Cantor Confusion
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From: Han de Bruijn on 18 Oct 2006 04:45 Virgil wrote: > In article <e82a9$4535d64e$82a1e228$22073(a)news1.tudelft.nl>, > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > >>Virgil wrote: >> >>>In article <7d12f$4534cca1$82a1e228$21528(a)news1.tudelft.nl>, >>> Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: >>> >>>>Virgil wrote: >>>> >>>>>In article <1161029391.305685.141910(a)b28g2000cwb.googlegroups.com>, >>>>>Han.deBruijn(a)DTO.TUDelft.NL wrote: >>>>> >>>>>>Sure, theories. Can't you talk about something else but "theories"? >>>>> >>>>>Isn't the point of physics to come up with theories? >>>> >>>>AND experiments. All physical theories are judged by experiments. >>>> >>>>>And now a physicist wants to outen them? >>> >>>Does HdB suggest that there are no standards by which to judge mental >>>theories? >> >>No. But these standards are not mental. > > How does HdB apply his Jo Blocks to mental theories? Huh? What are Jo Blocks? Han de Bruijn
From: jpalecek on 18 Oct 2006 05:02 mueckenh(a)rz.fh-augsburg.de wrote: > 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. No. Definition (from MathWorld): Constructible number: A number which can be represented by a finite number of additions, subtractions, multiplications, divisions, and finite square root extractions of integers. How do you represent the diagonal number (which is sort of a limit of a series) via FINITE number of +,-,*,/,sqrt( ) ? > Every list of reals can be shown incomplete in exactly the same way as > every list of contructible reals can be shown incomplete. No. Regards, JP
From: Sebastian Holzmann on 18 Oct 2006 05:13 mueckenh(a)rz.fh-augsburg.de <mueckenh(a)rz.fh-augsburg.de> wrote: > Further the result o the gedankenexperiment must not depend on > switching numbers. Removing balls 1, 11, 21, ... does not change the > quantities in fact, but according to set theory it does. Therefore set > theory has been contradicted. No. In mathematics, there is no such things as "balls" or "vases". You have tried to model some vaguely "real" problem in the world of mathematics. The outcome is not what you expected it to be. This is no fault of mathematics, but of the process of modelling. If you try to cook a stew using a roast recipe, you cannot blame the cookbook for giving "wrong" instructions.
From: Dik T. Winter on 18 Oct 2006 06:26 In article <1161159542.523113.318440(a)f16g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > 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). Let me ask. You *did* state that "The function f(t) = 9t is continuous, because the function 1/9t is continuous", yes? Do you not see that that reasoning is wrong? That if 1/f(t) is continuous that does not mean that f(t) itself is continuous? > > 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. Yes, so you can not talk about continuity. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 18 Oct 2006 06:29
In article <1161159675.931671.301310(a)m73g2000cwd.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > 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. You still do not understand. Any constructable list of constructable numbers gives a constructable diagonal number, so any *constructable* list of constructable is necessarily incomplete. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |