Cantor Confusion
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From: David Marcus on 16 Oct 2006 15:45 stephen(a)nomail.com wrote: > Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: > > stephen(a)nomail.com wrote: > > >> Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: > >> > >>>Dik T. Winter wrote: > >> > >>>>In article <1160857746.680029.319340(a)m7g2000cwm.googlegroups.com> > >>>>Han.deBruijn(a)DTO.TUDelft.NL writes: > >>>> > Virgil schreef: > >>>>... > >>>> > > I do not object to the constraints of the mathematics of physics when > >>>> > > doing physics, but why should I be so constrained when not doing physics? > >>>> > > >>>> > Because (empirical) physics is an absolute guarantee for consistency? > >>>> > >>>>Can you prove that? > >> > >>>Is it possible to live in a (physical) world that is inconsistent? > >> > >> Perhaps. How could we know? > > > How can we know, heh? Can things in the real world be true AND false > > (: definition of inconsistency) at the same time? > > > Han de Bruijn > > What does it mean for a thing in the real world to be true? > How do you know if a thing in the real world is true? > > Consider the twin slit experiment. Is the fact that none of > the following accurately describe the situation an inconsistency? > a) the photon goes through one slit > b) the photon goes through both slits > c) the photon goes through neither slit In Bohmian Mechanics (and similar theories), the photon goes through only one slit. Physicists could learn something about logical thinking from mathematicians. -- David Marcus
From: mueckenh on 16 Oct 2006 15:45 Dik T. Winter schrieb: > In article <1160858083.663838.195390(a)b28g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > In article <1160675848.377420.163220(a)b28g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > > Dik T. Winter schrieb: > > > > > In article <1160646886.830639.308620(a)c28g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > > > ... > > > > > > If every digit position is well defined, then 0.111... is > > > > > > covered "up to every position" by the list numbers, which > > > > > > are simply the natural indizes. I claim that covering "up > > > > > > to every" implies covering "every". > > > > > > > > > > Yes, you claim. Without proof. > > > > > > > > 1) "Covering up to n" > > > > means > > > > 2)"covering n" > > > > and "covering the predecessors of n". > > > > Therefore we need not prove (2) if (1) is true. > > > > > > Yes. But you claim: (3) "covering 0.111...", not covering n. > > > > Yes. But you claim 0.111... consists merey of finite n. That is the > > error. > > Yes, I know, you detest the axiom of infinity. > > > "...classical logic was abstracted from the mathematics of finite sets > > and their subsets...Forgetful of this limited origin, one afterwards > > mistook that logic for something above and prior to all mathematics, > > and finally applied it, without justification, to the mathematics of > > infinite sets. This is the Fall and original sin of [Cantor's] set > > theory ..." (Weyl) > > Oh, perhaps. What is the relevance to mathematics? > > > > > > You state it is true for each finite > > > > > sequence, so it is also true for the infinite sequence. > > > > > > > > It is true *for every finite position*. I do not at all care how many > > > > such positions there are. The obvious covering of (2) by (1) does not > > > > depend on frequency. > > > > > > But were is the "covering up to 0.111..."? > > > > If there are actually infinitely many positions, then 0.111... is not > > completely covered, hence not defined , > > Still unproven. > > > hence not existing, hence the > > "if there are actually infinitely many positions" contradicts itself. > > Pray, first show a *valid* mathematical proof of your statement above. It is impossible to show a "valid" mathematical proof against set theory. We have discussed the vase and I would not have believed in advance that anybody could maintain arguments here like Virgil and William and others. Therefore I am sure set theory will never be contradicted --- its proponents simply will die out. Regards, WM
From: David Marcus on 16 Oct 2006 15:46 Han de Bruijn wrote: > stephen(a)nomail.com wrote: > > > Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: > > > >>Dik T. Winter wrote: > > > >>>In article <1160857746.680029.319340(a)m7g2000cwm.googlegroups.com> > >>>Han.deBruijn(a)DTO.TUDelft.NL writes: > >>> > Virgil schreef: > >>>... > >>> > > I do not object to the constraints of the mathematics of physics when > >>> > > doing physics, but why should I be so constrained when not doing physics? > >>> > > >>> > Because (empirical) physics is an absolute guarantee for consistency? > >>> > >>>Can you prove that? > > > >>Is it possible to live in a (physical) world that is inconsistent? > > > > Perhaps. How could we know? > > How can we know, heh? Can things in the real world be true AND false > (: definition of inconsistency) at the same time? That is not the definition of "inconsistency" in Mathematics. On the other hand, I don't know of any statements in Mathematics that are both true and false. If you have one, please state it. -- David Marcus
From: Virgil on 16 Oct 2006 15:47 In article <1161006980.742157.216800(a)k70g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > > please give a statement using the language of ZFC. > > All mathematics is built on ZFC and derived from it. I think that this is an overstatement, but I am not aware of any mathematics, outside of the constructionists', that is in conflict with ZFC. > From ZFC we can > derive the result of the vase-balls problem too. Hence we can criticize > ZFC if its results are contradictory. Mathematics based on ZFC says > that the vase at noon is empty or not empty (by the way, what was your > result V(12)?). > > But whatever your result may be: Both statemets are in contradiction > with the foundations of ZFC. If the vase is empty then mathematics of > limits as derived from ZFC is wrong. The standard mathematics of limits is only wrong for those who claim all functions are continuous at all points. "Mueckenh" in another post has claimed things that imply that that 1/t must be continuous at t = 0, which is quite false in standard mathematics.
From: Han.deBruijn on 16 Oct 2006 15:47
stephen(a)nomail.com schreef: > Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: > > stephen(a)nomail.com wrote: > > >> Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: > >> > >>>Dik T. Winter wrote: > >> > >>>>In article <1160857746.680029.319340(a)m7g2000cwm.googlegroups.com> > >>>>Han.deBruijn(a)DTO.TUDelft.NL writes: > >>>> > Virgil schreef: > >>>>... > >>>> > > I do not object to the constraints of the mathematics of physics when > >>>> > > doing physics, but why should I be so constrained when not doing physics? > >>>> > > >>>> > Because (empirical) physics is an absolute guarantee for consistency? > >>>> > >>>>Can you prove that? > >> > >>>Is it possible to live in a (physical) world that is inconsistent? > >> > >> Perhaps. How could we know? > > > How can we know, heh? Can things in the real world be true AND false > > (: definition of inconsistency) at the same time? > > What does it mean for a thing in the real world to be true? > How do you know if a thing in the real world is true? Start studying something else but mathematics. A _science_, perhaps? > Consider the twin slit experiment. Is the fact that none of > the following accurately describe the situation an inconsistency? > a) the photon goes through one slit > b) the photon goes through both slits > c) the photon goes through neither slit Of course not. Is the "fact" that heat is phlogiston an inconsistency? Han de Bruijn |