Cantor Confusion
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From: Virgil on 14 Oct 2006 18:20 In article <1160856895.115824.134080(a)b28g2000cwb.googlegroups.com>, Han.deBruijn(a)DTO.TUDelft.NL wrote: > stephen(a)nomail.com schreef: > > > Dik T. Winter <Dik.Winter(a)cwi.nl> wrote: > > > In article <a1364$452f53c9$82a1e228$2828(a)news2.tudelft.nl> > > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> writes: > > > ... > > > > I don't know about Mueckenheim's opinion in these matters, but an old > > > > Greek philosopher - Heraclitos - has said: "panta rei kai ouden menei" > > > > (everything flows and nothing remains the same). And I agree with > > > > that. > > > > > Pray warn me when 2 has changed sufficiently to be the square of a > > > rational. > > > I would not like to miss that moment. > > > > The day the circle is squared cannot be to far behind. > > Come on, guys! You all know that, in the world of approximations, > 2 _is_ the square of a rational and the circle _is_ squared. What happens in that world is not mathematically relevant.
From: David Marcus on 14 Oct 2006 18:20 mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > Virgil wrote: > > > In article <1160650371.242557.284430(a)h48g2000cwc.googlegroups.com>, > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > Dik T. Winter schrieb: > > > > > In article <1160578706.221013.145300(a)c28g2000cwb.googlegroups.com> > > > > > mueckenh(a)rz.fh-augsburg.de writes: > > > > > > Dik T. Winter schrieb: > > > > > > > So the definition I gave for a limit of a sequence of sets you agree > > > > > > > with? Or not? I am seriously confused. With the definition I gave, > > > > > > > lim{n = 1 .. oo} {n + 1, ..., 10n} = {}. > > > > > > > > > > > > Sorry, I don't understand your definition. > > > > > > > > > > What part of the definition do you not understand? I will repeat it here: > > > > > > What *might* be a sensible definition of a limit for a sequence of sets > > > > > > of > > > > > > naturals is, that (given each A_n is a set of naturals), the limit > > > > > > lim{n = 1 ... oo} A_n = A > > > > > > exists if and only if for every p in N, there is an n0, such that either > > > > > > (1) p in A_n for n > n0 > > > > > > or > > > > > > (2) p !in A_n for n > n0. > > > > > > In the first case p is in A, in the second case p !in A. > > > > > Pray, read the complete definition before you give comments. > > > > > > > > I do not believe that definition (2) is of any relevance. > > > > Cantor uses Lim{n} n = omega witout much ado. > > > > omega is simply defined as the limit of the increasing natural numbers. > > > > In his first paper he uses even Wallis' symbol oo. What should there > > > > require a definition, if all natural did exist? > > > > This is what I use and write in modern form: Lim {n-->oo} {1,2,3,..,n} > > > > = N. > > > > > > Where in ZFC or NBG does "Mueckenh"find any definition of any such limit? > > > > Or, in what book does mueckenh find this? > > My name is Wolfgang Mueckenheim, briefly known as WM. "mueckenh" is > only used by Virgil due to his bad education or behaviour. The book I > recommend is the collected works of Cantor. But sometimes I dare to > write things not yet included in books. You might try putting your full name in your posts, e.g., in the "From" line. Since Cantor predates axiomatic set theory, he is not a reliable source for modern Mathematics. If you wish to only discuss history or the development of Mathematics, then you should say that is what you are doing and carefully define any terms you use, since the meanings may have changed over the last century. -- David Marcus
From: Virgil on 14 Oct 2006 18:23 In article <1160857302.229278.80800(a)f16g2000cwb.googlegroups.com>, Han.deBruijn(a)DTO.TUDelft.NL wrote: > Virgil schreef: > > > In article <ecef7$452f4eb8$82a1e228$2523(a)news2.tudelft.nl>, > > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > > > > > Virgil wrote: > > > > > > > In article <1160648741.707624.62340(a)m7g2000cwm.googlegroups.com>, > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > > >>Apply your knowledge to the balls of the vase. > > > > > > > > Which knowledge tells me that at noon each and every ball has been > > > > removed from the vase. > > > > > > Not AT noon but BEFORE noon. True then. And ten other balls have been > > > inserted. > > > > All of which will be removed before noon, as will any others inserted. > > > > > > > those times do not include noon or go past noon. > > > > > > True. Because those times are a FAKE. > > > > > > > You are assuming properties not given. > > > > > > No. YOU are assuming properties not given: _you_ are assuming that your > > > fake parameter falsely called time has the properties of physical time, > > > such that it can pass through noon, which it cannot. > > > > I am assuming time is a real valued variable which can take any real > > value including 0 and positive values. > > Time is much more complicated than this. That's why your time is fake. It is mathematical time, not physical time, and since there are so many different physical "times", that gives it a great advantage.
From: Virgil on 14 Oct 2006 18:26 In article <1160857324.535391.81550(a)f16g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > By this arguing there can never be any contradictions in set theory. > Any theory which cannot be falsified is a religion. Then "Mueckenh" is a devout worshiper of his own religion. And whether ZF or NBG CAN be falsified is unknown, and one hopes unknowable, but they have not been falsified as yet, though many have tried.
From: Virgil on 14 Oct 2006 18:28
In article <1160857629.657335.96340(a)f16g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > MoeBlee schrieb: > > > > > > > Han de Bruijn wrote: > > > > > Set Theory is simply not very useful. The main problem being that > > > > > finite > > > > > sets in your axiom system are STATIC. They can not grow. > > > > > > > > Set theory provides for capturing the notion of mathematical growth. > > > > Sets don't grow, but growth is expressible in set theory. If there is a > > > > mathematical notion that set theory cannot express, then please say > > > > what it is. > > > > > > Obviously the notion of "rational relation" as used in the binary tree > > > cannot be expressed by mathematical notion: > > > Consider the binary tree which has (no finite paths but only) infinite > > > paths representing the real numbers between 0 and 1. The edges (like a, > > > b, and c below) connect the nodes, i.e., the binary digits. The set of > > > edges is countable, because we can enumerate them > > > > > > 0. > > > /a \ > > > 0 1 > > > /b \c / \ > > > 0 1 0 1 > > > ............. > > > > > > Now we set up a relation between paths and edges. Relate edge a to all > > > paths which begin with 0.0. Relate edge b to all paths which begin with > > > 0.00 and relate edge c to all paths which begin with 0.01. Half of edge > > > a is inherited by all paths which begin with 0.00, the other half of > > > edge a is inherited by all paths which begin with 0.01. > > > > So each finite path of length N is related to > > > > 1 + 1/2 +1/4 + ... + 1/2^N > > > > edges > > > > > Continuing in this manner in infinity, > > > > > > we get a limit which may or may not be related to anything. > > This geometric sum is defined if any infinity is defined. > > > > > we see that every single infinite path is > > > related to 1 + 1/2 + 1/ 4 + ... = 2 edges, which are not related to any > > > other path. > > > > No, the statement that what holds for finite paths also > > holds for infinite paths needs proof. > > Your provide none. > > Who provides, if not I? No one has provided it at all. |