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uniformization set theory 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 references toggle the table of contents uniformization set theory 1 language bahasa indonesia 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 in set theory a branch of mathematics the axiom of uniformization is a weak form of the axiom of choice it states that if r displaystyle r is a subset of x y displaystyle x times y where x displaystyle x and y displaystyle y are polish spaces then there is a subset f displaystyle f of r displaystyle r that is a partial function from x displaystyle x to y displaystyle y and whose domain the set of all x displaystyle x such that f x displaystyle f x exists equals x x y y x y r displaystyle x in x mid exists y in y x y in r such a function is called a uniformizing function for r displaystyle r or a uniformization of r displaystyle r uniformization of relation r light blue by function f red to see the relationship with the axiom of choice observe that r displaystyle r can be thought of as associating to each element of x displaystyle x a subset of y displaystyle y a uniformization of r displaystyle r then picks exactly one element from each such subset whenever the subset is non empty thus allowing arbitrary sets x and y rather than just polish spaces would make the axiom of uniformization equivalent to the axiom of choice a pointclass γ displaystyle boldsymbol gamma is said to have the uniformization property if every relation r displaystyle r in γ displaystyle boldsymbol gamma can be uniformized by a partial function in γ displaystyle boldsymbol gamma the uniformization property is implied by the scale property at least for adequate pointclasses of a certain form it follows from zfc alone that π 1 1 displaystyle boldsymbol pi _ 1 1 and σ 2 1 displaystyle boldsymbol sigma _ 2 1 have the uniformization property it follows from the existence of sufficient large cardinals that π 2 n 1 1 displaystyle boldsymbol pi _ 2n 1 1 and σ 2 n 2 1 displaystyle boldsymbol sigma _ 2n 2 1 have the uniformization property for every natural number n displaystyle n therefore the collection of projective sets has the uniformization property every relation in l r can be uniformized but not necessarily by a function in l r in fact l r does not have the uniformization property equivalently l r does not satisfy the axiom of uniformization note it s trivial that every relation in l r can be uniformized in v assuming v satisfies the axiom of choice the point is that every such relation can be uniformized in some transitive inner model of v in which the axiom of determinacy holds references edit moschovakis yiannis n 1980 descriptive set theory north holland isbn 0 444 70199 0 retrieved from https en wikipedia org w index php title uniformization_ set_theory oldid 965015578 categories set theory descriptive set theory axiom of choice this page was last edited on 28 june 2020 at 22 25 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 uniformization set theory 1 language add topic