On 13 Mrz., 18:40, Frederick Williams <freddywilli...@btinternet.com> wrote: > Some mathematicians do reject the > axiom of choice, but I do not know if any have done so because of > Banach-Tarski.
Every mathematician does so! Notwithstanding any intermediate abracadabra the result is that v = 2V and that is wrong in mathematics. > > I suspect that a good many, on first hearing of the Banach-Tarski > paradox, thought 'Wow! How about that! Isn't mathematics fun?' And > perhaps: 'So what happens if Choice is false? Do any loopy things happen > in that case?' Meanwhile, note that if set theory is consistent, one > may safely assume either Choice or its negation.
ZF is not consistent if this result is correct.
Proof: With axiom of choice it is possible to choose uncountably many elements out of uncountably many sets. But non-material elements cannot be chosen other than by defining them. Further it is undisputed that only countably many definitions are available and that "uncountable" is larger than "countable".
These things are well-known to every mathematician who deserves this name. We have no choice.