Axiom of Choice: History and Controversy

Con­tro­versy: Math­e­mat­i­ci­ans Di­vided! Counter-In­tu­itive Re­sults, and The His­tory of the Ax­iom of Choice

Math­e­mat­i­ci­ans have been us­ing an in­tu­itive con­cept of a set for prob­a­bly as long as math­e­mat­ics has been prac­ticed. At first, math­e­mat­i­ci­ans as­sumed that the ax­iom of choice was sim­ply true (as in­deed it is for finite col­lec­tions of sets).

Ge­org Can­tor in­tro­duced the con­cept of trans­finite num­bers and differ­ent car­di­nal­ities of in­finity in a 1874 pa­per (which con­tains his in­fa­mous Di­ag­o­nal­iza­tion Ar­gu­ment) and along with this sparked the in­tro­duc­tion of set the­ory. In 1883, Can­tor in­tro­duced a prin­ci­ple called the ‘Well-Order­ing Princ­ple’ (dis­cussed fur­ther in a sec­tion be­low) which he called a ‘law of thought’ (i.e., in­tu­itively true). He at­tempted to prove this prin­ci­ple from his other prin­ci­ples, but found that he was un­able to do so.

Ernst Zer­melo at­tempted to de­velop an ax­io­matic treat­ment of set the­ory. He man­aged to prove the Well-Order­ing Prin­ci­ple in 1904 by in­tro­duc­ing a new prin­ci­ple: The Prin­ci­ple of Choice. This sparked much dis­cus­sion amongst math­e­mat­i­ci­ans. In 1908 pub­lished a pa­per con­tain­ing re­sponses to this de­bate, as well as a new for­mu­la­tion of the Ax­iom of Choice. In this year, he also pub­lished his first ver­sion of the set the­o­retic ax­ioms, known as the Zer­melo Ax­ioms of Set The­ory. Math­e­mat­i­ci­ans, Abra­ham Fraenkel and Tho­ralf Skolem im­proved this sys­tem (in­de­pen­dently of each other) into its mod­ern ver­sion, the Zer­melo Fraenkel Ax­ioms of Set The­ory.

In 1914, Felix Haus­dorff proved Haus­dorff’s para­dox. The ideas be­hind this proof were used in 1924 by Banach and Alfred Tarski to prove the more fa­mous Banach-Tarski para­dox (dis­cussed in more de­tail be­low). This lat­ter the­o­rem is of­ten quoted as ev­i­dence of the false­hood of the ax­iom of choice.

Between 1935 and 1938, Kurt Gödel proved that the Ax­iom of Choice is con­sis­tent with the rest of the ZF ax­ioms.

Fi­nally, in 1963, Paul Co­hen de­vel­oped a rev­olu­tion­ary math­e­mat­i­cal tech­nique called forc­ing, with which he proved that the ax­iom of choice could not be proven from the ZF ax­ioms (in par­tic­u­lar, that the nega­tion of AC is con­sis­tent with ZF). For this, and his proof of the con­sis­tency of the nega­tion of the Gen­er­al­ized Con­tinuum Hy­poth­e­sis from ZF, he was awarded a fields medal in 1966.

This ax­iom came to be ac­cepted in the gen­eral math­e­mat­i­cal com­mu­nity, but was re­jected by the con­struc­tive math­e­mat­i­ci­ans as be­ing fun­da­men­tally non-con­struc­tive. How­ever, it should be noted that in many forms of con­struc­tive math­e­mat­ics, there are prov­able ver­sions of the ax­iom of choice. The differ­ence is that in gen­eral in con­struc­tive math­e­mat­ics, ex­hibit­ing a set of non-empty sets (tech­ni­cally, in con­struc­tive set-the­ory, these should be ‘in­hab­ited’ sets) also amounts to ex­hibit­ing a proof that they are all non-empty, which amounts to ex­hibit­ing an el­e­ment for all of them, which amounts to ex­hibit­ing a func­tion choos­ing an el­e­ment in each. So in con­struc­tive math­e­mat­ics, to even state that you have a set of in­hab­ited sets re­quires stat­ing that you have a choice func­tion to these sets prov­ing they are all in­hab­ited.

Some ex­pla­na­tion of the his­tory of the ax­iom of choice (as well as some of its is­sues) can be found in the pa­per “100 years of Zer­melo’s ax­iom of choice: what was the prob­lem with it?” by the con­struc­tive math­e­mat­i­cian Per Martin-Löf at this web­page.

(Martin-Löf stud­ied un­der An­drey Kol­mogorov of Kol­mogorov com­plex­ity and has made con­tri­bu­tions to in­for­ma­tion the­ory, math­e­mat­i­cals­tatis­tics, and math­e­mat­i­cal<em>logic, in­clud­ing de­vel­op­ing a form of in­tu­ition­is­tic type the­ory).

A nice timeline is also sum­marised on Stan­ford En­cy­clopae­dia of Philos­o­phy.