tag:blogger.com,1999:blog-3891434218564545511.post5315765266173720314..comments2024-03-27T20:37:09.185-05:00Comments on Alexander Pruss's Blog: The Banach-Tarski Paradox and the Axiom of ChoiceAlexander R Prusshttp://www.blogger.com/profile/05989277655934827117noreply@blogger.comBlogger3125tag:blogger.com,1999:blog-3891434218564545511.post-92033087711146527222013-06-29T13:12:06.038-05:002013-06-29T13:12:06.038-05:00I liked Feynman's commentary on this paradox. ...I liked Feynman's commentary on this paradox. A topologist showed him the proof, and he agreed with it, saying something along the lines of "It's fine you can do it with 'continuous spheres', since there's no such thing. The important thing is you can't do it with oranges, because oranges are made of a finite number of indivisible parts."Paul Rimmerhttps://www.blogger.com/profile/05194127784395197998noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-27169060871468772502013-06-16T09:27:18.572-05:002013-06-16T09:27:18.572-05:00I think I came across that. But a lot of these AC...I think I came across that. But a lot of these AC-ish results have no intuitive plausibility over and beyond AC. The only reason I have to think BPI or HB are true is that I have reason to think AC is true--because it just seems true--and AC entails them. If I thought AC were false, I would have no reason to think BPI and HB are true.<br /><br />Dependent Choice (DC), and Choice for families of two-element sets (AC2), may be different--they may have intuitive plausibility over and beyond that of full AC.Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-22468954433968590252013-06-16T08:31:18.361-05:002013-06-16T08:31:18.361-05:00You probably know all this, but just in case, by r...You probably know all this, but just in case, by relatively recent results of Pawlikowski (1990's), Banach-Tarski needs considerably less than full AC, the Hahn-Banach theorem is enough. HB is weaker than the Boolean Prime Ideal theorem (BPI = existence of free ultrafilters, equivalent to the axiom of choice for families of *finite sets*), itself weaker than AC, and if memory serves me right, it is independent of (countable) dependent choice.<br /><br />The Rubin & Rubin monograph (or their website) is the go-to place to settle such matters.grodrigueshttps://www.blogger.com/profile/12366931909873380710noreply@blogger.com