Cantor Confusion
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From: Virgil on 23 Nov 2006 00:58 In article <45646A1E.1080606(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 11/21/2006 10:24 PM, Virgil wrote: > > In article <456304B0.70705(a)et.uni-magdeburg.de>, > > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > > > >> On 11/12/2006 8:28 PM, William Hughes wrote: > >> > mueckenh(a)rz.fh-augsburg.de wrote: > >> > >> > As you stated "Goes on forever" is a property of the natural > >> > numbers. The set {1,2,3,...} is just the natural numbers. So > >> > this set must go on forever. > >> > >> 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. > > > > When one represents the members of a set as members of a sequence, as > > {1,2,3,...} does, that little ambiguity should not really mislead anyone. > > That little ambiguity? Doesn't it make a categorical difference if a set > has been a priori set for good? When one speaks of the sequence {1,2,3,...} and another speaks of the set {1,2,3,...}, the notational ambiguity should not override the words "sequence" and "set".
From: Virgil on 23 Nov 2006 01:01 In article <45646BE7.6050307(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 11/22/2006 12:45 PM, 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. > > Perhaps you correctly learned set theory. So I guess, William Hughes was > correct and Virgil this time wrong. While all sequences are sets ( as functions from N to some set of values) not all sets are sequences.
From: Virgil on 23 Nov 2006 01:03 In article <1164210944.081091.234730(a)m7g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > William Hughes schrieb: > > > > > > > No. You can develop "potential set theory". Just define > > > > an element of a potential set to be an element of any one > > > > of the arbitrary sets that can be produced. Now go through > > > > set theory and add the word "potential" in front of each > > > > occurence of the word set. There is no difference between > > > > saying "a potentially infinite set exists" and saying "an actually > > > > infinite set exists". > > > > > > A potentially infinite set has no cardinal number. It cannot be in > > > bijection with another infinite set because it does not exist > > > completely. > > > > > > > Piffle. You need to study your potential-set theory. > > > > We have defined what it means to be an element of > > a potential set. > > We never have all elements available. Speak only for yourself, WM. > > > > Therefore we can define bijections between potentially > > infinite sets. > > We can never prove hat a bijection fails, like in Cantor's argument. Speak only for yourself, WM. > > > > Therefore we can define cardinalities of potentially infinite > > sets. > > Yes, we can define that a set is finite or that it is infinite, > denoting the latter case conveniently by oo. Speak only for yourself, WM. > That's all. And way too much.
From: Virgil on 23 Nov 2006 01:10 In article <45647BF3.7090907(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 11/21/2006 10:25 PM, Virgil wrote: > > In article <4563067B.5050805(a)et.uni-magdeburg.de>, > > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > > > >> On 10/29/2006 10:22 PM, mueckenh(a)rz.fh-augsburg.de wrote: > >> > >> > Every set of natural numbers has a superset of natural numbers which is > >> > finite. Every! > >> > >> > >> I am only aware of the unique natural numbers. If one imagines them > >> altogether like a set, then this bag may also be the only lonly one. > > > > Is that "lonely"? > > I meant: There is only one natural sequence of numbers: 1, 2, 3, ... > For instance 0, 1, 2, ... or 3, 4, 5, ... or 2, 4, 6, ... are something > else. "Every" implies that there are more than one. There is only one sequence of all natural numbers taken in natural order. But there are uncountably many different sequences of natural numbers if one is not required to include all of them nor to take them in any particular order.
From: Ralf Bader on 23 Nov 2006 02:04
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. >> >> 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). This is the point I alluded to when I wrote "Mückenheim took that example for the main issue, ascribing some other weird opinions to Cantor on the way.". That Cantor wanted to suggest that the removal of an uncountable set does not leave the plane connected is nothing but your phantasy. It is trivial to give examples of uncountable sets whose removal does not disconnect the plane. E.g. the points inside a circle or rectangle, and I'm pretty sure that Cantor was aware of this. Because uncountability has nothing to do with connectedness in this way, it is remarkable that countability has - it seems to be much more plausible to me that this is how Cantor thought. > 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. Yes, and this proof even seems to be correct, but it is trivial, that is: on the level of an easy exercise in a beginner's topology course. > See the appendix of http://arxiv.org/pdf/math.GM/0306200 It is telling that you consider such a crumb of dust worth to be published. You have no idea about the current state of the art. BTW, I am not obliged to explain your errors to your satisfaction. Ralf |