Cantor Confusion
in [Math]
|
Prev: Pi berechnen: Ramanujan oder BBP
Next: Group Theory
From: Virgil on 23 Nov 2006 00:24 In article <1164198054.373398.108010(a)e3g2000cwe.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > Consider > > "For all m in N there is an n in N such that D(m) in L_n" > > and > > "There is an n in N such that for all m in N D(m) in L_n." > > > > The former is trivial, the latter is false. > > > > This can more easily be seen in comparing > > "For all m in N there is n in N such that n > m" which is true > > with > > "There is an n in N such that for all m in N, n > m" which is false. > > This is false. In XF and NBG, 'Am e N, En e N, n > m' is true, 'En e N, Am e N, n > m' is false. WM has yet to produce a system which differs. > And by means of the EIT we can conclude WM "concluding' something does not make it so. > If you do not see it, then try to find two elements of the diagonal > which cannot belong to one single line. There are no two elements that can both be the last element of any one line. > If you seek long enough, > perhaps your eyes will be opened. WM's eyes are so far open that he keeps seeing things which are not there. I have no wish to have mine opened that far.
From: Virgil on 23 Nov 2006 00:29 In article <1164198305.561038.42210(a)f16g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1164143353.398396.215510(a)m73g2000cwd.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Franziska Neugebauer schrieb: > > > > > > Perhaps D. Marcus' understanding in > > > > <MPG.1fcbb0f0d873ccdd989977(a)news.rcn.com> > > > > may help you to cope with my sentence under discussion. > > > > > > > You wrote: > > > 1) "The cardinality of omega is |omega| not omega." > > > And after learning from me that this is wrong, > > > 2) "The cardinality of omega, also written as |omega|, is omega". > > > > > > Now I am interested whether or not this is a contradiction in your > > > eyes. > > > > WM is guilty of much more serious trangressions of both logic and common > > sense than is involved with whether omega and |omega| are the same. > > That is not the question any longer. It is certainly beyond question that WM is guilty of much more serious trangressions of both logic and common sense than is involved with whether omega and |omega| are the same. > The question is: Can a set theorist admit that she is in error? WM has yet to find one which is in anywhere near as much error as WM. So let us see WM admit his own many fatal errors before bringing up the purported errors of others. > > Regards, WM
From: Virgil on 23 Nov 2006 00:33 In article <1164198530.509417.22070(a)b28g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1164143519.034139.154960(a)h54g2000cwb.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Randy Poe schrieb: > > > > > > > > > > What about all contructible numbers or all words? They cannot be > > > > > mapped > > > > > on N though they are a countable set. > > > > > > > > The set of all finite words, the set of polynomials, or any > > > > countable set can be put in bijection with N. Where do you > > > > get your view that it "can't be mapped to N"? > > > > > > The list of all finite words cannot be constructed. > > > > It can be constructed inductively, which means that in ZF or NBG, it can > > be constructed. > > The natural numbers cannot be constructed inductively. Induction proofs > do not cover N, I was told. Then you were told wrong. > > > > > > > The list of all > > > constructible numbers cannot be constructed. > > > > > > The list of all naturals can be constructed inductively, which means > > that in ZF or NBG, it can be constructed. > > It can be proven inductively that all initial segments of the set of > natural numbers are finite. So, if the induction proof concerns all n > in N, then N is finite. WM claims that a quality of members of a set must be inherited by the set itself. All that WM can prove is that every MEMBER of N is finite, which says nothing about N itself.
From: Virgil on 23 Nov 2006 00:37 In article <456467B1.8040207(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 11/22/2006 12:13 PM, Han de Bruijn wrote: > > Eckard Blumschein wrote: > > > >> Presumably you got me wrong concerning what I consider the fundament of > >> modern mathematics. I consider set theory a toxic coat of modern > >> mathematics rather than its genuine fundament. [ ... snip ... ] > > > > It could be that we agree completely on this: > > > > http://groups.google.nl/group/sci.math/msg/9c2f02ed63157fcc?hl=nl& > > > > Han de Bruijn > > Having experienced non-capitalist systems and having close connections > to Muslem scientists, I shoud be able understand your suspicion. > > No. I do not share it. I consider set theory a toxic coat of modern > mathematics. This implies, set theory did not have a positive > stimulating effect on mathematics at all. It is not really the fundament > of modern mathematics. Free mathematics is merely an excuse for those > who are unable to envision usefully applicable and well matching > together mathematical structures. Economy is not merely an outdated > coat. Free market economy has and will have very drastic consequences. > > Let's either try to realistically elucidate some relationship between > politics and mathematics or strictly avoid this topic. It is definitely > true that there is no parity between the comparatively small Jewish > population and the many black people among mathematicians. Things may > possibly change towards more important mathematicians of Chinese descent > or from India. I see education at school very dangerous if it leads to > hypocrisy and blind obedience due to enforced believe in set theory. > I wonder why apparently nobody seems to be interested in the question > "Why at all do we consider sets instead of numbers?". My feeling says > apriorism is a special kind of stupidity and a very stubborn one. Those whose "feelings" tell them that they are Napolean Buonaparte have feelings as relevant to the structure of mathematics as EB's.
From: Virgil on 23 Nov 2006 00:55
In article <1164208580.112906.57510(a)b28g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <MPG.1fcd54d17c2f382e9899a4(a)news.rcn.com>, > > David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > Ralf Bader schrieb: > > > > > > > > > and Mueckenheim took that example for the main issue, ascribing > > > > > some other weird opinions to Cantor on the way. Although, > > > > > surprisingly, > > > > > Mueckenheim's simplified proof of a restricted assertion is in > > > > > principle > > > > > correct, it is also telling that he seems to consider it worth to be > > > > > published what actually is a mildly interesting exercise in basic > > > > > point > > > > > set > > > > > topology. > > > > > > > > Unfortunately you missed the clue. The simplified version has only been > > > > developed, because it can easily be extended to uncountable sets. > > > > > > > > I showed that the real plane even stays connected if an *uncountable* > > > > set of points is left out. So Cantor's theorem is useless. It doesn't > > > > show a difference between countable and uncountable, as Cantor might > > > > have tried to suggest. > > > > > > So, it turns out that you were proving a different theorem than Ralph > > > thought. That's reassuring. For a moment there, we thought you might > > > have given a correct proof of something. Care to show us your proof that > > > the plane is connected even if an uncountable number of points is > > > deleted? > > > > I will accept that it can be connected if the "right" uncountable set is > > removed, but I can think of uncountable sets of points whose removal at > > least appears to totally disconnect the set of those remaining. > > Of course, for example if you remove all points. The remaining empty set is trivially connected. > > > > For example if one removes all the points in the Cartesian plane which > > have either coordinate non-integral, what is left is about as > > disconnected as one can imagine. > > Cantor wanted to suggest that the removal of a countable set leaves the > plane connected while the removal of an uncountable set does not. Those who try to read Cantors mind often deceive themselves. And WM is foremost among those self-deceivers. > As an > example he chose the algebraic numbers (obviously in order to > distinguish them from the transcendental numbers). > > I proved that there is by far a simpler proof for the algebraic > numbers. > I proved that this proof can also be applied to the transcendental > numbers. > See the appendix of http://arxiv.org/pdf/math.GM/0306200 I have seen it. No serious mathematician would dare present such a sloppy paper for publication. Nor would any well-referreed mathematical journal publish it. The alleged proofs, at least as far as I bothered to read, are fatally flawed, and do not establish their claimed results. |