
Re: ZFC is inconsistent
Posted:
Mar 13, 2013 1:40 PM


spermatozon@yahoo.com wrote: >
> Now we have paradoxes like > Russells paradox > BanachTarskin paradox
Though Russell's paradox and the BanachTarski paradox (note the spelling in each case) are reasonably called paradoxes, they are paradoxes in two very different senses.
In the Russell case, a contradiction arises from a few very reasonable (or at least reasonableseeming) assumptions. Therefore one or more of those assumptions must be rejected.
In the BanachTarski case, assuming the axiom of choice leads not to a contradiction, but to something unexpected. If you find that unexpected thing actually repugnant, then reject the axiom of choice. Otherwise accept that the unexpected happens. Some mathematicians do reject the axiom of choice, but I do not know if any have done so because of BanachTarski.
I suspect that a good many, on first hearing of the BanachTarski 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.
 When a true genius appears in the world, you may know him by this sign, that the dunces are all in confederacy against him. Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting

