FOM: history of AC

Martin Davis martin at
Thu May 11 15:04:35 EDT 2000

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 
>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.


                           Martin Davis
                    Visiting Scholar UC Berkeley
                      Professor Emeritus, NYU
                          martin at
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