Cantor Confusion
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From: William Hughes on 22 Nov 2006 08:23 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. > > > > > > > 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. Piffle. The two statments i: P(n) is true for every n an element of N. ii: P(N) is true are not the same (trivial example P(x) is true iff x is an element of N). Induction can be used to prove statements of the form i. (eg all elements of N are finite). Induction cannot be used to prove statements of the form ii (e.g. N is finite). - William Hughes
From: Franziska Neugebauer on 22 Nov 2006 08:46 mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: >> In article <456304B0.70705(a)et.uni-magdeburg.de>, >> Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> > Being admittedly not very familiar with set theory, I nonetheless >> > wonder if sets are considered like something going on forever. >> >> Sequences do, sets do not. >> > Sequences are sets. Is the "identical sequence" n |-> n a set, too? F. N. -- xyz
From: Eckard Blumschein on 22 Nov 2006 10:07 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. Kind regards, Eckard
From: Eckard Blumschein on 22 Nov 2006 10:13 On 11/22/2006 12:16 AM, David Marcus wrote: > It is hard to be always wrong. In what, according to your opinion, WM is not wrong?
From: mueckenh on 22 Nov 2006 10:16
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. > > 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. 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 Regards, WM |