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axiom schema wikipedia jump to content main menu main menu move to sidebar hide navigation main page contents current events random article about wikipedia contact us contribute help learn to edit community portal recent changes upload file special pages search search appearance donate create account log in personal tools donate create account log in contents move to sidebar hide top 1 formal definition 2 examples toggle examples subsection 2 1 first order logic 2 2 arithmetic 2 3 set theory 3 finite axiomatization toggle finite axiomatization subsection 3 1 finitely axiomatized related theories 4 in higher order logic 5 see also 6 notes 7 references toggle the table of contents axiom schema 10 languages deutsch français hrvatski italiano 日本語 nederlands polski русский українська 中文 edit links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia template that specifies one or more axioms axiom schema of replacement the image f a displaystyle f a of the domain set a displaystyle a under the definable class function f displaystyle f is itself a set b displaystyle b 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 1 2 axiom schemata are commonly used to give finite descriptions of theories whose axioms include infinitely many formulas 3 formal definition edit let l displaystyle l be the object language of a formal theory an axiom schema is a metalinguistic expression or schema template containing one or more placeholders for expressions of l displaystyle l together with side conditions specifying the expressions that may replace the placeholders 1 2 the resulting object language formulas are the instances of the schema and are taken as axioms of the theory the placeholders in an axiom schema may stand for terms formulas predicates or relations depending on the formal system side conditions often require that a term be free for a variable in a formula that a variable occur free in a formula or that a variable not occur free in a specified formula 3 such conditions are part of the schema not additional axioms of the object language this distinguishes a schematic letter from an ordinary object language variable for example a schematic letter may be a placeholder for any formula of a language whereas an object language variable ranges over objects in the domain of interpretation 1 examples edit first order logic edit many hilbert style presentations of first order logic use axiom schemata for example the quantifier axiom schema x φ x φ t displaystyle forall x phi x rightarrow phi t has as instances the formulas obtained by replacing φ displaystyle phi with an object language formula and t displaystyle t with a term subject to the condition that t displaystyle t is free for x displaystyle x in φ displaystyle phi 3 rules or axiom schemata of this kind avoid listing separately every formula obtained by the permitted substitutions arithmetic edit a standard first order presentation of peano arithmetic includes the induction schema for every formula φ x y displaystyle varphi x vec y in the language of arithmetic with possible parameters y displaystyle vec y the schema has an instance of the form φ 0 y x φ x y φ s x y x φ x y displaystyle bigl varphi 0 vec y land forall x varphi x vec y rightarrow varphi sx vec y bigr rightarrow forall x varphi x vec y thus the induction principle is not a single first order axiom but a family of axioms one for each admissible formula φ displaystyle varphi 3 4 czesław ryll nardzewski proved that first order peano arithmetic is not finitely axiomatizable in its usual language 5 consequently the induction schema cannot be replaced in that language by a finite list of axioms with the same deductive consequences set theory edit the standard first order axiomatization zfc contains axiom schemata including the schema of separation and the schema of replacement in the replacement schema each formula that defines a functional relation gives an axiom asserting that the image of any set under that definable function is also a set 6 7 richard montague proved a non finite axiomatizability result for zf style set theories in particular assuming consistency zfc is not finitely axiomatizable in its ordinary first order language 8 hence the replacement and separation schemata cannot simply be eliminated from zfc by replacing them with finitely many axioms in the same language finite axiomatization edit a theory is finitely axiomatizable if there is a finite set of sentences whose deductive closure is exactly the theory 9 an axiom schema may be finitely describable while still specifying infinitely many axioms since the schema is a metatheoretic recipe for generating instances rather than a finite set of object language sentences 10 whether a theory is finitely axiomatizable depends on the language in which the theory is formulated a theory that is not finitely axiomatizable in one language may have a finitely axiomatized conservative extension in a richer language finitely axiomatized related theories edit von neumann bernays gödel set theory nbg extends the language of set theory by adding classes nbg is a conservative extension of zfc for statements about sets but unlike zfc it can be finitely axiomatized its class existence theorem is obtained from finitely many class existence axioms rather than from a schema 7 11 new foundations nf introduced by w v o quine is usually presented with extensionality and a stratified comprehension schema 12 theodore hailperin showed in 1944 that this comprehension principle is equivalent to a finite conjunction of instances giving a finite axiomatization of nf 13 later presentations of nf and related systems may take finite axioms as basic and prove stratified comprehension as a theorem 14 in higher order logic edit some axiom schemata in first order theories can be replaced by single axioms in a higher order language for example the first order induction schema for arithmetic has a second order analogue in which a variable quantifies over properties or classes directly f f 0 x f x f s x x f x displaystyle forall f bigl f 0 land forall x f x rightarrow f sx rightarrow forall x f x bigr in this second order sentence f displaystyle f is a genuine variable ranging over properties or classes not a metalinguistic placeholder 3 the second order induction axiom and the first order induction schema are therefore formally different and the second order axiom is stronger under the standard semantics for second order logic 3 analogously some first order set theoretic schemata can be represented by quantifying over classes or higher order objects in an expanded language this is one reason class theories such as nbg can give finite axiomatizations while remaining conservative over zfc for set theoretic sentences 7 11 see also edit axiom schema of predicative separation axiom schema of replacement axiom schema of specification notes edit 1 2 3 corcoran 2016 1 1 2 corcoran 2006 pp 219 220 1 2 3 4 5 6 corcoran 2016 2 mendelson 1997 ch 3 ryll nardzewski 1952 pp 239 263 potter 2004 chs 1 2 1 2 3 mendelson 1997 ch 4 montague 1961 pp 45 69 mendelson 1997 ch 2 corcoran 2016 1 2 1 2 potter 2004 ch 13 forster 2025 3 hailperin 1944 pp 1 19 holmes 1998 ch 8 references edit corcoran john 2006 schemata the concept of schema in the history of logic pdf bulletin of symbolic logic 12 2 219 240 doi 10 2178 bsl 1146620060 s2cid 6909703 corcoran john 2016 schema in zalta edward n ed stanford encyclopedia of philosophy fall 2016 ed issn 1095 5054 oclc 429049174 forster thomas 2025 quine s new foundations in zalta edward n ed stanford encyclopedia of philosophy fall 2025 ed issn 1095 5054 oclc 429049174 hailperin theodore 1944 a set of axioms for logic journal of symbolic logic 9 1 1 19 doi 10 2307 2267307 jstor 2267307 holmes m randall 1998 elementary set theory with a universal set cahiers du centre de logique vol 10 academia bruylant isbn 9782872094882 mendelson elliott 1997 an introduction to mathematical logic 4th ed chapman hall isbn 0 412 80830 7 montague richard 1961 semantical closure and non finite axiomatizability i infinitistic methods proceedings of the symposium on foundations of mathematics warsaw 2 9 september 1959 państwowe wydawnictwo naukowe and pergamon press pp 45 69 potter michael 2004 set theory and its philosophy oxford university press isbn 9780199269730 ryll nardzewski czesław 1952 the role of the axiom of induction in elementary arithmetic pdf fundamenta mathematicae 39 1 239 263 doi 10 4064 fm 39 1 239 263 v t e set theory overview set mathematics axioms adjunction choice countable dependent global constructibility v l determinacy projective extensionality infinity limitation of size pairing power set regularity union martin s axiom axiom schema replacement specification operations cartesian product complement i e set difference de morgan s laws disjoint union identities intersection power set symmetric difference union concepts methods almost cardinality cardinal number large class constructible universe continuum hypothesis diagonal argument element ordered pair tuple family forcing one to one correspondence ordinal number set builder notation transfinite induction venn diagram set types amorphous countable empty finite hereditarily filter base subbase ultrafilter fuzzy infinite dedekind infinite recursive singleton subset superset transitive uncountable universal theories alternative axiomatic constructive naive cantor s theorem zermelo general principia mathematica new foundations zermelo fraenkel von neumann bernays gödel morse kelley kripke platek tarski grothendieck paradoxes problems russell s paradox suslin s problem burali forti paradox set theorists paul bernays georg cantor paul cohen richard dedekind abraham fraenkel kurt gödel thomas jech john von neumann willard quine bertrand russell thoralf skolem ernst zermelo v t e mathematical logic general axiom list cardinality first order logic formal proof formal semantics foundations of mathematics information theory lemma logical consequence model theorem theory type theory theorems list paradoxes gödel s completeness incompleteness theorems tarski s undefinability banach tarski paradox cantor s theorem paradox diagonal argument compactness halting problem lindström s löwenheim skolem russell s paradox logics traditional classical logic logical truth tautology proposition inference logical equivalence consistency equiconsistency argument soundness validity syllogism square of opposition venn diagram propositional boolean algebra boolean functions logical connectives propositional calculus propositional formula truth tables many valued logic 3 finite predicate first order list second order monadic higher order fixed point free quantifiers predicate monadic predicate calculus set theory set hereditary class ur element ordinal number extensionality forcing relation equivalence partition set operations intersection union complement cartesian product power set identities types of sets countable uncountable empty inhabited singleton finite infinite transitive ultrafilter recursive fuzzy universal universe constructible grothendieck von neumann maps cardinality function map domain codomain image in sur bi jection schröder bernstein theorem isomorphism gödel numbering enumeration large cardinal inaccessible aleph number operation binary theories zermelo fraenkel axiom of choice continuum hypothesis general kripke platek morse kelley naive new foundations tarski grothendieck von neumann bernays gödel ackermann constructive formal systems list language syntax alphabet arity automata axiom schema expression ground extension by definition conservative relation formation rule grammar formula atomic closed ground open free bound variable language metalanguage logical connective predicate functional variable propositional variable proof quantifier rank sentence atomic spectrum signature string substitution symbol function logical constant non logical variable term theory list example axiomatic systems list of true arithmetic peano second order elementary function primitive recursive robinson skolem of the real numbers tarski s axiomatization of boolean algebras canonical minimal axioms of geometry euclidean elements hilbert s tarski s non euclidean principia mathematica proof theory formal proof natural deduction logical consequence rule of inference sequent calculus theorem systems axiomatic deductive hilbert list complete theory independence from zfc proof of impossibility ordinal analysis reverse mathematics self verifying theories model theory interpretation function of models model atomic equivalence finite prime saturated spectrum submodel non standard model of non standard arithmetic diagram elementary categorical theory model complete theory satisfiability semantics of logic strength theories of truth semantic tarski s kripke s t schema transfer principle truth predicate truth value type ultraproduct validity computability theory church encoding church turing thesis computably enumerable computable function computable set decision problem decidable undecidable p np p versus np problem kolmogorov complexity lambda calculus primitive recursive function recursion recursive set turing machine type theory related abstract logic algebraic logic automated theorem proving category theory concrete abstract category category of sets history of logic history of mathematical logic timeline logicism mathematical object philosophy of mathematics supertask mathematics portal retrieved from https en wikipedia org w index php title axiom_schema oldid 1367479324 categories formal systems mathematical axioms hidden categories articles with short description short description is different from wikidata this page was last edited on 3 august 2026 at 09 19 utc page was rendered with parsoid text is available under the creative commons attribution sharealike 4 0 license additional terms may apply by using this site you agree to the terms of use and privacy policy wikipedia is a registered trademark of the wikimedia foundation inc a non profit organization privacy policy about wikipedia disclaimers contact wikipedia legal safety contacts code of conduct developers statistics cookie statement mobile view search search toggle the table of contents axiom schema 10 languages add topic
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