Concept

Axiom schema

Overview

In mathematical logic, an axiom schema (plural: axiom schemata or axiom schemas) is a rule or template that specifies a family of axioms. A schema contains placeholders together with side conditions saying how those placeholders may be replaced; each permitted replacement is an instance of the schema. Axiom schemata are commonly used to give finite descriptions of theories whose axioms include infinitely many formulas.

Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).

Possible passages

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1. As an illustration of one possible logical system of geometry we may consider all relations of three terms which are analogous in certain formal respects to the relation "between" as it appears to be in actual space. A space is then defined by means of one such three-term relation.

   Bertrand Russell, Mysticism and Logic and Other Essays

   Close in meaning to Axiom schema; the term itself is not named.

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2. CV. In forming axioms, we must invent a different form of induction from that hitherto in use; not only for the proof and discovery of principles (as they are called), but also of minor, intermediate, and, in short, every kind of axioms. The induction which proceeds by simple enumeration is puerile, leads to uncertain conclusions, and is exposed to danger from one contradictory instance, deciding generally from too…

   Francis Bacon, Novum organum

   States what Axiom schema is.

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3. The exercise of logical reason is always concerned with these absolutely general conditions. In its broadest sense, the discovery of mathematics is the discovery that the totality of these general abstract conditions, which are concurrently applicable to the relationships among the entities of any one concrete occasion, are themselves inter-connected in the manner of a pattern with a key to it. This pattern of…

   Alfred North Whitehead, Science and the modern world

   States what Axiom schema is.

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Related ideas

• First-order logic
• Mathematical logic
• Peano axioms
• Willard Van Orman Quine
• Metalanguage
• Axiom schema of replacement
• Von Neumann–Bernays–Gödel set theory
• Richard Montague
• Axiom
• New Foundations

Place in the tradition

Part of Intuitionistic logic.

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