In mathematical logic, an axiom schema generalizes the notion of axiom. An axiom schema is a formula in the language of an axiomatic system, in which one or more schematic variables appear. These variables, which are metalinguistic constructs, stand for any term or subformula of the system, which may or may not be required to satisfy certain conditions. Often, such conditions require that certain variables be free, or that certain variables not appear in the subformula or term.

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  • In mathematical logic, an axiom schema generalizes the notion of axiom. An axiom schema is a formula in the language of an axiomatic system, in which one or more schematic variables appear. These variables, which are metalinguistic constructs, stand for any term or subformula of the system, which may or may not be required to satisfy certain conditions. Often, such conditions require that certain variables be free, or that certain variables not appear in the subformula or term. Given that the number of possible subformulas or terms that can be inserted in place of a schematic variable is countably infinite, an axiom schema stands for a countably infinite set of axioms. This set can usually be defined recursively. A theory that can be axiomatized without schemata is said to be finitely axiomatized. Theories that can be finitely axiomatized are seen as a bit more metamathematically elegant, even if they are less practical for deductive work. Two very well known instances of axiom schemata are the: Induction schema that is part of Peano's axioms for the arithmetic of the natural numbers; Axiom schema of replacement that is part of the standard ZFC axiomatization of set theory. It has been proved that these schemata cannot be eliminated. Hence Peano arithmetic and ZFC cannot be finitely axiomatized. This is also the case for quite a few other axiomatic theories in mathematics, philosophy, linguistics, etc. All theorems of ZFC are also theorems of von Neumann-Bernays-Gödel set theory, but the latter is, quite surprisingly, finitely axiomatized. The set theory New Foundations can be finitely axiomatized, but only with some loss of elegance. Schematic variables in first-order logic are usually trivially eliminable in second-order logic, because a schematic variable is often a placeholder for any property or relation over the individuals of the theory. This is the case with the schemata of Induction and Replacement mentioned above. Higher-order logic allows quantified variables to range over all possible properties or relations.
  • In logica matematica si chiama schema di assiomi una scrittura simbolica che rappresenta schematicamente delle regole di costruzione per un insieme (eventualmente infinito) di formule ben formate che si intende includere tra gli assiomi di una teoria proposizionale o del primo ordine. Le fbf che rientrano nello schema vengono chiamate istanze dello schema. Un esempio semplice è lo schema di assiomi: <math>(\mathcal A \to \mathcal B) \to \mathcal A</math> che ha come istanze un insieme infinito di fbf tra cui: <math>(P \to P) \to P</math> <math>(\to) \to (P \lor Q)</math> <math>(\to Q) \to (P \to Q)</math>
  • Een axiomaschema is in de wiskundige logica een veralgemenisering van een axioma. Bekende voorbeelden van axiomaschema's zijn: Het inductie-axioma als onderdeel van de axioma's van Peano voor de rekenkunde van natuurlijke getallen. Het axiomaschema van vervanging dat deel uitmaakt van de standaard ZFC-axiomatisering van de verzamelingenleer.
  • 在符号逻辑中，用有穷数目的公理表达公理系统有时是不方便或不可能的。因此，要使用公理模式。形式上说，公理模式是合式公式的集合，其中每个元素都被接受为一个公理。这个集合常常是递归构造的。一个周知的公理模式是替代公理模式。 在元数学家之间，关于包含公理模式的公理系统是否应当被看作是优雅的是有争议的。一些逻辑学家更偏好使用有穷数目的公理，如果可能的话。
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  • John Corcoran
  • Schema
  • schema
  • 2008-09-21 (xsd:date)
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  • In mathematical logic, an axiom schema generalizes the notion of axiom. An axiom schema is a formula in the language of an axiomatic system, in which one or more schematic variables appear. These variables, which are metalinguistic constructs, stand for any term or subformula of the system, which may or may not be required to satisfy certain conditions. Often, such conditions require that certain variables be free, or that certain variables not appear in the subformula or term.
  • In logica matematica si chiama schema di assiomi una scrittura simbolica che rappresenta schematicamente delle regole di costruzione per un insieme (eventualmente infinito) di formule ben formate che si intende includere tra gli assiomi di una teoria proposizionale o del primo ordine. Le fbf che rientrano nello schema vengono chiamate istanze dello schema.
  • Een axiomaschema is in de wiskundige logica een veralgemenisering van een axioma. Bekende voorbeelden van axiomaschema's zijn: Het inductie-axioma als onderdeel van de axioma's van Peano voor de rekenkunde van natuurlijke getallen. Het axiomaschema van vervanging dat deel uitmaakt van de standaard ZFC-axiomatisering van de verzamelingenleer.
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  • Axiom schema
  • Schema di assiomi
  • Axiomaschema
  • 公理模式
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