On 17 Mrz., 07:49, fom <fomJ...@nyms.net> wrote: > On 3/16/2013 1:25 PM, WM wrote: > > > On 16 Mrz., 18:17, fom <fomJ...@nyms.net> wrote: > > > An additional remark: > > >> No. Zermelo's AC requires that one name can be written > >> with certainty. > > > This statement is not Zermelo's original statement. It can be proven > > to hold, iff it was possible to choose, in practice, one element from > > every subset of T. If this was do-able. > > Well, the critical investigation of the > statement during the twentieth century > resulted in taking it as an axiom. > > Its provability is not the criterion > by which it is to be understood. > > > > > There have been many mathematicians criticizing Zermelo's axiom > > (Borel, Peano, Poincaré and others). Zermelo discusses a lot of > > objections in another 1908 paper. And the most amazing fact is, that > > at that time none of the arguments aims at the fact, that there are > > only countably many choices possible by theoretical reasons. > > > Zermelo agrees that the AC is not provable. He did not know, at that > > time, that it is disprovable by theoretical mathematics. > > Disprovable by belief, perhaps.
Zermelo created the axiom of choice because it was obvious to him that is is correct, i.e., that his choice can be done, at least in principle. Then he went on and "proved" from this axiom the well- ordering theorem. If he had known that the axiom of choice can be disproved by proving that at most countably many choiced can be executed, even in principle, why should he have used it? With same counterfactuality he could have inveted the axiom: Every set has a well-ordering.
So from historical context it is clear, that he invented AC because he believed in its possibility. This has later been disproved.
To take the sentence "the Cartesian product of non-empty sets is non- empty" as an argument against the fact that the logically equvalent choice cannot be done on uncountable sets witnesses a very poor understanding of mathematics. With the same justification you could state "there is a second prime triple beyond (3, 5, 7,)" as an axiom and refuse the counter proof.