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axiom schema of predicative separation 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 statement toggle statement subsection 1 1 motivation 2 theories toggle theories subsection 2 1 finite axiomatizability 3 see also 4 references toggle the table of contents axiom schema of predicative separation add languages add 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 schema of axioms in set theory this article needs more citations please help improve this article by adding citations to reliable sources unsourced material may be challenged and removed find sources axiom schema of predicative separation news newspapers books scholar jstor february 2024 learn how and when to remove this message in axiomatic set theory the axiom schema of predicative separation or of restricted or δ 0 separation is a schema of axioms that is a restriction of the usual axiom schema of separation in zermelo fraenkel set theory this name δ 0 stems from the lévy hierarchy in analogy with the arithmetic hierarchy statement edit the axiom asserts only the existence of a subset of a set if that subset can be defined without reference to the entire universe of sets the formal statement of this is the same as full separation schema but with a restriction on the formulas that may be used for any formula φ x y z z y z x φ z displaystyle forall x exists y forall z z in y leftrightarrow z in x wedge varphi z provided that φ contains only bounded quantifiers and as usual that the variable y is not free in it so all quantifiers in φ if any must appear in the forms u v ψ u displaystyle exists u in v psi u u v ψ u displaystyle forall u in v psi u for some sub formula ψ and of course the definition of v displaystyle v is bound to those rules as well motivation edit this restriction is necessary from a predicative point of view since the universe of all sets contains the set being defined if it were referenced in the definition of the set the definition would be circular theories edit the axiom appears in the systems of constructive set theory cst and czf as well as in the system of kripke platek set theory finite axiomatizability edit although the schema contains one axiom for each restricted formula φ it is possible in czf to replace this schema with a finite number of axioms 1 see also edit constructive set theory axiom schema of separation references edit aczel peter rathjen michael august 19 2010 cst book draft pdf p 97 archived from the original pdf on february 5 2024 tennant neil 2021 does choice really imply excluded middle part ii historical philosophical and foundational reflections on the goodman myhill result philosophia mathematica 29 28 63 doi 10 1093 philmat nkaa010 this set theory related article is a stub you can help wikipedia by adding missing information v t e retrieved from https en wikipedia org w index php title axiom_schema_of_predicative_separation oldid 1347484559 categories constructivism philosophy of mathematics axioms of set theory set theory stubs hidden categories articles with short description short description is different from wikidata articles needing additional references from february 2024 all articles needing additional references all stub articles this page was last edited on 7 april 2026 at 00 36 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 of predicative separation add languages add topic