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wiki_600_chunk_0 | 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 whi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_601_chunk_0 | 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_602_chunk_0 | Function letter | 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 genera... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_603_chunk_0 | 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 co... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_604_chunk_0 | O-minimal theory | 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_605_chunk_0 | 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 requi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_606_chunk_0 | 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_607_chunk_0 | 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 structu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_608_chunk_0 | Fuzzy concept | 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. Co... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_609_chunk_0 | Fuzzy concept | 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_610_chunk_0 | Fuzzy concept | 8. Specifying relevant conditions to which the concept applies, as a procedure (computer programming, formal concept analysis). 9. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_611_chunk_0 | Fuzzy concept | 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 pri... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_612_chunk_0 | Fuzzy concept | 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). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_613_chunk_0 | Fuzzy concept | 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_614_chunk_0 | Fuzzy concept | 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). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_615_chunk_0 | Fuzzy concept | 19. Specifying a series of logical operators or inferential system which captures all or most cases to which the concept applies (algorithm). 20. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_616_chunk_0 | Fuzzy concept | 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 t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_617_chunk_0 | 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_618_chunk_0 | 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_619_chunk_0 | 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_620_chunk_0 | Unstable fixed point | 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_621_chunk_0 | Least 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 stud... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_622_chunk_0 | 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 red... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_623_chunk_0 | Equiconsistency | 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 th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_624_chunk_0 | Formation rule | 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, ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_625_chunk_0 | 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_626_chunk_0 | Internal set | 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 ra... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_627_chunk_0 | 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_628_chunk_0 | 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 equivale... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_629_chunk_0 | 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 co... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_630_chunk_0 | 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 , {\... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_631_chunk_0 | 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 . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_632_chunk_0 | 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_633_chunk_0 | 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 ) ) ) ) = ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_634_chunk_0 | 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_635_chunk_0 | 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 natur... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_636_chunk_0 | Monadic second-order | 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 f... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_637_chunk_0 | 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 Höhl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_638_chunk_0 | Extension by definitions | 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 form... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_639_chunk_0 | 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_640_chunk_0 | 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 s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_641_chunk_0 | 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 represe... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_642_chunk_0 | 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 {\d... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_643_chunk_0 | 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_644_chunk_0 | 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 synta... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_645_chunk_0 | 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 fo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_646_chunk_0 | 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, realizabili... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_647_chunk_0 | 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 in... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_648_chunk_0 | 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".Sec... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_649_chunk_0 | 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 sub... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_650_chunk_0 | 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_651_chunk_0 | 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). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_652_chunk_0 | Sequent calculus | Gentzen style. Every line is a conditional tautology (or theorem) with zero or more conditions on the left. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_653_chunk_0 | 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 mor... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_654_chunk_0 | 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_655_chunk_0 | Computable real function | A suitable generalization of the first definition is: Let D {\displaystyle D} be a subset of R n . {\displaystyle \mathbb {R} ^{n}.} | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_656_chunk_0 | Computable real function | A function f: D → R {\displaystyle f\colon D\to \mathbb {R} } is sequentially computable if, for every n {\displaystyle n} -tuplet ( { x i 1 } i = 1 ∞ , … { x i n } i = 1 ∞ ) {\displaystyle \left(\{x_{i\,1}\}_{i=1}^{\infty },\ldots \{x_{i\,n}\}_{i=1}^{\infty }\right)} of computable sequences of real numbers such that (... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_657_chunk_0 | Fraïssé limit | In mathematical logic, specifically in the discipline of model theory, the Fraïssé limit (also called the Fraïssé construction or Fraïssé amalgamation) is a method used to construct (infinite) mathematical structures from their (finite) substructures. It is a special example of the more general concept of a direct limi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_658_chunk_0 | Fraïssé limit | Given a class K {\displaystyle \mathbf {K} } of finite relational structures, if K {\displaystyle \mathbf {K} } satisfies certain properties (described below), then there exists a unique countable structure Flim ( K ) {\displaystyle \operatorname {Flim} (\mathbf {K} )} , called the Fraïssé limit of K {\displaystyle \... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_659_chunk_0 | Stratified formula | In mathematical logic, stratification is any consistent assignment of numbers to predicate symbols guaranteeing that a unique formal interpretation of a logical theory exists. Specifically, we say that a set of clauses of the form Q 1 ∧ ⋯ ∧ Q n ∧ ¬ Q n + 1 ∧ ⋯ ∧ ¬ Q n + m → P {\displaystyle Q_{1}\wedge \dots \wedge Q_{... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_660_chunk_0 | Display logic | In mathematical logic, structural proof theory is the subdiscipline of proof theory that studies proof calculi that support a notion of analytic proof, a kind of proof whose semantic properties are exposed. When all the theorems of a logic formalised in a structural proof theory have analytic proofs, then the proof the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_661_chunk_0 | Integer-valued function | In mathematical logic, such concepts as primitive recursive functions and μ-recursive functions represent integer-valued functions of several natural variables or, in other words, functions on Nn. Gödel numbering, defined on well-formed formulae of some formal language, is a natural-valued function. Computability theor... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_662_chunk_0 | Barwise compactness theorem | In mathematical logic, the Barwise compactness theorem, named after Jon Barwise, is a generalization of the usual compactness theorem for first-order logic to a certain class of infinitary languages. It was stated and proved by Barwise in 1967. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_663_chunk_0 | Borel hierarchy | In mathematical logic, the Borel hierarchy is a stratification of the Borel algebra generated by the open subsets of a Polish space; elements of this algebra are called Borel sets. Each Borel set is assigned a unique countable ordinal number called the rank of the Borel set. The Borel hierarchy is of particular interes... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_664_chunk_0 | Cantor–Dedekind axiom | In mathematical logic, the Cantor–Dedekind axiom is the thesis that the real numbers are order-isomorphic to the linear continuum of geometry. In other words, the axiom states that there is a one-to-one correspondence between real numbers and points on a line. This axiom became a theorem proved by Emil Artin in his boo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_665_chunk_0 | Cantor–Dedekind axiom | Analytic geometry was developped from the Cartesian coordinate system introduced by René Descartes. It implicitly assumed this axiom by blending the distinct concepts of real numbers and points on a line, sometimes referred to as the real number line. Artin's proof, not only makes this blend explicity, but also that an... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_666_chunk_0 | De Bruijn index | In mathematical logic, the De Bruijn index is a tool invented by the Dutch mathematician Nicolaas Govert de Bruijn for representing terms of lambda calculus without naming the bound variables. Terms written using these indices are invariant with respect to α-conversion, so the check for α-equivalence is the same as tha... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_667_chunk_0 | De Bruijn index | y (λx. x)) (λx. z x) is λ (λ 1 (λ 1)) (λ 2 1). See the following illustration, where the binders are coloured and the references are shown with arrows. De Bruijn indices are commonly used in higher-order reasoning systems such as automated theorem provers and logic programming systems. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_668_chunk_0 | De Bruijn notation | In mathematical logic, the De Bruijn notation is a syntax for terms in the λ calculus invented by the Dutch mathematician Nicolaas Govert de Bruijn. It can be seen as a reversal of the usual syntax for the λ calculus where the argument in an application is placed next to its corresponding binder in the function instead... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_669_chunk_0 | Friedberg–Muchnik theorem | In mathematical logic, the Friedberg–Muchnik theorem is a theorem about Turing reductions that was proven independently by Albert Muchnik and Richard Friedberg in the middle of the 1950s. It is a more general view of the Kleene–Post theorem. The Kleene–Post theorem states that there exist incomparable languages A and B... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_670_chunk_0 | Hilbert–Bernays provability conditions | In mathematical logic, the Hilbert–Bernays provability conditions, named after David Hilbert and Paul Bernays, are a set of requirements for formalized provability predicates in formal theories of arithmetic (Smith 2007:224). These conditions are used in many proofs of Kurt Gödel's second incompleteness theorem. They a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_671_chunk_0 | Kanamori–McAloon theorem | In mathematical logic, the Kanamori–McAloon theorem, due to Kanamori & McAloon (1987), gives an example of an incompleteness in Peano arithmetic, similar to that of the Paris–Harrington theorem. They showed that a certain finitistic theorem in Ramsey theory is not provable in Peano arithmetic (PA). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_672_chunk_0 | Lindenbaum algebra | In mathematical logic, the Lindenbaum–Tarski algebra (or Lindenbaum algebra) of a logical theory T consists of the equivalence classes of sentences of the theory (i.e., the quotient, under the equivalence relation ~ defined such that p ~ q exactly when p and q are provably equivalent in T). That is, two sentences are e... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_673_chunk_0 | Lindenbaum algebra | The algebra is named for logicians Adolf Lindenbaum and Alfred Tarski. Starting in the academic year 1926-1927, Lindenbaum pioneered his method in Jan Łukasiewicz's mathematical logic seminar, and the method was popularized and generalized in subsequent decades through work by Tarski. The Lindenbaum–Tarski algebra is c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_674_chunk_0 | Löwenheim–Skolem theorem | In mathematical logic, the Löwenheim–Skolem theorem is a theorem on the existence and cardinality of models, named after Leopold Löwenheim and Thoralf Skolem. The precise formulation is given below. It implies that if a countable first-order theory has an infinite model, then for every infinite cardinal number κ it has... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_675_chunk_0 | Löwenheim–Skolem theorem | As a consequence, first-order theories are unable to control the cardinality of their infinite models. The (downward) Löwenheim–Skolem theorem is one of the two key properties, along with the compactness theorem, that are used in Lindström's theorem to characterize first-order logic. In general, the Löwenheim–Skolem th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_676_chunk_0 | Mostowski collapse | In mathematical logic, the Mostowski collapse lemma, also known as the Shepherdson–Mostowski collapse, is a theorem of set theory introduced by Andrzej Mostowski (1949, theorem 3) and John Shepherdson (1953). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_677_chunk_0 | Paris–Harrington theorem | In mathematical logic, the Paris–Harrington theorem states that a certain combinatorial principle in Ramsey theory, namely the strengthened finite Ramsey theorem, which is expressible in Peano arithmetic, is not provable in this system. The combinatorial principle is however provable in slightly stronger systems. This ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_678_chunk_0 | Peano axiom | In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th-century Italian mathematician Giuseppe Peano. These axioms have been used nearly unchanged in a number of metamathematical investigations, including research... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_679_chunk_0 | Peano axiom | The first axiom asserts the existence of at least one member of the set of natural numbers. The next four are general statements about equality; in modern treatments these are often not taken as part of the Peano axioms, but rather as axioms of the "underlying logic". The next three axioms are first-order statements ab... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_680_chunk_0 | Scott–Curry theorem | In mathematical logic, the Scott–Curry theorem is a result in lambda calculus stating that if two non-empty sets of lambda terms A and B are closed under beta-convertibility then they are recursively inseparable. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_681_chunk_0 | Arithmetical reducibility | In mathematical logic, the arithmetical hierarchy, arithmetic hierarchy or Kleene–Mostowski hierarchy (after mathematicians Stephen Cole Kleene and Andrzej Mostowski) classifies certain sets based on the complexity of formulas that define them. Any set that receives a classification is called arithmetical. The arithmet... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_682_chunk_0 | Compactness theorem | In mathematical logic, the compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model. This theorem is an important tool in model theory, as it provides a useful (but generally not effective) method for constructing models of any set of sentences that... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_683_chunk_0 | Compactness theorem | Likewise, it is analogous to the finite intersection property characterization of compactness in topological spaces: a collection of closed sets in a compact space has a non-empty intersection if every finite subcollection has a non-empty intersection. The compactness theorem is one of the two key properties, along wit... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_684_chunk_0 | Cut rule | In mathematical logic, the cut rule is an inference rule of sequent calculus. It is a generalisation of the classical modus ponens inference rule. Its meaning is that, if a formula A appears as a conclusion in one proof and a hypothesis in another, then another proof in which the formula A does not appear can be deduce... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_685_chunk_0 | Diagonal lemma | In mathematical logic, the diagonal lemma (also known as diagonalization lemma, self-reference lemma or fixed point theorem) establishes the existence of self-referential sentences in certain formal theories of the natural numbers—specifically those theories that are strong enough to represent all computable functions.... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_686_chunk_0 | Intersection type discipline | ( x x ) {\displaystyle \lambda x.\! (x\;x)} can be assigned the type ( ( α → β ) ∩ α ) → β {\displaystyle ((\alpha \to \beta )\cap \alpha )\to \beta } in most intersection type systems, assuming for the term variable x {\displaystyle x} both the function type α → β {\displaystyle \alpha \to \beta } and the correspondin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_687_chunk_0 | Intersection type discipline | Prominent intersection type systems include the Coppo–Dezani type assignment system, the Barendregt-Coppo–Dezani type assignment system, and the essential intersection type assignment system. Most strikingly, intersection type systems are closely related to (and often exactly characterize) normalization properties of λ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_688_chunk_0 | Primitive recursive functional | In mathematical logic, the primitive recursive functionals are a generalization of primitive recursive functions into higher type theory. They consist of a collection of functions in all pure finite types. The primitive recursive functionals are important in proof theory and constructive mathematics. They are a central... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_689_chunk_0 | Quantifier rank | In mathematical logic, the quantifier rank of a formula is the depth of nesting of its quantifiers. It plays an essential role in model theory. Notice that the quantifier rank is a property of the formula itself (i.e. the expression in a language). Thus two logically equivalent formulae can have different quantifier ra... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_690_chunk_0 | Rules of passage (logic) | In mathematical logic, the rules of passage govern how quantifiers distribute over the basic logical connectives of first-order logic. The rules of passage govern the "passage" (translation) from any formula of first-order logic to the equivalent formula in prenex normal form, and vice versa. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_691_chunk_0 | Spectrum of a sentence | In mathematical logic, the spectrum of a sentence is the set of natural numbers occurring as the size of a finite model in which a given sentence is true. By a result in descriptive complexity, a set of natural numbers is a spectrum if and only if it can be recognized in non-deterministic exponential time. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_692_chunk_0 | Controversy over Cantor's theory | In mathematical logic, the theory of infinite sets was first developed by Georg Cantor. Although this work has become a thoroughly standard fixture of classical set theory, it has been criticized in several areas by mathematicians and philosophers. Cantor's theorem implies that there are sets having cardinality greater... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_693_chunk_0 | Controversy over Cantor's theory | This argument can be improved by using a definition he gave later. The resulting argument uses only five axioms of set theory. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_694_chunk_0 | Controversy over Cantor's theory | Cantor's set theory was controversial at the start, but later became largely accepted. Most modern mathematics textbooks implicitly use Cantor's views on mathematical infinity. For example, a line is generally presented as the infinite set of its points, and it is commonly taught that there are more real numbers than r... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_695_chunk_0 | Decidable sublanguages of set theory | In mathematical logic, various sublanguages of set theory are decidable. These include: Sets with Monotone, Additive, and Multiplicative Functions. Sets with restricted quantifiers. == References == | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_696_chunk_0 | Weak interpretability | In mathematical logic, weak interpretability is a notion of translation of logical theories, introduced together with interpretability by Alfred Tarski in 1953. Let T and S be formal theories. Slightly simplified, T is said to be weakly interpretable in S if, and only if, the language of T can be translated into the la... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_697_chunk_0 | Ham sandwich theorem | In mathematical measure theory, for every positive integer n the ham sandwich theorem states that given n measurable "objects" in n-dimensional Euclidean space, it is possible to divide each one of them in half (with respect to their measure, e.g. volume) with a single (n − 1)-dimensional hyperplane. This is even possi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_698_chunk_0 | Link-centric preferential attachment | In mathematical modeling of social networks, link-centric preferential attachment is a node's propensity to re-establish links to nodes it has previously been in contact with in time-varying networks. This preferential attachment model relies on nodes keeping memory of previous neighbors up to the current time. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_699_chunk_0 | Guess value | In mathematical modeling, a guess value is more commonly called a starting value or initial value. These are necessary for most optimization problems which use search algorithms, because those algorithms are mainly deterministic and iterative, and they need to start somewhere. One common type of application is nonlinea... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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