Cantor Confusion
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From: mueckenh on 18 Oct 2006 04:13 Virgil schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > > I warned you that this point is new: The edges are split in shares of > > 1/2, 1/4, and so on. But when fractions were introduced in mathematics, > > most people may have had the same problems as you today. I am sure you > > can understand it from the written text above. (Many others have > > already understood it.) > > Name one. William Hughes. Regards, WM
From: mueckenh on 18 Oct 2006 04:15 Alan Morgan schrieb: > In article <1161027684.800946.299570(a)m73g2000cwd.googlegroups.com>, > <mueckenh(a)rz.fh-augsburg.de> wrote: > > > >Alan Morgan schrieb: > > > >> In article <1160944143.122919.243860(a)h48g2000cwc.googlegroups.com>, > >> <mueckenh(a)rz.fh-augsburg.de> wrote: > >> > > >> >William Hughes schrieb: > >> > > >> >> However, you wish to do more. You want to show > >> >> that claiming "N does not have an upper bound and > >> >> N exists as a complete set" leads to a contradiction.] > >> >> > >> >That is true too. And it is easy to see: If we define Lim [n-->oo] > >> >{1,2,3,...,n} = N, then we can see it easily: > >> > > >> >For all n e N we have {2,4,6,...,2n} contains larger natural numbers > >> >than |{2,4,6,...,2n}| = n. > >> > >> Agreed. > > > >Do you know what "for all n e N" means? There are *not any* further > >natural numbers. There is no chance to increase the cardinal number. > > I'm not the one claiming that aleph_0 is a natural number. You are. > At least, I think you are. > > >> >There is no larger natural number than aleph_0 = |{2,4,6,...}|. > >> >Contradiction, because there are only natural numbers in {2,4,6,...}. > >> > >> That would be a contradiction only if Aleph0 e N, but it isn't. Your > >> statement above is true for finite n. Showing that it isn't true for > >> infinite n (or in the limit or whatever terminology you choose to use) > >> does not produce a contradiction. > > > >There are no infinite n, whatever terminology you coose. And aleph_0 is > >considered larger than any finite n. That is simply impossible. > > Impossible? That's practically the *definition* of aleph_0. > > You've stated something that is true for all natural numbers. It is > false for aleph_0. That would be a problem if aleph_0 were a natural > number. It isn't. So, no problem. > > >There must be finite even numbers X, larger than 2n, which complete the > >set > >{2,4,6,..., 2n, X }. > > I have no idea what you mean here. What is "complete" the set? For > any fininte n there are even numbers > 2n. What is your point? > Let it be. Consider only the balls in the vase. It is the same problem but easier to understand. Regards, WM
From: mueckenh on 18 Oct 2006 04:19 Dik T. Winter schrieb: > In article <1161007554.513186.56640(a)i42g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > ... > > > > The function f(t) = 9t is continuous, because the function 1/9t is > > > > continuous. > > > > > > Yes, but that is not the number of balls in the vase. > > > > > For the t-th transaction 9t is the number of balls in the vase. > > Let me clarify. WM apparently asserts that if a function g(x) is continuous > at some point, so is 1/g(x). That is (obviously) false. sin(x) is > continuous at x = 0, while 1/sin(x) is not. > I used but a simple definition of the improper limit oo of the function x which is given by the fact that the function 1/x has the proper limit 0 (fo x --> oo in both cases). > Moreover, when we let t go to infinity, 1/9t is *not* continuous at > infinity (whatever that may mean). We can only define the limit, > not the function value. We can only define limits in *all* cases concerning the infinite. Nothing else is possible. Regards WM
From: mueckenh on 18 Oct 2006 04:21 Dik T. Winter schrieb: > In article <1161008572.469763.93200(a)f16g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > jpalecek(a)web.de schrieb: > ... > > > The fact that you cannot compute a list of all computable reals does > > > not mean that there is no list of all computable numbers. There is one, > > > and it is not computable. > > > > > The fact that you cannot compute a list of all reals does not mean that > > there is no list of all reals. There is one, but it is not possible to > > publish this list. > > You are seriously wrong. You should have noted that this was an ironic reply. But in order to avoid machines and undecidabilities: The set of constructible numbers is countable. Any diagonal number is a constructed and hence constructible number. Every list of reals can be shown incomplete in exactly the same way as every list of contructible reals can be shown incomplete. Regards, WM
From: Han de Bruijn on 18 Oct 2006 04:44
Virgil wrote: > In article <a745$4535d5f4$82a1e228$22073(a)news1.tudelft.nl>, > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > >>Virgil wrote: >> >>>In article <78269$4534cb2a$82a1e228$21528(a)news1.tudelft.nl>, >>> Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: >>> >>>>_part_ of the question: do INFINITIES exist or not. Are they true or are >>>>they false? And IMO _that_ can be decided _now_, without rocket science. >>> >>>Do physical infinities exist? Probably not. >> >>Right! >> >>>Do physical triangles exist? Almost certainly not. >> >>Wrong! >> >>>Therefore, according to HdB's thesis, we should abolish trigonometry. >> >>The difference between (actual) infinities and (idealized) triangles is >>that the former can NOT be coarsened to something in physics that can be >>measured eventually, while the latter CAN be coarsened to something in >>physics that IS measurable. > > The idealized infiniteness of ideal triangle sizes are coarsened to the > finitely many things in physics that can be measured. Right! Thus the ideal triangle sizes can be materialized (as I say) into something from the real world. Now do the same with aleph_0. And tell us when you have been successful. Han de Bruijn |