On Fri, 14 Jul 2006 22:00:26 -0600, Virgil <email@example.com> wrote:
>In article <firstname.lastname@example.org>, > Lester Zick <DontBother@nowhere.net> wrote: > >> On Fri, 14 Jul 2006 18:41:42 -0300, "Shmuel (Seymour J.) Metz" >> <email@example.com> wrote: >> >> >In <firstname.lastname@example.org>, on 07/13/2006 >> > at 09:42 AM, email@example.com said: >> > >> >>However typical set theory definitions which run >> >>along the lines of a "set of all points which . . ." do turn out to >> >>be a joke because they invariably rely on various geometric >> >>assumptions regarding figures such as planes, lines, etc. >> > >> >Nonsense. Set Theory neither relies on geometric assumptions nor does >> >it talk about figures. Now, you can certainly apply Set Theory to >> >Geometry, but then what you have are geometric definitions, not Set >> >Theory definitions. >> >> Are you an idiot? If you define a circle as the "set of all points >> equidistant from any point" you are not using a set theory definition >> for a geometric figure? > >As no set theory of my acquaintance requires any definition of >distances between points, I would say that any definition which does >refer to distances is certainly not purely set theoretic, and is at >least partly geometric.
Of course it's geometric. That's the whole problem with "set of all points . . ." definitions in set theory. They just describe what is already known implicitly from other geometric assumptions and reasoning. They don't originate with set theory so claiming set theory in that form as a universal basis for all math is absurd. Yet set theorists go right ahead and proclaim such things all the time.
The other problem is that whatever the foundation of set theory may be every constituent definition is a critical part of the edifice. A chain is no stronger than its weakest link. What is it that proves "set of all points . . ." definitions true? I've already gone round and round with mathematikers who conflate transcendental numbers with irrational numbers for no better reason that both are described in rational terms with non repeating fractional expansions when in point of fact one describes curves and the other straight lines.