Cantor Confusion
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From: Virgil on 7 Dec 2006 15:32 In article <4577FC31.9000700(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/7/2006 4:38 AM, Virgil wrote: > > In article <4576DF19.7070005(a)et.uni-magdeburg.de>, > > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > > > >> On 12/5/2006 11:01 PM, Virgil wrote: > >> > In article <4575B727.6070006(a)et.uni-magdeburg.de>, > >> > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> > > >> >> On 12/5/2006 3:36 PM, Georg Kreyerhoff wrote: > >> >> > Eckard Blumschein schrieb: > >> >> > > >> >> >> Do not confuse Cantor's virtue of belief in god given sets with my > >> >> >> power > >> >> >> of abstraction. > >> >> > > >> >> > Your power of abstraction is nonexistant. You're not even able to > >> >> > distinguish > >> >> > between representations of numbers and the abstract concept of > >> >> > numbers. > >> >> > > >> >> > Georg > >> >> > >> >> Really? > >> > > >> > Really! > >> > >> No. Maybe some mistakable wording gave rise for a false impression. > >> More likely, you and Georg do not understand that there is no number > >> outside an appropriate representation. > > > > Who gave you the power to dictate what a "number" is, EB? > > I just realized how brutally Cantor raped the old and correct notion of > number and feel safe with Gauss all the others. Where does one find that "old and correct notion of number"? And why does "old" make it any more correct than "new". There are lots of new "numbers", that are just as correct as old ones. > > > > > A "number" in mathematics is what the majority of mathematicians agree > > it is, regardless of what anti-mathematicians like EB try to dictate. > > The majority of really important mathematics perhaps lived before Cantor > or did not take issue towards his at best somewhat strange and > absolutely unfounded violation of the notion number. The vast majority of extant mathematics was created since Cantor's death. No one right now can say with any certainty how important that new math will untimately prove to be. So that EB is claiming what he cannot demonstrate. > > > >> The abstract concept of numbers > >> must not be misused as to declare rationals and embeded rationals > >> likewise existent. > > > > The "abstract concept of number" can be used in any way that > > mathematicians choose to use it, > > If there was really general agreement among mathematicians, then there > would be an acceptable printed definition. Since Cantor's definition of > set has been declared untennable without substitute, By whom? > I do not expect a > clean definition of number either.
From: Virgil on 7 Dec 2006 15:37 In article <1165491428.331603.244610(a)l12g2000cwl.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > > > But WM's construction requires an infinite sequence of branches for each > > object to be paired with a path. And we already know that there are > > uncountably many such infinite sequences constructable from countably > > many branches(or nodes), so WM's pairing scheme is phony. > > I will try to make it simpler for you: There is no way you can make such a a falsehood simple enough to appear true. WM finds a bijection between infinite sequences of branches and paths, which is not surprising because each path IS an infinite sequence of branches. Why he maintains this this somehow bijects the individual branches with paths, is not clear.
From: Virgil on 7 Dec 2006 15:38 In article <1165492505.490385.276090(a)73g2000cwn.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <4576BC30.2000101(a)et.uni-magdeburg.de>, > > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > > > > > > > It is evident that EB understands "infinity" even less that Cantor. > > That is a hard attack! To understand the infinite even less than > Cantor??? > > I don't know anybody who understood the infinite better than Cantor, > not even approximately as well. Then my comment does not disparage EB.
From: Virgil on 7 Dec 2006 15:44 In article <1165492756.322548.255040(a)j72g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > In article <1165421463.339178.48680(a)j44g2000cwa.googlegroups.com> > > mueckenh(a)rz.fh-augsburg.de writes: > > > > > > William Hughes schrieb: > > ... > > > > Consider A= [0,1), the set of real > > > > numbers in greater than or equal to 0 and less than 1. > > > > This set is totally ordered. This set is > > > > composed of finite elements. We have the true statement > > > > > > > > For every element r in A, there exists an element > > > > s in A such that r < s. > > > > > > > > However, if we simple reverse the quantifiers we get > > > > the false statement > > > > > > This set is not finite. > > > > The set in the quantifiers you are using is not finite either. The > > quantifier are not over a single line, but over the set of natural > > numbers. > > For finite natural numbers, we have finite lines only. It is not the > question of a single line. Every line is finite. Therefore there is no > line where quantifier reversal could not be applied. It is not the contents of a single line but the set of all lines that is being quantified, so that the finiteness of lines is irrelevant. Consider the real interval [0,oo) = { x in R : x >= 0} and the set of all finite ordinals O = {0,1,2,...} and let N = [0,oo) intersect O. then you are quantifying over this infinite N
From: Virgil on 7 Dec 2006 15:51
In article <1165494365.893586.202650(a)f1g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > It appears that the negation of "the set of all even numbers has > > cardinal greater than any even number" would have to be " there is an > > even natural number as great as the cardinality of the set of all even > > natural numbers." > > Of course. Even greater even numbers must exist. And therefore, there > is no set of all even numbers. there is also the alternative that the "number" of even inetegers is larger than any even integer. It is certainly true for "even" ordinals. > > > > > > By their fruits ye shall know them. Mathematics based on infinite sets > > > > has produced useful and even indispensible results. Try getting modern > > > > physics without it. > > > > > > For applications the usual way to interpret things is sufficient. > > > > Absent accepting the infinitenesses of analysis, a lot of analysis goes > > missing. > > The infinity of standard analysis is only potential and will remain > after ZF will have perished. Such faith in the illogical is charming, but doomed. ZF, or NBG, or something like them, will be around long after we are all gone. |