Cantor Confusion
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From: Virgil on 13 Dec 2006 14:55 In article <1166010770.689924.190700(a)79g2000cws.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Tony Orlow schrieb: > > > > That's fine. If you don't like irrationals, ignore them. > > Oh I would certainly love them but I never met one. I met only names, > different lengths in squares and circles, cubes and spheres, Schall und > Rauch. > > >There are an > > actually infinite number of rationals in a unit interval. > > How "are" they? Thriving! Along with their irrational compatriots! Thank you for asking > > > Physics used to be more continuous, but atoms and quantum effects have > > been discovered. Time and space may even be discrete. Mathematics can > > reflect that, or treat things as continuous. Mathematics can do both. > > I don't think we've > > determined for sure that nothing is continuous. Do you? > > What *in principle* can't be measured, is not existing. There are all sorts of things in most people's worlds that cannot be measured: beauty, love, faithfulness, etc. Apparently WM's world lacks them.
From: Virgil on 13 Dec 2006 14:57 In article <1166011032.914983.204230(a)79g2000cws.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Han de Bruijn schrieb: > > > William Hughes wrote: > > > > > Tony Orlow wrote: > > > > > >>Well, the proof is simple. Any finite number of subdivisions of any > > >>finite interval will only identify a finite number of real midpoints in > > >>that interval, between any two of which will remain more real midpoints. > > >>Therefore, there are more than any finite number of real points in the > > >>interval. > > > > > > This just shows that the number of real points is unbounded. > > > It does not show it is infinite (unless of course you use the > > > fact that any unbounded set of natural numbers is infinite). > > > > Isn't unbounded the same as infinite, i.e. = not finite = unlimited = > > without a limit? > > Unbounded is potentially infinite but it is not necessarily actually > infinite. > AS in honest mathematics, the one implies the other, in honest math one has neither or one has both. In ZFC and NBG, both.
From: Franziska Neugebauer on 13 Dec 2006 15:04 mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: >> mueckenh(a)rz.fh-augsburg.de wrote: >> > Franziska Neugebauer schrieb: >> >> mueckenh(a)rz.fh-augsburg.de wrote: >> >> > Franziska Neugebauer schrieb: >> >> >> mueckenh(a)rz.fh-augsburg.de wrote: >> >> >> > Dik T. Winter schrieb: >> >> >> [...] >> >> >> >> Again you have provided neither a definition of "number", >> >> >> >> nor of "grow". >> >> >> >> Are you unable to do so? In common parlance, but that is >> >> >> >> not mathematics. In mathematics functions can grow in >> >> >> >> relation to their argument, but not the entities they >> >> >> >> denote. >> >> >> > >> >> >> > Functions cannot grow, according to modern mathematics. >> >> >> >> >> >> Wrong. I have provided a definition: >> >> >> >> >> >> ,----[ >> >> >> <45742128$0$97220$892e7fe2(a)authen.yellow.readfreenews.net> ] >> >> >> | Definition: A function f: A |-> B grows iff there exist a1 < >> >> >> | a2 of dom(f) and f(a1) < f(a2). We use the abbreviation "f >> >> >> | grows" for of "the function f grows". >> >> >> `---- >> >> > >> >> > The natural number n is a particular set of n elements. >> >> > If the number n can take a value n_1 and can take a value n_2 >> >> > with n_1 =/= n_2 then the number n can vary. >> >> >> >> How is that related to your sentence that "functions cannot grow"? >> > >> > It is the usual set theoretic view and as such in direct >> > opposistion to my view. >> >> Can't see any support for your claim, that "functions cannot grow". > > In *set theory* a set does not grow. In set theory a function is a set. In set theory a function cannot grow. _IN_ Z set theory there is no _notion_ of /growing/ in the first place. Without _defining_ what "to grow" shall mean sets and function do neither grow nor not grow. Without a definition of "to grow" it is just _meaningless_ to speak of "growing" sets or of sets/functions which "can" or "cannot grow". Hence my definition, which enables us to meaningfully speak of "growing" functions (but not of sets in general) in order to model in particular "political" functions like the number of states of the EC at given time. > In my view a function can grow. Without a proper _definition_ of "to grow" it is meaningless to Z set theoretically write about function which "can grow". Which _definition_ do you have in mind? > A function can be abbreviated as number variable or as variable > number. Could you explain, what "to abbreviate" means in the present context? F. N. -- xyz
From: Virgil on 13 Dec 2006 15:07 In article <1166025551.974752.93070(a)80g2000cwy.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > Unbounded is potentially infinite but it is not necessarily actually > > > infinite. > > > > > > > Please give an example of a set you consider to be > > actually infinite. > > > > Impossile. There is no actual infinty as is shown by our discusson abou > the IET. So I had to lie if being forced to name an actually infinite > set. > > Regards, WM In ZFC and NBG, every inductive set,and the axioms require that there be some, is actually infinite. What is the axiom system that WM espouses? I'll bet it includes a statement that there are no "actually infinite" sets, as one cannot prove that without assuming it.
From: Virgil on 13 Dec 2006 15:07
In article <1166011440.201958.55080(a)73g2000cwn.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Why isn't it sufficient to collect the shares of two edegs for every > path? That shows that there are not more paths than edges. Can you > explain your objection to factions? But that is not what WM does. What WM did was to have an infinite sequence of edges (matched with an infinite sequence of rationals) for each path. But the "number" of such infinite sequences is itself uncountable, so that there is no bijection between the set of edges and the set of such sequences of edges. |