FOM: history of AC

[Arnon Avron]

> Arnon Avron wrote (referring to Friedman's comparison of alrge cardianl 
> axioms with AC):
> 
> 
> 
> >I believe that the comparison with AC here is inappropriate. The big 
> >difference
> >is that AC has implicitly been used by mathematicians (at least in a weak
> >form, like DC) long before it was explicitly formulated by Zermelo and Russel.
> 
> This history wasn't so smooth. See Moore's excellent monograph "Zermelo's 
> Axiom of Choice". Peano's first reaction to AC was that it was evidently 
> false. The French "finitists" (Borel, Lebesgue, de la Valle Poussin), made 
> a big point of avoiding it. De la Valle Poussin's monograph on Lebesgue 
> integration (from which I learned the subject) in its second edition even 
> refused to state the theorem that the union of a countable set of countable 
> sets is countable without including the proviso that the implied mappings 
> are explicitly given.

I know this (and I read Moore's monograph). But all these discussions
started AFTER the explicit formulation. The fact remains that the axiom
has been used before that implicitly, and even by those who later objected
to it, like Lebesgue (according to Moore's book!).

By the way, personally even today I try to avoid the axiom of choice 
whenever I can, and explicitly give note when I do use it (usually in the 
form of Zorn Lemma).


Arnon Avron
Computer Science Department
Tel-Aviv University