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In mathematical logic (a subtopic within the field of formal logic), two formulae are equisatisfiable if the first formula is satisfiable whenever the second is and vice versa; in other words, either both formulae are satisfiable or both are not. Equisatisfiable formulae may disagree, however, for a particular choice o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical logic and computer science the symbol ⊢ ( ⊢ {\displaystyle \vdash } ) has taken the name turnstile because of its resemblance to a typical turnstile if viewed from above. It is also referred to as tee and is often read as "yields", "proves", "satisfies" or "entails".
In mathematical logic and computer ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
However, not every total recursive function is a primitive recursive function—the most famous example is the Ackermann function. Other equivalent classes of functions are the functions of lambda calculus and the functions that can be computed by Markov algorithms. The subset of all total recursive functions with values... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical logic and computer science, some type theories and type systems include a top type that is commonly denoted with top or the symbol ⊤. The top type is sometimes called also universal type, or universal supertype as all other types in the type system of interest are subtypes of it, and in most cases, it c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
If V {\displaystyle V} is a set of strings, then V ∗ {\displaystyle V^{*}} is defined as the smallest superset of V {\displaystyle V} that contains the empty string ε {\displaystyle \varepsilon } and is closed under the string concatenation operation. If V {\displaystyle V} is a set of symbols or characters, then V ∗ {... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical logic and computer science, the lambda-mu calculus is an extension of the lambda calculus introduced by M. Parigot. It introduces two new operators: the μ operator (which is completely different both from the μ operator found in computability theory and from the μ operator of modal μ-calculus) and the b... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical logic and descriptive set theory, the analytical hierarchy is an extension of the arithmetical hierarchy. The analytical hierarchy of formulas includes formulas in the language of second-order arithmetic, which can have quantifiers over both the set of natural numbers, N {\displaystyle \mathbb {N} } , a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical logic and logic programming, a Horn clause is a logical formula of a particular rule-like form which gives it useful properties for use in logic programming, formal specification, and model theory. Horn clauses are named for the logician Alfred Horn, who first pointed out their significance in 1951.
In... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
295). Skolem's paradox is that every countable axiomatisation of set theory in first-order logic, if it is consistent, has a model that is countable. This appears contradictory because it is possible to prove, from those same axioms, a sentence that intuitively says (or that precisely says in the standard model of the ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical logic and set theory, an ordinal collapsing function (or projection function) is a technique for defining (notations for) certain recursive large countable ordinals, whose principle is to give names to certain ordinals much larger than the one being defined, perhaps even large cardinals (though they can... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical logic and set theory, an ordinal notation is a partial function mapping the set of all finite sequences of symbols, themselves members of a finite alphabet, to a countable set of ordinals. A Gödel numbering is a function mapping the set of well-formed formulae (a finite sequence of symbols on which the ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical logic and theoretical computer science, a register machine is a generic class of abstract machines used in a manner similar to a Turing machine. All the models are Turing equivalent.
In mathematical logic and theoretical computer science, an abstract rewriting system (also (abstract) reduction system o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical logic and type theory, the λ-cube (also written lambda cube) is a framework introduced by Henk Barendregt to investigate the different dimensions in which the calculus of constructions is a generalization of the simply typed λ-calculus. Each dimension of the cube corresponds to a new kind of dependency ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical logic the theory of pure equality is a first-order theory. It has a signature consisting of only the equality relation symbol, and includes no non-logical axioms at all.This theory is consistent but incomplete, as a non-empty set with the usual equality relation provides an interpretation making certain... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical logic, Craig's theorem (also known as Craig's trick) states that any recursively enumerable set of well-formed formulas of a first-order language is (primitively) recursively axiomatizable. This result is not related to the well-known Craig interpolation theorem, although both results are named after th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Frege's PC and standard PC share two common axioms: THEN-1 and THEN-2. Notice that axioms THEN-1 through THEN-3 only make use of (and define) the implication operator, whereas axioms FRG-1 through FRG-3 define the negation operator.
The following theorems will aim to find the remaining nine axioms of standard PC withi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Kirby and Paris introduced a graph-theoretic hydra game with behavior similar to that of Goodstein sequences: the "Hydra" (named for the mythological multi-headed Hydra of Lerna) is a rooted tree, and a move consists of cutting off one of its "heads" (a branch of the tree), to which the hydra responds by growing a fini... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical logic, Lindenbaum's lemma, named after Adolf Lindenbaum, states that any consistent theory of predicate logic can be extended to a complete consistent theory. The lemma is a special case of the ultrafilter lemma for Boolean algebras, applied to the Lindenbaum algebra of a theory.
In mathematical logic,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical logic, Löb's theorem states that in Peano arithmetic (PA) (or any formal system including PA), for any formula P, if it is provable in PA that "if P is provable in PA then P is true", then P is provable in PA. If Prov(P) means that the formula P is provable, we may express this more formally as If P A ⊢... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
New Foundations has a universal set, so it is a non-well-founded set theory. That is to say, it is an axiomatic set theory that allows infinite descending chains of membership, such as ... xn ∈ xn-1 ∈ ... ∈ x2 ∈ x1. It avoids Russell's paradox by permitting only stratifiable formulas to be defined using the axiom schem... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical logic, Russell's paradox (also known as Russell's antinomy) is a set-theoretic paradox published by the British philosopher and mathematician Bertrand Russell in 1901. Russell's paradox shows that every set theory that contains an unrestricted comprehension principle leads to contradictions. The paradox... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In symbols: Let R = { x ∣ x ∉ x } , then R ∈ R ⟺ R ∉ R {\displaystyle {\text{Let }}R=\{x\mid x\not \in x\}{\text{, then }}R\in R\iff R\not \in R} Russell also showed that a version of the paradox could be derived in the axiomatic system constructed by the German philosopher and mathematician Gottlob Frege, hence underm... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Unlike Peano arithmetic, Skolem arithmetic is a decidable theory. This means it is possible to effectively determine, for any sentence in the language of Skolem arithmetic, whether that sentence is provable from the axioms of Skolem arithmetic. The asymptotic running-time computational complexity of this decision probl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical logic, a Boolean-valued model is a generalization of the ordinary Tarskian notion of structure from model theory. In a Boolean-valued model, the truth values of propositions are not limited to "true" and "false", but instead take values in some fixed complete Boolean algebra. Boolean-valued models were ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
The importance of Herbrand interpretations is that, if any interpretation satisfies a given set of clauses S then there is a Herbrand interpretation that satisfies them. Moreover, Herbrand's theorem states that if S is unsatisfiable then there is a finite unsatisfiable set of ground instances from the Herbrand universe... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical logic, a conservative extension is a supertheory of a theory which is often convenient for proving theorems, but proves no new theorems about the language of the original theory. Similarly, a non-conservative extension is a supertheory which is not conservative, and can prove more theorems than the orig... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
This can also be seen as a methodology for writing and structuring large theories: start with a theory, T 0 {\displaystyle T_{0}} , that is known (or assumed) to be consistent, and successively build conservative extensions T 1 {\displaystyle T_{1}} , T 2 {\displaystyle T_{2}} , ... of it. Recently, conservative extens... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In the special case where Δ {\displaystyle \Delta } is the empty set, the deduction theorem claim can be more compactly written as: A ⊢ B {\displaystyle A\vdash B} implies ⊢ A → B {\displaystyle \vdash A\to B} . The deduction theorem for predicate logic is similar, but comes with some extra constraints (that would for ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical logic, a first-order Gödel logic is a member of a family of finite- or infinite-valued logics in which the sets of truth values V are closed subsets of the unit interval containing both 0 and 1. Different such sets V in general determine different Gödel logics. The concept is named after Kurt Gödel. ==... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Tarski's exponential function problem concerns the extension of this theory to another primitive operation, the exponential function. It is an open problem whether this theory is decidable, but if Schanuel's conjecture holds then the decidability of this theory would follow. In contrast, the extension of the theory of ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical logic, a formal calculation, or formal operation, is a calculation that is systematic but without a rigorous justification. It involves manipulating symbols in an expression using a generic substitution without proving that the necessary conditions hold. Essentially, it involves the form of an expressio... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
204): A ⇒ B → ¬ A ∨ B A ⇔ B → ( ¬ A ∨ B ) ∧ ( A ∨ ¬ B ) ¬ ( A ∨ B ) → ¬ A ∧ ¬ B ¬ ( A ∧ B ) → ¬ A ∨ ¬ B ¬ ¬ A → A ¬ ∃ x A → ∀ x ¬ A ¬ ∀ x A → ∃ x ¬ A {\displaystyle {\begin{aligned}A\Rightarrow B&~\to ~\lnot A\lor B\\A\Leftrightarrow B&~\to ~(\lnot A\lor B)\land (A\lor \lnot B)\\\lnot (A\lor B)&~\to ~\lnot A\land \lnot... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Repeated application of distributivity may exponentially increase the size of a formula. In the classical propositional logic, transformation to negation normal form does not impact computational properties: the satisfiability problem continues to be NP-complete, and the validity problem continues to be co-NP-complete.... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In set theory, the issue of which properties of sets are absolute is well studied. The Shoenfield absoluteness theorem, due to Joseph Shoenfield (1961), establishes the absoluteness of a large class of formulas between a model of set theory and its constructible universe, with important methodological consequences. The... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
While this allows non-standard interpretations of symbols such as + {\displaystyle +} , one can restrict their meaning by providing additional axioms. The satisfiability modulo theories problem considers satisfiability of a formula with respect to a formal theory, which is a (finite or infinite) set of axioms.
Satisfi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
The problem of determining whether a formula in propositional logic is satisfiable is decidable, and is known as the Boolean satisfiability problem, or SAT. In general, the problem of determining whether a sentence of first-order logic is satisfiable is not decidable. In universal algebra, equational theory, and automa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
An important problem in computational logic is to determine fragments of well-known logics such as first-order logic that are as expressive as possible yet are decidable or more strongly have low computational complexity. The field of descriptive complexity theory aims at establishing a link between logics and computat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Judgments are used in formalizing deduction systems: a logical axiom expresses a judgment, premises of a rule of inference are formed as a sequence of judgments, and their conclusion is a judgment as well (thus, hypotheses and conclusions of proofs are judgments). A characteristic feature of the variants of Hilbert-sty... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In logics with double negation elimination (where ¬ ¬ x ≡ x {\displaystyle \lnot \lnot x\equiv x} ) the complementary literal or complement of a literal l {\displaystyle l} can be defined as the literal corresponding to the negation of l {\displaystyle l} . We can write l ¯ {\displaystyle {\bar {l}}} to denote the comp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical logic, a logic L has the finite model property (fmp for short) if any non-theorem of L is falsified by some finite model of L. Another way of putting this is to say that L has the fmp if for every formula A of L, A is an L-theorem if and only if A is a theorem of the theory of finite models of L. If L i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical logic, a predicate variable is a predicate letter which functions as a "placeholder" for a relation (between terms), but which has not been specifically assigned any particular relation (or meaning). Common symbols for denoting predicate variables include capital roman letters such as P {\displaystyle P... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical logic, a redundant proof is a proof that has a subset that is a shorter proof of the same result. In other words, a proof is redundant if it has more proof steps than are actually necessary to prove the result. Formally, a proof ψ {\displaystyle \psi } of κ {\displaystyle \kappa } is considered redundan... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Thus the predicate "is not satisfied" attributes something to the concept "is a Bosnian philosopher", and is thus a second-level predicate. This idea is the basis of Frege's theory of number. == References ==
In mathematical logic, a sentence (or closed formula) of a predicate logic is a Boolean-valued well-formed for... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
To properly evaluate the truth (or falsehood) of a sentence, one must make reference to an interpretation of the theory. For first-order theories, interpretations are commonly called structures. Given a structure or interpretation, a sentence will have a fixed truth value. A theory is satisfiable when it is possible to... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical logic, a set T {\displaystyle {\mathcal {T}}} of logical formulae is deductively closed if it contains every formula φ {\displaystyle \varphi } that can be logically deduced from T {\displaystyle {\mathcal {T}}} , formally: if T ⊢ φ {\displaystyle {\mathcal {T}}\vdash \varphi } always implies φ ∈ T {\di... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In logic, a formula is satisfiable if it is true under at least one interpretation, and thus a tautology is a formula whose negation is unsatisfiable. In other words, it cannot be false. It cannot be untrue.
Unsatisfiable statements, both through negation and affirmation, are known formally as contradictions. A formul... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
The tee symbol ⊤ {\displaystyle \top } is sometimes used to denote an arbitrary tautology, with the dual symbol ⊥ {\displaystyle \bot } (falsum) representing an arbitrary contradiction; in any symbolism, a tautology may be substituted for the truth value "true", as symbolized, for instance, by "1".Tautologies are a key... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical logic, a term denotes a mathematical object while a formula denotes a mathematical fact. In particular, terms appear as components of a formula. This is analogous to natural language, where a noun phrase refers to an object and a whole sentence refers to a fact. A first-order term is recursively constru... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical logic, a theory can be extended with new constants or function names under certain conditions with assurance that the extension will introduce no contradiction. Extension by definitions is perhaps the best-known approach, but it requires unique existence of an object with the desired property. Addition ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Let T 1 {\displaystyle T_{1}} be a theory obtained from T {\displaystyle T} by extending its language with new constants a 1 , … , a m {\displaystyle a_{1},\ldots ,a_{m}} and adding a new axiom φ ( a 1 , … , a m ) {\displaystyle \varphi (a_{1},\ldots ,a_{m})} .Then T 1 {\displaystyle T_{1}} is a conservative extension ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Then T 1 {\displaystyle T_{1}} is a conservative extension of T {\displaystyle T} , i.e. the theories T {\displaystyle T} and T 1 {\displaystyle T_{1}} prove the same theorems not involving the functional symbol f {\displaystyle f} ). Shoenfield states the theorem in the form for a new function name, and constants are ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In model theory, the notion of a categorical theory is refined with respect to cardinality. A theory is κ-categorical (or categorical in κ) if it has exactly one model of cardinality κ up to isomorphism. Morley's categoricity theorem is a theorem of Michael D. Morley (1965) stating that if a first-order theory in a cou... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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\i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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 me... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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.
In mathematical logic, abstract model theory is a generalization of model theory ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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.
In mathematical logic, an arithmetical set (or arithmetic set)... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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. Compou... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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 impor... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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 |
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 |
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 |
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 |
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 |
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 lar... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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.
In mathematical logic, interpretability is a relation between formal theories that expresses the possibility of interpreting or translating ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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 |
{\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 .
{\displaystyle x|(y\mid (x\mid z))=((z\mid y)\mid y)\m... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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 |
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 |
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 |
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 |
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 |
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 |
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.
... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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 |
Natural deduction. Every (conditional) line has exactly one asserted proposition on the right. 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-st... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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 |
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 |
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 |
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 |
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 |
The term λx. λy. λz.
x z (y z) (the S combinator), with De Bruijn indices, is λ λ λ 3 1 (2 1). The term λz. (λy.
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 r... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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 |
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 |
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 |
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).
In mathematical logic, the Paris–Harrington theorem states that a certain combinatorial principle in Ramsey th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In 1881, Charles Sanders Peirce provided an axiomatization of natural-number arithmetic. In 1888, Richard Dedekind proposed another axiomatization of natural-number arithmetic, and in 1889, Peano published a simplified version of them as a collection of axioms in his book The principles of arithmetic presented by a new... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical logic, the ancestral relation (often shortened to ancestral) of a binary relation R is its transitive closure, however defined in a different way, see below. Ancestral relations make their first appearance in Frege's Begriffsschrift. Frege later employed them in his Grundgesetze as part of his definitio... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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 |
In mathematical logic, the disjunction and existence properties are the "hallmarks" of constructive theories such as Heyting arithmetic and constructive set theories (Rathjen 2005).
In mathematical logic, the implicational propositional calculus is a version of classical propositional calculus which uses only one conn... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
( 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 |
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 |
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 |
In mathematical logic, true arithmetic is the set of all true first-order statements about the arithmetic of natural numbers. This is the theory associated with the standard model of the Peano axioms in the language of the first-order Peano axioms. True arithmetic is occasionally called Skolem arithmetic, though this t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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 |
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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