Axiom of Choice: Introduction

“The Ax­iom of Choice is nec­es­sary to se­lect a set from an in­finite num­ber of pairs of socks, but not an in­finite num­ber of pairs of shoes.” — Ber­trand Rus­sell, In­tro­duc­tion to math­e­mat­i­cal philosophy

“Tarski told me the fol­low­ing story. He tried to pub­lish his the­o­rem [the equiv­alence be­tween the Ax­iom of Choice and the state­ment ‘ev­ery in­finite set A has the same car­di­nal­ity as AxA’] in the Comptes Ren­dus Acad. Sci. Paris but Fréchet and Lebesgue re­fused to pre­sent it. Fréchet wrote that an im­pli­ca­tion be­tween two well known propo­si­tions is not a new re­sult. Lebesgue wrote that an im­pli­ca­tion be­tween two false propo­si­tions is of no in­ter­est. And Tarski said that af­ter this mis­ad­ven­ture he never again tried to pub­lish in the Comptes Ren­dus.”

  • Jan My­ciel­ski, A Sys­tem of Ax­ioms of Set The­ory for the Rationalists

Obli­ga­tory Introduction

The Ax­iom of Choice, the most con­tro­ver­sial ax­iom of the 20th Cen­tury.

The ax­iom states that a cer­tain kind of func­tion, called a `choice’ func­tion, always ex­ists. It is called a choice func­tion, be­cause, given a col­lec­tion of non-empty sets, the func­tion ‘chooses’ a sin­gle el­e­ment from each of the sets. It is a pow­er­ful and use­ful ax­iom, as­sert­ing the ex­is­tence of use­ful math­e­mat­i­cal struc­tures (such as bases for vec­tor spaces of ar­bi­trary di­men­sion, and ul­tra­prod­ucts). It is a gen­er­ally ac­cepted ax­iom, and is in wide use by math­e­mat­i­ci­ans. In fact, ac­cord­ing to Elliott Men­del­son in In­tro­duc­tion to Math­e­mat­i­cal Logic (1964) “The sta­tus of the Ax­iom of Choice has be­come less con­tro­ver­sial in re­cent years. To most math­e­mat­i­ci­ans it seems quite plau­si­ble and it has so many im­por­tant ap­pli­ca­tions in prac­ti­cally all branches of math­e­mat­ics that not to ac­cept it would seem to be a wilful hob­bling of the prac­tic­ing math­e­mat­i­cian. ”

Nev­er­less, be­ing an ax­iom, it can­not be proven and must in­stead be as­sumed. In par­tic­u­lar, it is an ax­iom of set the­ory and it is not prov­able from the other ax­ioms (the Zer­melo-Fraenkel ax­ioms of Set The­ory). In fact many math­e­mat­i­ci­ans, in par­tic­u­lar con­struc­tive math­e­mat­i­ci­ans, re­ject the ax­iom, stat­ing that it does not cap­ture a ‘real’ or ‘phys­i­cal’ prop­erty, but is in­stead just a math­e­mat­i­cal odd­ity, an arte­fact of the math­e­mat­ics used to ap­prox­i­mate re­al­ity, rather than re­al­ity it­self. In the words of the LessWrong com­mu­nity: the con­struc­tive math­e­mat­i­ci­ans would claim it is a state­ment about the map, and not the ter­ri­tory.

His­tor­i­cally, the ax­iom has ex­pe­rienced much con­tro­versy. Be­fore it was shown to be in­de­pen­dent of the other ax­ioms, it was be­lieved ei­ther to fol­low from them (i.e., be ‘True’) or lead to a con­tra­dic­tion (i.e., be ‘False’). Its in­de­pen­dence from the other ax­ioms was, in fact, a very sur­pris­ing re­sult at the time.