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Russell paradox

Summary

Russel's_paradox

At the end of the 1890s, Georg Cantor – considered the founder of modern set theory – had already realized that his theory would lead to a contradiction, as he told Hilbert and Richard Dedekind by letter.According to the unrestricted comprehension principle, for any sufficiently well-defined property, there is the set of all and only the objects that have that property. Let R be the set of all sets that are not members of themselves. (This set is sometimes called "the Russell set".)

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Russell paradox

Summary

Russel's_paradox

If R is not a member of itself, then its definition entails that it is a member of itself; yet, if it is a member of itself, then it is not a member of itself, since it is the set of all sets that are not members of themselves. The resulting contradiction is Russell's paradox.

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Russell paradox

Summary

Russel's_paradox

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 undermining Frege's attempt to reduce mathematics to logic and questioning the logicist programme. Two influential ways of avoiding the paradox were both proposed in 1908: Russell's own type theory and the Zermelo set theory. In particular, Zermelo's axioms restricted the unlimited comprehension principle.

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Russell paradox

Summary

Russel's_paradox

With the additional contributions of Abraham Fraenkel, Zermelo set theory developed into the now-standard Zermelo–Fraenkel set theory (commonly known as ZFC when including the axiom of choice). The main difference between Russell's and Zermelo's solution to the paradox is that Zermelo modified the axioms of set theory while maintaining a standard logical language, while Russell modified the logical language itself. The language of ZFC, with the help of Thoralf Skolem, turned out to be that of first-order logic.

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Skolem arithmetic

Summary

Skolem_arithmetic

In mathematical logic, Skolem arithmetic is the first-order theory of the natural numbers with multiplication, named in honor of Thoralf Skolem. The signature of Skolem arithmetic contains only the multiplication operation and equality, omitting the addition operation entirely. Skolem arithmetic is weaker than Peano arithmetic, which includes both addition and multiplication operations.

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Skolem arithmetic

Summary

Skolem_arithmetic

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 problem is triply exponential.

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Type in type

Summary

System_U

In mathematical logic, System U and System U− are pure type systems, i.e. special forms of a typed lambda calculus with an arbitrary number of sorts, axioms and rules (or dependencies between the sorts). They were both proved inconsistent by Jean-Yves Girard in 1972. This result led to the realization that Martin-Löf's original 1971 type theory was inconsistent as it allowed the same "Type in Type" behaviour that Girard's paradox exploits.

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Tarski's high school algebra problem

Summary

Tarski's_high_school_algebra_problem

In mathematical logic, Tarski's high school algebra problem was a question posed by Alfred Tarski. It asks whether there are identities involving addition, multiplication, and exponentiation over the positive integers that cannot be proved using eleven axioms about these operations that are taught in high-school-level mathematics. The question was solved in 1980 by Alex Wilkie, who showed that such unprovable identities do exist.

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Boolean-valued model

Summary

Boolean-valued_model

In mathematical logic, a Boolean-valued model is a generalization of the ordinary Tarskian notion of structure from model theory. In a Boolean-valued model, the truth values of propositions are not limited to "true" and "false", but instead take values in some fixed complete Boolean algebra. Boolean-valued models were introduced by Dana Scott, Robert M. Solovay, and Petr Vopěnka in the 1960s in order to help understand Paul Cohen's method of forcing. They are also related to Heyting algebra semantics in intuitionistic logic.

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Gödel encoding

Summary

Gödel_encoding

In mathematical logic, a Gödel numbering is a function that assigns to each symbol and well-formed formula of some formal language a unique natural number, called its Gödel number. The concept was developed by Kurt Gödel for the proof of his incompleteness theorems. (Gödel 1931) A Gödel numbering can be interpreted as an encoding in which a number is assigned to each symbol of a mathematical notation, after which a sequence of natural numbers can then represent a sequence of symbols. These sequences of natural numbers can again be represented by single natural numbers, facilitating their manipulation in formal theories of arithmetic. Since the publishing of Gödel's paper in 1931, the term "Gödel numbering" or "Gödel code" has been used to refer to more general assignments of natural numbers to mathematical objects.

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Herbrand interpretation

Summary

Herbrand_interpretation

In mathematical logic, a Herbrand interpretation is an interpretation in which all constants and function symbols are assigned very simple meanings. Specifically, every constant is interpreted as itself, and every function symbol is interpreted as the function that applies it. The interpretation also defines predicate symbols as denoting a subset of the relevant Herbrand base, effectively specifying which ground atoms are true in the interpretation. This allows the symbols in a set of clauses to be interpreted in a purely syntactic way, separated from any real instantiation.

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Herbrand interpretation

Summary

Herbrand_interpretation

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 defined by S. Since this set is finite, its unsatisfiability can be verified in finite time. However, there may be an infinite number of such sets to check. It is named after Jacques Herbrand.

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Hintikka set

Summary

Hintikka_set

In mathematical logic, a Hintikka set is a set of logical formulas whose elements satisfy the following properties: An atom or its conjugate can appear in the set but not both, If a formula in the set has a main operator that is of "conjuctive-type", then its two operands appear in the set, If a formula in the set has a main operator that is of "disjuntive-type", then at least one of its two operands appears in the set.The exact meaning of "conjuctive-type" and "disjunctive-type" is defined by the method of semantic tableaux. Hintikka sets arise when attempting to prove completeness of propositional logic using semantic tableaux. They are named after Jaakko Hintikka.

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Lindström quantifier

Summary

Lindström_quantifier

In mathematical logic, a Lindström quantifier is a generalized polyadic quantifier. Lindström quantifiers generalize first-order quantifiers, such as the existential quantifier, the universal quantifier, and the counting quantifiers. They were introduced by Per Lindström in 1966. They were later studied for their applications in logic in computer science and database query languages.

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Conservative extension

Summary

Non-conservative_extension

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 original. More formally stated, a theory T 2 {\displaystyle T_{2}} is a (proof theoretic) conservative extension of a theory T 1 {\displaystyle T_{1}} if every theorem of T 1 {\displaystyle T_{1}} is a theorem of T 2 {\displaystyle T_{2}} , and any theorem of T 2 {\displaystyle T_{2}} in the language of T 1 {\displaystyle T_{1}} is already a theorem of T 1 {\displaystyle T_{1}} . More generally, if Γ {\displaystyle \Gamma } is a set of formulas in the common language of T 1 {\displaystyle T_{1}} and T 2 {\displaystyle T_{2}} , then T 2 {\displaystyle T_{2}} is Γ {\displaystyle \Gamma } -conservative over T 1 {\displaystyle T_{1}} if every formula from Γ {\displaystyle \Gamma } provable in T 2 {\displaystyle T_{2}} is also provable in T 1 {\displaystyle T_{1}} .

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Conservative extension

Summary

Non-conservative_extension

Note that a conservative extension of a consistent theory is consistent. If it were not, then by the principle of explosion, every formula in the language of T 2 {\displaystyle T_{2}} would be a theorem of T 2 {\displaystyle T_{2}} , so every formula in the language of T 1 {\displaystyle T_{1}} would be a theorem of T 1 {\displaystyle T_{1}} , so T 1 {\displaystyle T_{1}} would not be consistent. Hence, conservative extensions do not bear the risk of introducing new inconsistencies.

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Conservative extension

Summary

Non-conservative_extension

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 extensions have been used for defining a notion of module for ontologies: if an ontology is formalized as a logical theory, a subtheory is a module if the whole ontology is a conservative extension of the subtheory. An extension which is not conservative may be called a proper extension.

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Deduction theorem

Summary

Deduction_theorem

In mathematical logic, a deduction theorem is a metatheorem that justifies doing conditional proofs from a hypothesis in systems that do not explicitly axiomatize that hypothesis, i.e. to prove an implication A → B, it is sufficient to assume A as an hypothesis and then proceed to derive B. Deduction theorems exist for both propositional logic and first-order logic. The deduction theorem is an important tool in Hilbert-style deduction systems because it permits one to write more comprehensible and usually much shorter proofs than would be possible without it. In certain other formal proof systems the same conveniency is provided by an explicit inference rule; for example natural deduction calls it implication introduction. In more detail, the propositional logic deduction theorem states that if a formula B {\displaystyle B} is deducible from a set of assumptions Δ ∪ { A } {\displaystyle \Delta \cup \{A\}} then the implication A → B {\displaystyle A\to B} is deducible from Δ {\displaystyle \Delta } ; in symbols, Δ ∪ { A } ⊢ B {\displaystyle \Delta \cup \{A\}\vdash B} implies Δ ⊢ A → B {\displaystyle \Delta \vdash A\to B} .

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Deduction theorem

Summary

Deduction_theorem

In the special case where Δ {\displaystyle \Delta } is the empty set, the deduction theorem claim can be more compactly written as: A ⊢ B {\displaystyle A\vdash B} implies ⊢ A → B {\displaystyle \vdash A\to B} . The deduction theorem for predicate logic is similar, but comes with some extra constraints (that would for example be satisfied if A {\displaystyle A} is a closed formula). In general a deduction theorem needs to take into account all logical details of the theory under consideration, so each logical system technically needs its own deduction theorem, although the differences are usually minor.

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Deduction theorem

Summary

Deduction_theorem

The deduction theorem holds for all first-order theories with the usual deductive systems for first-order logic. However, there are first-order systems in which new inference rules are added for which the deduction theorem fails. Most notably, the deduction theorem fails to hold in Birkhoff–von Neumann quantum logic, because the linear subspaces of a Hilbert space form a non-distributive lattice.

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Definable set

Summary

Definable_set

In mathematical logic, a definable set is an n-ary relation on the domain of a structure whose elements satisfy some formula in the first-order language of that structure. A set can be defined with or without parameters, which are elements of the domain that can be referenced in the formula defining the relation.

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Gödel logic

Summary

Gödel_logic

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. == References ==

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Decidability of first-order theories of the real numbers

Summary

Decidability_of_first-order_theories_of_the_real_numbers

In mathematical logic, a first-order language of the real numbers is the set of all well-formed sentences of first-order logic that involve universal and existential quantifiers and logical combinations of equalities and inequalities of expressions over real variables. The corresponding first-order theory is the set of sentences that are actually true of the real numbers. There are several different such theories, with different expressive power, depending on the primitive operations that are allowed to be used in the expression. A fundamental question in the study of these theories is whether they are decidable: that is, is there an algorithm that can take a sentence as input and produce as output an answer "yes" or "no" to the question of whether the sentence is true in the theory.

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Decidability of first-order theories of the real numbers

Summary

Decidability_of_first-order_theories_of_the_real_numbers

The theory of real closed fields is the theory in which the primitive operations are multiplication and addition; this implies that, in this theory, the only numbers that can be defined are the real algebraic numbers. As proven by Tarski, this theory is decidable; see Tarski–Seidenberg theorem and Quantifier elimination. Current implementations of decision procedures for the theory of real closed fields are often based on quantifier elimination by cylindrical algebraic decomposition.

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Decidability of first-order theories of the real numbers

Summary

Decidability_of_first-order_theories_of_the_real_numbers

Tarski's exponential function problem concerns the extension of this theory to another primitive operation, the exponential function. It is an open problem whether this theory is decidable, but if Schanuel's conjecture holds then the decidability of this theory would follow. In contrast, the extension of the theory of real closed fields with the sine function is undecidable since this allows encoding of the undecidable theory of integers (see Richardson's theorem).

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Decidability of first-order theories of the real numbers

Summary

Decidability_of_first-order_theories_of_the_real_numbers

Still, one can handle the undecidable case with functions such as sine by using algorithms that do not necessarily terminate always. In particular, one can design algorithms that are only required to terminate for input formulas that are robust, that is, formulas whose satisfiability does not change if the formula is slightly perturbed. Alternatively, it is also possible to use purely heuristic approaches.

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One-place predicate

Summary

First-order_predicate

In mathematical logic, a first-order predicate is a predicate that takes only individual(s) constants or variables as argument(s). Compare second-order predicate and higher-order predicate. This is not to be confused with a one-place predicate or monad, which is a predicate that takes only one argument. For example, the expression "is a planet" is a one-place predicate, while the expression "is father of" is a two-place predicate.

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Formal calculation

Summary

Formal_calculation

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 expression without considering its underlying meaning. This reasoning can either serve as positive evidence that some statement is true when it is difficult or unnecessary to provide proof or as an inspiration for the creation of new (completely rigorous) definitions. However, this interpretation of the term formal is not universally accepted, and some consider it to mean quite the opposite: a completely rigorous argument, as in formal mathematical logic.

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Formal language theory

Formal theories, systems, and proofs

Formal_expression > Applications > Formal theories, systems, and proofs

In mathematical logic, a formal theory is a set of sentences expressed in a formal language. A formal system (also called a logical calculus, or a logical system) consists of a formal language together with a deductive apparatus (also called a deductive system). The deductive apparatus may consist of a set of transformation rules, which may be interpreted as valid rules of inference, or a set of axioms, or have both.

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Formal language theory

Formal theories, systems, and proofs

Formal_expression > Applications > Formal theories, systems, and proofs

A formal system is used to derive one expression from one or more other expressions. Although a formal language can be identified with its formulas, a formal system cannot be likewise identified by its theorems. Two formal systems F S {\displaystyle {\mathcal {FS}}} and F S ′ {\displaystyle {\mathcal {FS'}}} may have all the same theorems and yet differ in some significant proof-theoretic way (a formula A may be a syntactic consequence of a formula B in one but not another for instance).

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Formal language theory

Formal theories, systems, and proofs

Formal_expression > Applications > Formal theories, systems, and proofs

A formal proof or derivation is a finite sequence of well-formed formulas (which may be interpreted as sentences, or propositions) each of which is an axiom or follows from the preceding formulas in the sequence by a rule of inference. The last sentence in the sequence is a theorem of a formal system. Formal proofs are useful because their theorems can be interpreted as true propositions.

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Mathematical theorem

Theorems in logic

Formal_theorem > Theorems in logic

In mathematical logic, a formal theory is a set of sentences within a formal language. A sentence is a well-formed formula with no free variables. A sentence that is a member of a theory is one of its theorems, and the theory is the set of its theorems. Usually a theory is understood to be closed under the relation of logical consequence.

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Mathematical theorem

Theorems in logic

Formal_theorem > Theorems in logic

Some accounts define a theory to be closed under the semantic consequence relation ( ⊨ {\displaystyle \models } ), while others define it to be closed under the syntactic consequence, or derivability relation ( ⊢ {\displaystyle \vdash } ). For a theory to be closed under a derivability relation, it must be associated with a deductive system that specifies how the theorems are derived. The deductive system may be stated explicitly, or it may be clear from the context.

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Mathematical theorem

Theorems in logic

Formal_theorem > Theorems in logic

The closure of the empty set under the relation of logical consequence yields the set that contains just those sentences that are the theorems of the deductive system. In the broad sense in which the term is used within logic, a theorem does not have to be true, since the theory that contains it may be unsound relative to a given semantics, or relative to the standard interpretation of the underlying language. A theory that is inconsistent has all sentences as theorems.

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Mathematical theorem

Theorems in logic

Formal_theorem > Theorems in logic

The definition of theorems as sentences of a formal language is useful within proof theory, which is a branch of mathematics that studies the structure of formal proofs and the structure of provable formulas. It is also important in model theory, which is concerned with the relationship between formal theories and structures that are able to provide a semantics for them through interpretation. Although theorems may be uninterpreted sentences, in practice mathematicians are more interested in the meanings of the sentences, i.e. in the propositions they express.

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Mathematical theorem

Theorems in logic

Formal_theorem > Theorems in logic

What makes formal theorems useful and interesting is that they may be interpreted as true propositions and their derivations may be interpreted as a proof of their truth. A theorem whose interpretation is a true statement about a formal system (as opposed to within a formal system) is called a metatheorem. Some important theorems in mathematical logic are: Compactness of first-order logic Completeness of first-order logic Gödel's incompleteness theorems of first-order arithmetic Consistency of first-order arithmetic Tarski's undefinability theorem Church-Turing theorem of undecidability Löb's theorem Löwenheim–Skolem theorem Lindström's theorem Craig's theorem Cut-elimination theorem

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Formula

In mathematical logic

Mathematical_formula > In mathematics > In mathematical logic

In mathematical logic, a formula (often referred to as a well-formed formula) is an entity constructed using the symbols and formation rules of a given logical language. For example, in first-order logic, ∀ x ∀ y ( P ( f ( x ) ) → ¬ ( P ( x ) → Q ( f ( y ) , x , z ) ) ) {\displaystyle \forall x\forall y(P(f(x))\rightarrow \neg (P(x)\rightarrow Q(f(y),x,z)))} is a formula, provided that f {\displaystyle f} is a unary function symbol, P {\displaystyle P} a unary predicate symbol, and Q {\displaystyle Q} a ternary predicate symbol.

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Negation normal form

Summary

Negation_normal_form

In mathematical logic, a formula is in negation normal form (NNF) if the negation operator ( ¬ {\displaystyle \lnot } , not) is only applied to variables and the only other allowed Boolean operators are conjunction ( ∧ {\displaystyle \land } , and) and disjunction ( ∨ {\displaystyle \lor } , or). Negation normal form is not a canonical form: for example, a ∧ ( b ∨ ¬ c ) {\displaystyle a\land (b\lor \lnot c)} and ( a ∧ b ) ∨ ( a ∧ ¬ c ) {\displaystyle (a\land b)\lor (a\land \lnot c)} are equivalent, and are both in negation normal form. In classical logic and many modal logics, every formula can be brought into this form by replacing implications and equivalences by their definitions, using De Morgan's laws to push negation inwards, and eliminating double negations. This process can be represented using the following rewrite rules (Handbook of Automated Reasoning 1, p.

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Negation normal form

Summary

Negation_normal_form

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 B\\\lnot (A\land B)&~\to ~\lnot A\lor \lnot B\\\lnot \lnot A&~\to ~A\\\lnot \exists xA&~\to ~\forall x\lnot A\\\lnot \forall xA&~\to ~\exists x\lnot A\end{aligned}}} (In these rules, the ⇒ {\displaystyle \Rightarrow } symbol indicates logical implication in the formula being rewritten, and → {\displaystyle \to } is the rewriting operation.) Transformation into negation normal form can increase the size of a formula only linearly: the number of occurrences of atomic formulas remains the same, the total number of occurrences of ∧ {\displaystyle \land } and ∨ {\displaystyle \lor } is unchanged, and the number of occurrences of ¬ {\displaystyle \lnot } in the normal form is bounded by the length of the original formula. A formula in negation normal form can be put into the stronger conjunctive normal form or disjunctive normal form by applying distributivity.

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Negation normal form

Summary

Negation_normal_form

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. For formulas in conjunctive normal form, the validity problem is solvable in polynomial time, and for formulas in disjunctive normal form, the satisfiability problem is solvable in polynomial time.

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Shoenfield absoluteness theorem

Summary

Absoluteness_(logic)

In mathematical logic, a formula is said to be absolute to some class of structures (also called models), if it has the same truth value in each of the members of that class. One can also speak of absoluteness of a formula between two structures, if it is absolute to some class which contains both of them.. Theorems about absoluteness typically establish relationships between the absoluteness of formulas and their syntactic form. There are two weaker forms of partial absoluteness. If the truth of a formula in each substructure N of a structure M follows from its truth in M, the formula is downward absolute.

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Shoenfield absoluteness theorem

Summary

Absoluteness_(logic)

If the truth of a formula in a structure N implies its truth in each structure M extending N, the formula is upward absolute. Issues of absoluteness are particularly important in set theory and model theory, fields where multiple structures are considered simultaneously. In model theory, several basic results and definitions are motivated by absoluteness.

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Shoenfield absoluteness theorem

Summary

Absoluteness_(logic)

In set theory, the issue of which properties of sets are absolute is well studied. The Shoenfield absoluteness theorem, due to Joseph Shoenfield (1961), establishes the absoluteness of a large class of formulas between a model of set theory and its constructible universe, with important methodological consequences. The absoluteness of large cardinal axioms is also studied, with positive and negative results known.

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Satisfiability problem

Summary

Satisfiability_problem

In mathematical logic, a formula is satisfiable if it is true under some assignment of values to its variables. For example, the formula x + 3 = y {\displaystyle x+3=y} is satisfiable because it is true when x = 3 {\displaystyle x=3} and y = 6 {\displaystyle y=6} , while the formula x + 1 = x {\displaystyle x+1=x} is not satisfiable over the integers. The dual concept to satisfiability is validity; a formula is valid if every assignment of values to its variables makes the formula true. For example, x + 3 = 3 + x {\displaystyle x+3=3+x} is valid over the integers, but x + 3 = y {\displaystyle x+3=y} is not.

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Satisfiability problem

Summary

Satisfiability_problem

Formally, satisfiability is studied with respect to a fixed logic defining the syntax of allowed symbols, such as first-order logic, second-order logic or propositional logic. Rather than being syntactic, however, satisfiability is a semantic property because it relates to the meaning of the symbols, for example, the meaning of + {\displaystyle +} in a formula such as x + 1 = x {\displaystyle x+1=x} . Formally, we define an interpretation (or model) to be an assignment of values to the variables and an assignment of meaning to all other non-logical symbols, and a formula is said to be satisfiable if there is some interpretation which makes it true.

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Satisfiability problem

Summary

Satisfiability_problem

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.

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Satisfiability problem

Summary

Satisfiability_problem

Satisfiability and validity are defined for a single formula, but can be generalized to an arbitrary theory or set of formulas: a theory is satisfiable if at least one interpretation makes every formula in the theory true, and valid if every formula is true in every interpretation. For example, theories of arithmetic such as Peano arithmetic are satisfiable because they are true in the natural numbers.

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Satisfiability problem

Summary

Satisfiability_problem

This concept is closely related to the consistency of a theory, and in fact is equivalent to consistency for first-order logic, a result known as Gödel's completeness theorem. The negation of satisfiability is unsatisfiability, and the negation of validity is invalidity. These four concepts are related to each other in a manner exactly analogous to Aristotle's square of opposition.

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Satisfiability problem

Summary

Satisfiability_problem

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 automated theorem proving, the methods of term rewriting, congruence closure and unification are used to attempt to decide satisfiability. Whether a particular theory is decidable or not depends whether the theory is variable-free and on other conditions.

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Skolem function

Summary

Skolem_hull

In mathematical logic, a formula of first-order logic is in Skolem normal form if it is in prenex normal form with only universal first-order quantifiers. Every first-order formula may be converted into Skolem normal form while not changing its satisfiability via a process called Skolemization (sometimes spelled Skolemnization). The resulting formula is not necessarily equivalent to the original one, but is equisatisfiable with it: it is satisfiable if and only if the original one is satisfiable.Reduction to Skolem normal form is a method for removing existential quantifiers from formal logic statements, often performed as the first step in an automated theorem prover.

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Fragment (logic)

Summary

Fragment_(logics)

In mathematical logic, a fragment of a logical language or theory is a subset of this logical language obtained by imposing syntactical restrictions on the language. Hence, the well-formed formulae of the fragment are a subset of those in the original logic. However, the semantics of the formulae in the fragment and in the logic coincide, and any formula of the fragment can be expressed in the original logic. The computational complexity of tasks such as satisfiability or model checking for the logical fragment can be no higher than the same tasks in the original logic, as there is a reduction from the first problem to the other.

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Fragment (logic)

Summary

Fragment_(logics)

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 computational complexity theory, by identifying logical fragments that exactly capture certain complexity classes. == References ==

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Ground expression

Summary

Ground_sentence

In mathematical logic, a ground term of a formal system is a term that does not contain any variables. Similarly, a ground formula is a formula that does not contain any variables. In first-order logic with identity with constant symbols a {\displaystyle a} and b {\displaystyle b} , the sentence Q ( a ) ∨ P ( b ) {\displaystyle Q(a)\lor P(b)} is a ground formula. A ground expression is a ground term or ground formula.

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Judgment (mathematical logic)

Summary

Logical_assertion

In mathematical logic, a judgment (or judgement) or assertion is a statement or enunciation in a metalanguage. For example, typical judgments in first-order logic would be that a string is a well-formed formula, or that a proposition is true. Similarly, a judgment may assert the occurrence of a free variable in an expression of the object language, or the provability of a proposition. In general, a judgment may be any inductively definable assertion in the metatheory.

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Judgment (mathematical logic)

Summary

Logical_assertion

Judgments are used in formalizing deduction systems: a logical axiom expresses a judgment, premises of a rule of inference are formed as a sequence of judgments, and their conclusion is a judgment as well (thus, hypotheses and conclusions of proofs are judgments). A characteristic feature of the variants of Hilbert-style deduction systems is that the context is not changed in any of their rules of inference, while both natural deduction and sequent calculus contain some context-changing rules. Thus, if we are interested only in the derivability of tautologies, not hypothetical judgments, then we can formalize the Hilbert-style deduction system in such a way that its rules of inference contain only judgments of a rather simple form.

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Judgment (mathematical logic)

Summary

Logical_assertion

The same cannot be done with the other two deductions systems: as context is changed in some of their rules of inferences, they cannot be formalized so that hypothetical judgments could be avoided—not even if we want to use them just for proving derivability of tautologies. This basic diversity among the various calculi allows such difference, that the same basic thought (e.g. deduction theorem) must be proven as a metatheorem in Hilbert-style deduction system, while it can be declared explicitly as a rule of inference in natural deduction. In type theory, some analogous notions are used as in mathematical logic (giving rise to connections between the two fields, e.g. Curry–Howard correspondence). The abstraction in the notion of judgment in mathematical logic can be exploited also in foundation of type theory as well.

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Boolean literal

Summary

Literal_(mathematical_logic)

In mathematical logic, a literal is an atomic formula (also known as an atom or prime formula) or its negation. The definition mostly appears in proof theory (of classical logic), e.g. in conjunctive normal form and the method of resolution. Literals can be divided into two types: A positive literal is just an atom (e.g., x {\displaystyle x} ). A negative literal is the negation of an atom (e.g., ¬ x {\displaystyle \lnot x} ).The polarity of a literal is positive or negative depending on whether it is a positive or negative literal.

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Boolean literal

Summary

Literal_(mathematical_logic)

In logics with double negation elimination (where ¬ ¬ x ≡ x {\displaystyle \lnot \lnot x\equiv x} ) the complementary literal or complement of a literal l {\displaystyle l} can be defined as the literal corresponding to the negation of l {\displaystyle l} . We can write l ¯ {\displaystyle {\bar {l}}} to denote the complementary literal of l {\displaystyle l} . More precisely, if l ≡ x {\displaystyle l\equiv x} then l ¯ {\displaystyle {\bar {l}}} is ¬ x {\displaystyle \lnot x} and if l ≡ ¬ x {\displaystyle l\equiv \lnot x} then l ¯ {\displaystyle {\bar {l}}} is x {\displaystyle x} .

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Boolean literal

Summary

Literal_(mathematical_logic)

Double negation elimination occurs in classical logics but not in intuitionistic logic. In the context of a formula in the conjunctive normal form, a literal is pure if the literal's complement does not appear in the formula. In Boolean functions, each separate occurrence of a variable, either in inverse or uncomplemented form, is a literal. For example, if A {\displaystyle A} , B {\displaystyle B} and C {\displaystyle C} are variables then the expression A ¯ B C {\displaystyle {\bar {A}}BC} contains three literals and the expression A ¯ C + B ¯ C ¯ {\displaystyle {\bar {A}}C+{\bar {B}}{\bar {C}}} contains four literals. However, the expression A ¯ C + B ¯ C {\displaystyle {\bar {A}}C+{\bar {B}}C} would also be said to contain four literals, because although two of the literals are identical ( C {\displaystyle C} appears twice) these qualify as two separate occurrences.

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Finite model property

Summary

Finite_model_property

In mathematical logic, a logic L has the finite model property (fmp for short) if any non-theorem of L is falsified by some finite model of L. Another way of putting this is to say that L has the fmp if for every formula A of L, A is an L-theorem if and only if A is a theorem of the theory of finite models of L. If L is finitely axiomatizable (and has a recursive set of inference rules) and has the fmp, then it is decidable. However, the result does not hold if L is merely recursively axiomatizable. Even if there are only finitely many finite models to choose from (up to isomorphism) there is still the problem of checking whether the underlying frames of such models validate the logic, and this may not be decidable when the logic is not finitely axiomatizable, even when it is recursively axiomatizable. (Note that a logic is recursively enumerable if and only if it is recursively axiomatizable, a result known as Craig's theorem.)

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Soundness

Logical systems

Soundness > Use in mathematical logic > Logical systems

In mathematical logic, a logical system has the soundness property if every formula that can be proved in the system is logically valid with respect to the semantics of the system. In most cases, this comes down to its rules having the property of preserving truth. The converse of soundness is known as completeness. A logical system with syntactic entailment ⊢ {\displaystyle \vdash } and semantic entailment ⊨ {\displaystyle \models } is sound if for any sequence A 1 , A 2 , .

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Soundness

Logical systems

Soundness > Use in mathematical logic > Logical systems

, A n {\displaystyle A_{1},A_{2},...,A_{n}} of sentences in its language, if A 1 , A 2 , . . .

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Soundness

Logical systems

Soundness > Use in mathematical logic > Logical systems

, A n ⊢ C {\displaystyle A_{1},A_{2},...,A_{n}\vdash C} , then A 1 , A 2 , . . .

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Soundness

Logical systems

Soundness > Use in mathematical logic > Logical systems

, A n ⊨ C {\displaystyle A_{1},A_{2},...,A_{n}\models C} . In other words, a system is sound when all of its theorems are tautologies. Soundness is among the most fundamental properties of mathematical logic.

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Soundness

Logical systems

Soundness > Use in mathematical logic > Logical systems

The soundness property provides the initial reason for counting a logical system as desirable. The completeness property means that every validity (truth) is provable. Together they imply that all and only validities are provable.

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Soundness

Logical systems

Soundness > Use in mathematical logic > Logical systems

Most proofs of soundness are trivial. For example, in an axiomatic system, proof of soundness amounts to verifying the validity of the axioms and that the rules of inference preserve validity (or the weaker property, truth). If the system allows Hilbert-style deduction, it requires only verifying the validity of the axioms and one rule of inference, namely modus ponens. (and sometimes substitution) Soundness properties come in two main varieties: weak and strong soundness, of which the former is a restricted form of the latter.

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Epistemic structural realism

Definition of structure

Ontic_structural_realism > Definition of structure

In mathematical logic, a mathematical structure is a standard concept. A mathematical structure is a set of abstract entities with relations between them. The natural numbers under arithmetic constitute a structure, with relations such as "is evenly divisible by" and "is greater than". Here the relation "is greater than" includes the element (3, 4), but not the element (4, 3).

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Epistemic structural realism

Definition of structure

Ontic_structural_realism > Definition of structure

Points in space and the real numbers under Euclidean geometry are another structure, with relations such as "the distance between point P1 and point P2 is real number R1"; equivalently, the "distance" relation includes the element (P1, P2, R1). Other structures include the Riemann space of general relativity and the Hilbert space of quantum mechanics. The entities in a mathematical structure do not have any independent existence outside their participation in relations. Two descriptions of a structure are considered equivalent, and to be describing the same underlying structure, if there is a correspondence between the descriptions that preserves all relations.Many proponents of structural realism formally or informally ascribe "properties" to the abstract objects; some argue that such properties, while they can perhaps be "shoehorned" into the formalism of relations, should instead be considered distinct from relations.

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Non-standard number

Summary

Non-standard_model_of_arithmetic

In mathematical logic, a non-standard model of arithmetic is a model of (first-order) Peano arithmetic that contains non-standard numbers. The term standard model of arithmetic refers to the standard natural numbers 0, 1, 2, …. The elements of any model of Peano arithmetic are linearly ordered and possess an initial segment isomorphic to the standard natural numbers. A non-standard model is one that has additional elements outside this initial segment. The construction of such models is due to Thoralf Skolem (1934).

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Predicate symbol

Summary

Predicate_symbol

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} , Q {\displaystyle Q} and R {\displaystyle R} , or lower case roman letters, e.g., x {\displaystyle x} . In first-order logic, they can be more properly called metalinguistic variables. In higher-order logic, predicate variables correspond to propositional variables which can stand for well-formed formulas of the same logic, and such variables can be quantified by means of (at least) second-order quantifiers.

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Proof system

Summary

Proof_calculus

In mathematical logic, a proof calculus or a proof system is built to prove statements.

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Propositional variable

Summary

Propositional_variable

In mathematical logic, a propositional variable (also called a sentential variable or sentential letter) is an input variable (that can either be true or false) of a truth function. Propositional variables are the basic building-blocks of propositional formulas, used in propositional logic and higher-order logics.

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Redundant proof

Summary

Redundant_proof

In mathematical logic, a redundant proof is a proof that has a subset that is a shorter proof of the same result. In other words, a proof is redundant if it has more proof steps than are actually necessary to prove the result. Formally, a proof ψ {\displaystyle \psi } of κ {\displaystyle \kappa } is considered redundant if there exists another proof ψ ′ {\displaystyle \psi ^{\prime }} of κ ′ {\displaystyle \kappa ^{\prime }} such that κ ′ ⊆ κ {\displaystyle \kappa ^{\prime }\subseteq \kappa } (i.e. κ ′ subsumes κ {\displaystyle \kappa ^{\prime }\;{\text{subsumes}}\;\kappa } ) and | ψ ′ | < | ψ | {\displaystyle |\psi ^{\prime }|<|\psi |} where | φ | {\displaystyle |\varphi |} is the number of nodes in φ {\displaystyle \varphi } .

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Second-order predicate

Summary

Second-order_predicate

In mathematical logic, a second-order predicate is a predicate that takes a first-order predicate as an argument. Compare higher-order predicate. The idea of second order predication was introduced by the German mathematician and philosopher Frege.

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Second-order predicate

Summary

Second-order_predicate

It is based on his idea that a predicate such as "is a philosopher" designates a concept, rather than an object. Sometimes a concept can itself be the subject of a proposition, such as in "There are no Bosnian philosophers". In this case, we are not saying anything of any Bosnian philosophers, but of the concept "is a Bosnian philosopher" that it is not satisfied.

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Second-order predicate

Summary

Second-order_predicate

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 ==

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Sentence (mathematical logic)

Summary

Sentence_(mathematical_logic)

In mathematical logic, a sentence (or closed formula) of a predicate logic is a Boolean-valued well-formed formula with no free variables. A sentence can be viewed as expressing a proposition, something that must be true or false. The restriction of having no free variables is needed to make sure that sentences can have concrete, fixed truth values: as the free variables of a (general) formula can range over several values, the truth value of such a formula may vary. Sentences without any logical connectives or quantifiers in them are known as atomic sentences; by analogy to atomic formula.

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Sentence (mathematical logic)

Summary

Sentence_(mathematical_logic)

Sentences are then built up out of atomic formulas by applying connectives and quantifiers. A set of sentences is called a theory; thus, individual sentences may be called theorems.

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Sentence (mathematical logic)

Summary

Sentence_(mathematical_logic)

To properly evaluate the truth (or falsehood) of a sentence, one must make reference to an interpretation of the theory. For first-order theories, interpretations are commonly called structures. Given a structure or interpretation, a sentence will have a fixed truth value. A theory is satisfiable when it is possible to present an interpretation in which all of its sentences are true. The study of algorithms to automatically discover interpretations of theories that render all sentences as being true is known as the satisfiability modulo theories problem.

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Sequent

Summary

Sequent

In mathematical logic, a sequent is a very general kind of conditional assertion. A 1 , … , A m ⊢ B 1 , … , B n . {\displaystyle A_{1},\,\dots ,A_{m}\,\vdash \,B_{1},\,\dots ,B_{n}.} A sequent may have any number m of condition formulas Ai (called "antecedents") and any number n of asserted formulas Bj (called "succedents" or "consequents"). A sequent is understood to mean that if all of the antecedent conditions are true, then at least one of the consequent formulas is true. This style of conditional assertion is almost always associated with the conceptual framework of sequent calculus.

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Deductive closure

Summary

Deductive_closure

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 {\displaystyle \varphi \in {\mathcal {T}}} . If T {\displaystyle T} is a set of formulae, the deductive closure of T {\displaystyle T} is its smallest superset that is deductively closed. The deductive closure of a theory T {\displaystyle {\mathcal {T}}} is often denoted Ded ⁡ ( T ) {\displaystyle \operatorname {Ded} ({\mathcal {T}})} or Th ⁡ ( T ) {\displaystyle \operatorname {Th} ({\mathcal {T}})} . This is a special case of the more general mathematical concept of closure — in particular, the deductive closure of T {\displaystyle {\mathcal {T}}} is exactly the closure of T {\displaystyle {\mathcal {T}}} with respect to the operation of logical consequence ( ⊢ {\displaystyle \vdash } ).

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Superintuitionistic logic

Summary

Superintuitionistic_logic

In mathematical logic, a superintuitionistic logic is a propositional logic extending intuitionistic logic. Classical logic is the strongest consistent superintuitionistic logic; thus, consistent superintuitionistic logics are called intermediate logics (the logics are intermediate between intuitionistic logic and classical logic).

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Tautological implication

Summary

Tautology_(logic)

In mathematical logic, a tautology (from Greek: ταυτολογία) is a formula or assertion that is true in every possible interpretation. An example is "x=y or x≠y". Similarly, "either the ball is green, or the ball is not green" is always true, regardless of the colour of the ball. The philosopher Ludwig Wittgenstein first applied the term to redundancies of propositional logic in 1921, borrowing from rhetoric, where a tautology is a repetitive statement.

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Tautological implication

Summary

Tautology_(logic)

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.

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Tautological implication

Summary

Tautology_(logic)

Unsatisfiable statements, both through negation and affirmation, are known formally as contradictions. A formula that is neither a tautology nor a contradiction is said to be logically contingent.

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Tautological implication

Summary

Tautology_(logic)

Such a formula can be made either true or false based on the values assigned to its propositional variables. The double turnstile notation ⊨ S {\displaystyle \vDash S} is used to indicate that S is a tautology. Tautology is sometimes symbolized by "Vpq", and contradiction by "Opq".

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Tautological implication

Summary

Tautology_(logic)

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 concept in propositional logic, where a tautology is defined as a propositional formula that is true under any possible Boolean valuation of its propositional variables. A key property of tautologies in propositional logic is that an effective method exists for testing whether a given formula is always satisfied (equiv., whether its negation is unsatisfiable). The definition of tautology can be extended to sentences in predicate logic, which may contain quantifiers—a feature absent from sentences of propositional logic.

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Tautological implication

Summary

Tautology_(logic)

Indeed, in propositional logic, there is no distinction between a tautology and a logically valid formula. In the context of predicate logic, many authors define a tautology to be a sentence that can be obtained by taking a tautology of propositional logic, and uniformly replacing each propositional variable by a first-order formula (one formula per propositional variable). The set of such formulas is a proper subset of the set of logically valid sentences of predicate logic (i.e., sentences that are true in every model).

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Finite terms

Summary

Finite_terms

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 constructed from constant symbols, variables and function symbols. An expression formed by applying a predicate symbol to an appropriate number of terms is called an atomic formula, which evaluates to true or false in bivalent logics, given an interpretation. For example, ( x + 1 ) ∗ ( x + 1 ) {\displaystyle (x+1)*(x+1)} is a term built from the constant 1, the variable x, and the binary function symbols + {\displaystyle +} and ∗ {\displaystyle *} ; it is part of the atomic formula ( x + 1 ) ∗ ( x + 1 ) ≥ 0 {\displaystyle (x+1)*(x+1)\geq 0} which evaluates to true for each real-numbered value of x. Besides in logic, terms play important roles in universal algebra, and rewriting systems.

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Theory (mathematical logic)

Summary

First-order_theory

In mathematical logic, a theory (also called a formal theory) is a set of sentences in a formal language. In most scenarios a deductive system is first understood from context, after which an element ϕ ∈ T {\displaystyle \phi \in T} of a deductively closed theory T {\displaystyle T} is then called a theorem of the theory. In many deductive systems there is usually a subset Σ ⊆ T {\displaystyle \Sigma \subseteq T} that is called "the set of axioms" of the theory T {\displaystyle T} , in which case the deductive system is also called an "axiomatic system". By definition, every axiom is automatically a theorem. A first-order theory is a set of first-order sentences (theorems) recursively obtained by the inference rules of the system applied to the set of axioms.

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Extension by new constant and function names

Summary

Extension_by_new_constant_and_function_names

In mathematical logic, a theory can be extended with new constants or function names under certain conditions with assurance that the extension will introduce no contradiction. Extension by definitions is perhaps the best-known approach, but it requires unique existence of an object with the desired property. Addition of new names can also be done safely without uniqueness. Suppose that a closed formula ∃ x 1 … ∃ x m φ ( x 1 , … , x m ) {\displaystyle \exists x_{1}\ldots \exists x_{m}\,\varphi (x_{1},\ldots ,x_{m})} is a theorem of a first-order theory T {\displaystyle T} .

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Extension by new constant and function names

Summary

Extension_by_new_constant_and_function_names

Let T 1 {\displaystyle T_{1}} be a theory obtained from T {\displaystyle T} by extending its language with new constants a 1 , … , a m {\displaystyle a_{1},\ldots ,a_{m}} and adding a new axiom φ ( a 1 , … , a m ) {\displaystyle \varphi (a_{1},\ldots ,a_{m})} .Then T 1 {\displaystyle T_{1}} is a conservative extension of T {\displaystyle T} , which means that the theory T 1 {\displaystyle T_{1}} has the same set of theorems in the original language (i.e., without constants a i {\displaystyle a_{i}} ) as the theory T {\displaystyle T} . Such a theory can also be conservatively extended by introducing a new functional symbol:Suppose that a closed formula ∀ x → ∃ y φ ( y , x → ) {\displaystyle \forall {\vec {x}}\,\exists y\,\!\,\varphi (y,{\vec {x}})} is a theorem of a first-order theory T {\displaystyle T} , where we denote x → := ( x 1 , … , x n ) {\displaystyle {\vec {x}}:=(x_{1},\ldots ,x_{n})} . Let T 1 {\displaystyle T_{1}} be a theory obtained from T {\displaystyle T} by extending its language with a new functional symbol f {\displaystyle f} (of arity n {\displaystyle n} ) and adding a new axiom ∀ x → φ ( f ( x → ) , x → ) {\displaystyle \forall {\vec {x}}\,\varphi (f({\vec {x}}),{\vec {x}})} .

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Extension by new constant and function names

Summary

Extension_by_new_constant_and_function_names

Then T 1 {\displaystyle T_{1}} is a conservative extension of T {\displaystyle T} , i.e. the theories T {\displaystyle T} and T 1 {\displaystyle T_{1}} prove the same theorems not involving the functional symbol f {\displaystyle f} ). Shoenfield states the theorem in the form for a new function name, and constants are the same as functions of zero arguments. In formal systems that admit ordered tuples, extension by multiple constants as shown here can be accomplished by addition of a new constant tuple and the new constant names having the values of elements of the tuple.

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Existential theory of the reals

Background

Existential_theory_of_the_reals > Background

In mathematical logic, a theory is a formal language consisting of a set of sentences written using a fixed set of symbols. The first-order theory of real closed fields has the following symbols: the constants 0 and 1, a countable collection of variables X i {\displaystyle X_{i}} , the addition, subtraction, multiplication, and (optionally) division operations, symbols <, ≤, =, ≥, >, and ≠ for comparisons of real values, the logical connectives ∧, ∨, ¬, and ⇔, parentheses, and the universal quantifier ∀ and the existential quantifier ∃A sequence of these symbols forms a sentence that belongs to the first-order theory of the reals if it is grammatically well formed, all its variables are properly quantified, and (when interpreted as a mathematical statement about the real numbers) it is a true statement. As Tarski showed, this theory can be described by an axiom schema and a decision procedure that is complete and effective: for every fully quantified and grammatical sentence, either the sentence or its negation (but not both) can be derived from the axioms. The same theory describes every real closed field, not just the real numbers.

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Existential theory of the reals

Background

Existential_theory_of_the_reals > Background

However, there are other number systems that are not accurately described by these axioms; in particular, the theory defined in the same way for integers instead of real numbers is undecidable, even for existential sentences (Diophantine equations) by Matiyasevich's theorem.The existential theory of the reals is the fragment of the first-order theory consisting of sentences in which all the quantifiers are existential and they appear before any of the other symbols. That is, it is the set of all true sentences of the form where F ( X 1 , … X n ) {\displaystyle F(X_{1},\dots X_{n})} is a quantifier-free formula involving equalities and inequalities of real polynomials. The decision problem for the existential theory of the reals is the algorithmic problem of testing whether a given sentence belongs to this theory; equivalently, for strings that pass the basic syntactical checks (they use the correct symbols with the correct syntax, and have no unquantified variables) it is the problem of testing whether the sentence is a true statement about the real numbers.

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Existential theory of the reals

Background

Existential_theory_of_the_reals > Background

The set of n {\displaystyle n} -tuples of real numbers ( X 1 , … X n ) {\displaystyle (X_{1},\dots X_{n})} for which F ( X 1 , … X n ) {\displaystyle F(X_{1},\dots X_{n})} is true is called a semialgebraic set, so the decision problem for the existential theory of the reals can equivalently be rephrased as testing whether a given semialgebraic set is nonempty.In determining the time complexity of algorithms for the decision problem for the existential theory of the reals, it is important to have a measure of the size of the input. The simplest measure of this type is the length of a sentence: that is, the number of symbols it contains. However, in order to achieve a more precise analysis of the behavior of algorithms for this problem, it is convenient to break down the input size into several variables, separating out the number of variables to be quantified, the number of polynomials within the sentence, and the degree of these polynomials.

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Categorical (model theory)

Summary

Categorical_theory

In mathematical logic, a theory is categorical if it has exactly one model (up to isomorphism). Such a theory can be viewed as defining its model, uniquely characterizing the model's structure. In first-order logic, only theories with a finite model can be categorical.

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Categorical (model theory)

Summary

Categorical_theory

Higher-order logic contains categorical theories with an infinite model. For example, the second-order Peano axioms are categorical, having a unique model whose domain is the set of natural numbers N . {\displaystyle \mathbb {N} .}

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Categorical (model theory)

Summary

Categorical_theory

In model theory, the notion of a categorical theory is refined with respect to cardinality. A theory is κ-categorical (or categorical in κ) if it has exactly one model of cardinality κ up to isomorphism. Morley's categoricity theorem is a theorem of Michael D. Morley (1965) stating that if a first-order theory in a countable language is categorical in some uncountable cardinality, then it is categorical in all uncountable cardinalities. Saharon Shelah (1974) extended Morley's theorem to uncountable languages: if the language has cardinality κ and a theory is categorical in some uncountable cardinal greater than or equal to κ then it is categorical in all cardinalities greater than κ.

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Complete theory

Summary

Maximal_consistent_set

In mathematical logic, a theory is complete if it is consistent and for every closed formula in the theory's language, either that formula or its negation is provable. That is, for every sentence φ , {\displaystyle \varphi ,} the theory T {\displaystyle T} contains the sentence or its negation but not both (that is, either T ⊢ φ {\displaystyle T\vdash \varphi } or T ⊢ ¬ φ {\displaystyle T\vdash \neg \varphi } ). Recursively axiomatizable first-order theories that are consistent and rich enough to allow general mathematical reasoning to be formulated cannot be complete, as demonstrated by Gödel's first incompleteness theorem. This sense of complete is distinct from the notion of a complete logic, which asserts that for every theory that can be formulated in the logic, all semantically valid statements are provable theorems (for an appropriate sense of "semantically valid").