (i.e. what the strings mean). (See also formal grammar). |
c_ze4apcqkor01 | Geometric logic | Summary | Geometric_logic | In mathematical logic, geometric logic is an infinitary generalisation of coherent logic, a restriction of first-order logic due to Skolem that is proof-theoretically tractable. Geometric logic is capable of expressing many mathematical theories and has close connections to topos theory. |
c_ppk0mvjww8tj | Internal set | Summary | Internal_function | In mathematical logic, in particular in model theory and nonstandard analysis, an internal set is a set that is a member of a model. The concept of internal sets is a tool in formulating the transfer principle, which concerns the logical relation between the properties of the real numbers R, and the properties of a larger field denoted *R called the hyperreal numbers. The field *R includes, in particular, infinitesimal ("infinitely small") numbers, providing a rigorous mathematical justification for their use. |
c_mkqi11n4x5z8 | Internal set | Summary | Internal_function | Roughly speaking, the idea is to express analysis over R in a suitable language of mathematical logic, and then point out that this language applies equally well to *R. This turns out to be possible because at the set-theoretic level, the propositions in such a language are interpreted to apply only to internal sets rather than to all sets (note that the term "language" is used in a loose sense in the above). Edward Nelson's internal set theory is an axiomatic approach to nonstandard analysis (see also Palmgren at constructive nonstandard analysis). Conventional infinitary accounts of nonstandard analysis also use the concept of internal sets. |
c_jtzsrbmmqxxr | Logically independent | Summary | Logically_independent | In mathematical logic, independence is the unprovability of a sentence from other sentences. A sentence σ is independent of a given first-order theory T if T neither proves nor refutes σ; that is, it is impossible to prove σ from T, and it is also impossible to prove from T that σ is false. Sometimes, σ is said (synonymously) to be undecidable from T; this is not the same meaning of "decidability" as in a decision problem. A theory T is independent if each axiom in T is not provable from the remaining axioms in T. A theory for which there is an independent set of axioms is independently axiomatizable. |
c_dnrtnlcp2i1k | Indiscernibles | Summary | Set_of_indiscernibles | In mathematical logic, indiscernibles are objects that cannot be distinguished by any property or relation defined by a formula. Usually only first-order formulas are considered. |
c_6vc17lmbbx2j | Interpretability | Summary | Interpretability | In mathematical logic, interpretability is a relation between formal theories that expresses the possibility of interpreting or translating one into the other. |
c_p68gbfuglkro | Minimal axioms for Boolean algebra | Summary | Minimal_axioms_for_Boolean_algebra | In mathematical logic, minimal axioms for Boolean algebra are assumptions which are equivalent to the axioms of Boolean algebra (or propositional calculus), chosen to be as short as possible. For example, if one chooses to take commutativity for granted, an axiom with six NAND operations and three variables is equivalent to Boolean algebra: ( ( a ∣ b ) ∣ c ) ∣ ( a ∣ ( ( a ∣ c ) ∣ a ) ) = c {\displaystyle ((a\mid b)\mid c)\mid (a\mid ((a\mid c)\mid a))=c} where the vertical bar represents the NAND logical operation (also known as the Sheffer stroke). It is one of 25 candidate axioms for this property identified by Stephen Wolfram, by enumerating the Sheffer identities of length less or equal to 15 elements (excluding mirror images) that have no noncommutative models with four or fewer variables, and was first proven equivalent by William McCune, Branden Fitelson, and Larry Wos. MathWorld, a site associated with Wolfram, has named the axiom the "Wolfram axiom". |
c_83hzy7k50teu | Minimal axioms for Boolean algebra | Summary | Minimal_axioms_for_Boolean_algebra | McCune et al. also found a longer single axiom for Boolean algebra based on disjunction and negation.In 1933, Edward Vermilye Huntington identified the axiom ¬ ( ¬ x ∨ y ) ∨ ¬ ( ¬ x ∨ ¬ y ) = x {\displaystyle {\neg ({\neg x}\lor {y})}\lor {\neg ({\neg x}\lor {\neg y})}=x} as being equivalent to Boolean algebra, when combined with the commutativity of the OR operation, x ∨ y = y ∨ x {\displaystyle x\lor y=y\lor x} , and the assumption of associativity, ( x ∨ y ) ∨ z = x ∨ ( y ∨ z ) {\displaystyle (x\lor y)\lor z=x\lor (y\lor z)} . Herbert Robbins conjectured that Huntington's axiom could be replaced by ¬ ( ¬ ( x ∨ y ) ∨ ¬ ( x ∨ ¬ y ) ) = x , {\displaystyle \neg (\neg (x\lor y)\lor \neg (x\lor {\neg y}))=x,} which requires one fewer use of the logical negation operator ¬ {\displaystyle \neg } . Neither Robbins nor Huntington could prove this conjecture; nor could Alfred Tarski, who took considerable interest in it later. |
c_6hsb7jr7zv07 | Minimal axioms for Boolean algebra | Summary | Minimal_axioms_for_Boolean_algebra | The conjecture was eventually proved in 1996 with the aid of theorem-proving software. This proof established that the Robbins axiom, together with associativity and commutativity, form a 3-basis for Boolean algebra. The existence of a 2-basis was established in 1967 by Carew Arthur Meredith: ¬ ( ¬ x ∨ y ) ∨ x = x , {\displaystyle \neg ({\neg x}\lor y)\lor x=x,} ¬ ( ¬ x ∨ y ) ∨ ( z ∨ y ) = y ∨ ( z ∨ x ) . |
c_xhn703xmuq4s | Minimal axioms for Boolean algebra | Summary | Minimal_axioms_for_Boolean_algebra | {\displaystyle \neg ({\neg x}\lor y)\lor (z\lor y)=y\lor (z\lor x).} The following year, Meredith found a 2-basis in terms of the Sheffer stroke: ( x ∣ x ) ∣ ( y ∣ x ) = x , {\displaystyle (x\mid x)\mid (y\mid x)=x,} x | ( y ∣ ( x ∣ z ) ) = ( ( z ∣ y ) ∣ y ) ∣ x . |
c_26ggungobjiy | Minimal axioms for Boolean algebra | Summary | Minimal_axioms_for_Boolean_algebra | {\displaystyle x|(y\mid (x\mid z))=((z\mid y)\mid y)\mid x.} In 1973, Padmanabhan and Quackenbush demonstrated a method that, in principle, would yield a 1-basis for Boolean algebra. |
c_k5k0wep2ez4c | Minimal axioms for Boolean algebra | Summary | Minimal_axioms_for_Boolean_algebra | Applying this method in a straightforward manner yielded "axioms of enormous length", thereby prompting the question of how shorter axioms might be found. This search yielded the 1-basis in terms of the Sheffer stroke given above, as well as the 1-basis ¬ ( ¬ ( ¬ ( x ∨ y ) ∨ z ) ∨ ¬ ( x ∨ ¬ ( ¬ z ∨ ¬ ( z ∨ u ) ) ) ) = z , {\displaystyle \neg (\neg (\neg (x\lor y)\lor z)\lor \neg (x\lor \neg (\neg z\lor \neg (z\lor u))))=z,} which is written in terms of OR and NOT. == References == |
c_fuzcmrl8egej | Homogeneous model | Summary | Homogeneous_model | In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the statements of the theory hold). The aspects investigated include the number and size of models of a theory, the relationship of different models to each other, and their interaction with the formal language itself. In particular, model theorists also investigate the sets that can be defined in a model of a theory, and the relationship of such definable sets to each other. As a separate discipline, model theory goes back to Alfred Tarski, who first used the term "Theory of Models" in publication in 1954. |
c_f4tngc2bkmy2 | Homogeneous model | Summary | Homogeneous_model | Since the 1970s, the subject has been shaped decisively by Saharon Shelah's stability theory. Compared to other areas of mathematical logic such as proof theory, model theory is often less concerned with formal rigour and closer in spirit to classical mathematics. This has prompted the comment that "if proof theory is about the sacred, then model theory is about the profane". |
c_lfq6v9mfkor0 | Homogeneous model | Summary | Homogeneous_model | The applications of model theory to algebraic and Diophantine geometry reflect this proximity to classical mathematics, as they often involve an integration of algebraic and model-theoretic results and techniques. Consequently, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature. The most prominent scholarly organization in the field of model theory is the Association for Symbolic Logic. |
c_dm0kwse7wka6 | Monadic second-order | Summary | Monadic_second-order_logic | In mathematical logic, monadic second-order logic (MSO) is the fragment of second-order logic where the second-order quantification is limited to quantification over sets. It is particularly important in the logic of graphs, because of Courcelle's theorem, which provides algorithms for evaluating monadic second-order formulas over graphs of bounded treewidth. It is also of fundamental importance in automata theory, where the Büchi–Elgot–Trakhtenbrot theorem gives a logical characterization of the regular languages. |
c_yr7qypifwq1d | Monadic second-order | Summary | Monadic_second-order_logic | Second-order logic allows quantification over predicates. However, MSO is the fragment in which second-order quantification is limited to monadic predicates (predicates having a single argument). This is often described as quantification over "sets" because monadic predicates are equivalent in expressive power to sets (the set of elements for which the predicate is true). |
c_n8aroljf0kw9 | Monoidal t-norm logic | Summary | Monoidal_t-norm_logic | In mathematical logic, monoidal t-norm based logic (or shortly MTL), the logic of left-continuous t-norms, is one of the t-norm fuzzy logics. It belongs to the broader class of substructural logics, or logics of residuated lattices; it extends the logic of commutative bounded integral residuated lattices (known as Höhle's monoidal logic, Ono's FLew, or intuitionistic logic without contraction) by the axiom of prelinearity. |
c_139uh0kuvdnk | Extension by definitions | Summary | Definitional_extension | In mathematical logic, more specifically in the proof theory of first-order theories, extensions by definitions formalize the introduction of new symbols by means of a definition. For example, it is common in naive set theory to introduce a symbol ∅ {\displaystyle \emptyset } for the set that has no member. In the formal setting of first-order theories, this can be done by adding to the theory a new constant ∅ {\displaystyle \emptyset } and the new axiom ∀ x ( x ∉ ∅ ) {\displaystyle \forall x(x\notin \emptyset )} , meaning "for all x, x is not a member of ∅ {\displaystyle \emptyset } ". It can then be proved that doing so adds essentially nothing to the old theory, as should be expected from a definition. More precisely, the new theory is a conservative extension of the old one. |
c_o7ax95anzhqt | Positive formula | Summary | Positive_formula | In mathematical logic, positive set theory is the name for a class of alternative set theories in which the axiom of comprehension holds for at least the positive formulas ϕ {\displaystyle \phi } (the smallest class of formulas containing atomic membership and equality formulas and closed under conjunction, disjunction, existential and universal quantification). Typically, the motivation for these theories is topological: the sets are the classes which are closed under a certain topology. The closure conditions for the various constructions allowed in building positive formulas are readily motivated (and one can further justify the use of universal quantifiers bounded in sets to get generalized positive comprehension): the justification of the existential quantifier seems to require that the topology be compact. |
c_sqeh945pv78m | Predicate functor logic | Summary | Predicate_functor_logic | In mathematical logic, predicate functor logic (PFL) is one of several ways to express first-order logic (also known as predicate logic) by purely algebraic means, i.e., without quantified variables. PFL employs a small number of algebraic devices called predicate functors (or predicate modifiers) that operate on terms to yield terms. PFL is mostly the invention of the logician and philosopher Willard Quine. |
c_he3oe69w10c7 | Projective determinacy | Summary | Projective_determinacy | In mathematical logic, projective determinacy is the special case of the axiom of determinacy applying only to projective sets. The axiom of projective determinacy, abbreviated PD, states that for any two-player infinite game of perfect information of length ω in which the players play natural numbers, if the victory set (for either player, since the projective sets are closed under complementation) is projective, then one player or the other has a winning strategy. The axiom is not a theorem of ZFC (assuming ZFC is consistent), but unlike the full axiom of determinacy (AD), which contradicts the axiom of choice, it is not known to be inconsistent with ZFC. |
c_1h340761ij9p | Projective determinacy | Summary | Projective_determinacy | PD follows from certain large cardinal axioms, such as the existence of infinitely many Woodin cardinals. PD implies that all projective sets are Lebesgue measurable (in fact, universally measurable) and have the perfect set property and the property of Baire. It also implies that every projective binary relation may be uniformized by a projective set. |
c_2kenefz5ck0m | RecycleUnits | Summary | RecycleUnits | In mathematical logic, proof compression by RecycleUnits is a method for compressing propositional logic resolution proofs. Its main idea is to make use of intermediate (e.g. non input) proof results being unit clauses, i.e. clauses containing only one literal. Certain proof nodes can be replaced with the nodes representing these unit clauses. After this operation the obtained graph is transformed into a valid proof. The output proof is shorter than the original while being equivalent or stronger. |
c_rvhceqq7pbh5 | Resolution proof compression by splitting | Summary | Resolution_proof_compression_by_splitting | In mathematical logic, proof compression by splitting is an algorithm that operates as a post-process on resolution proofs. It was proposed by Scott Cotton in his paper "Two Techniques for Minimizing Resolution Proof".The Splitting algorithm is based on the following observation: Given a proof of unsatisfiability π {\displaystyle \pi } and a variable x {\displaystyle x} , it is easy to re-arrange (split) the proof in a proof of x {\displaystyle x} and a proof of ¬ x {\displaystyle \neg x} and the recombination of these two proofs (by an additional resolution step) may result in a proof smaller than the original. Note that applying Splitting in a proof π {\displaystyle \pi } using a variable x {\displaystyle x} does not invalidates a latter application of the algorithm using a differente variable y {\displaystyle y} . Actually, the method proposed by Cotton generates a sequence of proofs π 1 π 2 … {\displaystyle \pi _{1}\pi _{2}\ldots } , where each proof π i + 1 {\displaystyle \pi _{i+1}} is the result of applying Splitting to π i {\displaystyle \pi _{i}} . |
c_s3ui364xm1sv | Resolution proof compression by splitting | Summary | Resolution_proof_compression_by_splitting | During the construction of the sequence, if a proof π j {\displaystyle \pi _{j}} happens to be too large, π j + 1 {\displaystyle \pi _{j+1}} is set to be the smallest proof in { π 1 , π 2 , … , π j } {\displaystyle \{\pi _{1},\pi _{2},\ldots ,\pi _{j}\}} . For achieving a better compression/time ratio, a heuristic for variable selection is desirable. For this purpose, Cotton defines the "additivity" of a resolution step (with antecedents p {\displaystyle p} and n {\displaystyle n} and resolvent r {\displaystyle r} ): add ( r ) := max ( | r | − max ( | p | , | n | ) , 0 ) {\displaystyle \operatorname {add} (r):=\max(|r|-\max(|p|,|n|),0)} Then, for each variable v {\displaystyle v} , a score is calculated summing the additivity of all the resolution steps in π {\displaystyle \pi } with pivot v {\displaystyle v} together with the number of these resolution steps. |
c_598dmyx185g4 | Well-formed formulas | Summary | Well-formed_formulas | In mathematical logic, propositional logic and predicate logic, a well-formed formula, abbreviated WFF or wff, often simply formula, is a finite sequence of symbols from a given alphabet that is part of a formal language. A formal language can be identified with the set of formulas in the language. A formula is a syntactic object that can be given a semantic meaning by means of an interpretation. Two key uses of formulas are in propositional logic and predicate logic. |
c_3rvdl4exqwxu | Realizability | Summary | Realizability | In mathematical logic, realizability is a collection of methods in proof theory used to study constructive proofs and extract additional information from them. Formulas from a formal theory are "realized" by objects, known as "realizers", in a way that knowledge of the realizer gives knowledge about the truth of the formula. There are many variations of realizability; exactly which class of formulas is studied and which objects are realizers differ from one variation to another. Realizability can be seen as a formalization of the BHK interpretation of intuitionistic logic; in realizability the notion of "proof" (which is left undefined in the BHK interpretation) is replaced with a formal notion of "realizer". |
c_za5kdc65a6fy | Realizability | Summary | Realizability | Most variants of realizability begin with a theorem that any statement that is provable in the formal system being studied is realizable. The realizer, however, usually gives more information about the formula than a formal proof would directly provide. Beyond giving insight into intuitionistic provability, realizability can be applied to prove the disjunction and existence properties for intuitionistic theories and to extract programs from proofs, as in proof mining. It is also related to topos theory via realizability topoi. |
c_kp1h63yhaqau | Second-order arithmetic | Summary | Second-order_arithmetic | In mathematical logic, second-order arithmetic is a collection of axiomatic systems that formalize the natural numbers and their subsets. It is an alternative to axiomatic set theory as a foundation for much, but not all, of mathematics. A precursor to second-order arithmetic that involves third-order parameters was introduced by David Hilbert and Paul Bernays in their book Grundlagen der Mathematik. |
c_y17h15n0zzrt | Second-order arithmetic | Summary | Second-order_arithmetic | The standard axiomatization of second-order arithmetic is denoted by Z2. Second-order arithmetic includes, but is significantly stronger than, its first-order counterpart Peano arithmetic. Unlike Peano arithmetic, second-order arithmetic allows quantification over sets of natural numbers as well as numbers themselves. |
c_3cdhzurwj3kg | Second-order arithmetic | Summary | Second-order_arithmetic | Because real numbers can be represented as (infinite) sets of natural numbers in well-known ways, and because second-order arithmetic allows quantification over such sets, it is possible to formalize the real numbers in second-order arithmetic. For this reason, second-order arithmetic is sometimes called "analysis".Second-order arithmetic can also be seen as a weak version of set theory in which every element is either a natural number or a set of natural numbers. Although it is much weaker than Zermelo–Fraenkel set theory, second-order arithmetic can prove essentially all of the results of classical mathematics expressible in its language. |
c_vwh8o3zntpqc | Second-order arithmetic | Summary | Second-order_arithmetic | A subsystem of second-order arithmetic is a theory in the language of second-order arithmetic each axiom of which is a theorem of full second-order arithmetic (Z2). Such subsystems are essential to reverse mathematics, a research program investigating how much of classical mathematics can be derived in certain weak subsystems of varying strength. Much of core mathematics can be formalized in these weak subsystems, some of which are defined below. Reverse mathematics also clarifies the extent and manner in which classical mathematics is nonconstructive. |
c_pnaaqoykxcur | Sequent calculus | Summary | Sequent_calculus | In mathematical logic, sequent calculus is a style of formal logical argumentation in which every line of a proof is a conditional tautology (called a sequent by Gerhard Gentzen) instead of an unconditional tautology. Each conditional tautology is inferred from other conditional tautologies on earlier lines in a formal argument according to rules and procedures of inference, giving a better approximation to the natural style of deduction used by mathematicians than to David Hilbert's earlier style of formal logic, in which every line was an unconditional tautology. More subtle distinctions may exist; for example, propositions may implicitly depend upon non-logical axioms. In that case, sequents signify conditional theorems in a first-order language rather than conditional tautologies. |
c_ipx6cwznrosw | Sequent calculus | Summary | Sequent_calculus | Sequent calculus is one of several extant styles of proof calculus for expressing line-by-line logical arguments. Hilbert style. Every line is an unconditional tautology (or theorem). |
c_6hgwl26mmcym | Sequent calculus | Summary | Sequent_calculus | Gentzen style. Every line is a conditional tautology (or theorem) with zero or more conditions on the left. |
c_lr9fdafo3beg | Sequent calculus | Summary | Sequent_calculus | Natural deduction. Every (conditional) line has exactly one asserted proposition on the right. Sequent calculus. |
c_jci5v4ly3ysi | Sequent calculus | Summary | Sequent_calculus | Every (conditional) line has zero or more asserted propositions on the right.In other words, natural deduction and sequent calculus systems are particular distinct kinds of Gentzen-style systems. Hilbert-style systems typically have a very small number of inference rules, relying more on sets of axioms. Gentzen-style systems typically have very few axioms, if any, relying more on sets of rules. |
c_ipuyxhnjkrz0 | Sequent calculus | Summary | Sequent_calculus | Gentzen-style systems have significant practical and theoretical advantages compared to Hilbert-style systems. For example, both natural deduction and sequent calculus systems facilitate the elimination and introduction of universal and existential quantifiers so that unquantified logical expressions can be manipulated according to the much simpler rules of propositional calculus. In a typical argument, quantifiers are eliminated, then propositional calculus is applied to unquantified expressions (which typically contain free variables), and then the quantifiers are reintroduced. |
c_66ps7mymrjwz | Sequent calculus | Summary | Sequent_calculus | This very much parallels the way in which mathematical proofs are carried out in practice by mathematicians. Predicate calculus proofs are generally much easier to discover with this approach, and are often shorter. Natural deduction systems are more suited to practical theorem-proving. Sequent calculus systems are more suited to theoretical analysis. |
c_arb6475fzq1u | Computable real function | Summary | Computable_real_function | In mathematical logic, specifically computability theory, a function f: R → R {\displaystyle f\colon \mathbb {R} \to \mathbb {R} } is sequentially computable if, for every computable sequence { x i } i = 1 ∞ {\displaystyle \{x_{i}\}_{i=1}^{\infty }} of real numbers, the sequence { f ( x i ) } i = 1 ∞ {\displaystyle \{f(x_{i})\}_{i=1}^{\infty }} is also computable. A function f: R → R {\displaystyle f\colon \mathbb {R} \to \mathbb {R} } is effectively uniformly continuous if there exists a recursive function d: N → N {\displaystyle d\colon \mathbb {N} \to \mathbb {N} } such that, if | x − y | < 1 d ( n ) {\displaystyle |x-y|<{1 \over d(n)}} then | f ( x ) − f ( y ) | < 1 n {\displaystyle |f(x)-f(y)|<{1 \over n}} A real function is computable if it is both sequentially computable and effectively uniformly continuous,These definitions can be generalized to functions of more than one variable or functions only defined on a subset of R n . {\displaystyle \mathbb {R} ^{n}.} The generalizations of the latter two need not be restated. |