Cantor Confusion
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From: Virgil on 6 Dec 2006 22:38 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? A "number" in mathematics is what the majority of mathematicians agree it is, regardless of what anti-mathematicians like EB try to dictate. > 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, and anti-mathematical pipsqueeks like EB have no power to dictate what mathematicians are or are not allowed to do.
From: Dik T. Winter on 6 Dec 2006 22:38 In article <virgil-6BADE6.20103906122006(a)comcast.dca.giganews.com> Virgil <virgil(a)comcast.net> writes: > In article <4576D4C7.4070207(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: .... > > Dedekind as well as Cantor started from this idea. Cantor even > > fabricated the notion cardinality in order to quantify the putative > > difference in size. Considering the rationals a subset of the reals, > > dull people conclude that there must be more reals than rationals. > > There certainly cannot be fewer, or even just as many, so what other > option is open/ Why not just as many? In a lattice definition of < between sets there are obviously fewer. In a cardinality definition there are not obviously fewer, it could be just as many. It all depends on the definitions of more and fewer and equally many. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Virgil on 6 Dec 2006 22:41 In article <4576DFFC.1000807(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/5/2006 11:04 PM, Virgil wrote: > > In article <4575B9F3.7080107(a)et.uni-magdeburg.de>, > > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> > >> Reals according to DA2 are fictitious > > > > No one mathematically competent who is at all familiar with Cantor's 2nd > > proof finds any such thing falsehoods in it. > > > > It is EB who is fictitious. > > Set theorist may wish this. No my arguments are real and unrefuted. In mathematics they are neither real nor unrefuted. In some sort of EB wonderland EB may be able to dictate what is or is not allowable, but in mathematics, he is outside the pale, and his dictates are of no consequence..
From: Virgil on 6 Dec 2006 22:53 In article <1165418973.725100.179680(a)n67g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Dik T. Winter schrieb: > > >> > Everybody knows what the number of ther EC states is. > > [...] > > > The number of EC states is "the number of EC states". > > > > This is hardly a definition. > > > > > It is simply a notion which can be equal to a natural number. > > > > Which may evaluate to a number. > > No. It evaluates to a number as little as 6 evaluates to a number. It > *is* a number, though not a fixed number. That is a matter of > definition of the word "number". "Un-fixed numbers" are not numbers. They may be variables whose values are niumbes, but they are not numbers. > > > Without explicitly or implicitly > > providing a context (year) there is no definite answer in terms of > > natural numbers. Mathematically the number of EC states can be modelled > > as function of time. > > > > > The set of prime numbers does not contain the number 1. > > > > According to a widespread defintion of "prime number" the set of prime > > numers refers to a set which does not contain the number 1. > > > > > But once upon a time it did contain it. > > > > There may have been a time, when a different set (one containing > > the 1) was referred to by the named "set of prime numbers". > > That is a matter of definition. It was *the* set of prime numbers. Unless you specify a definition of primeness and a ring, that claim is specious. What constitutes the set of prime numbers depends on the definition of prime numbers being used and the ring in which you are operating. 5 and 13 are not prime in the ring of Gaussian integers. For any given definition of what is a prime, and in what ring it is to be a prime, the set of primes is fixed and forever immutable. Change either the definition or the ring, and you get may get a different set. This is not a matter of time but of definition.
From: Virgil on 6 Dec 2006 23:10
In article <1165421742.266029.197420(a)79g2000cws.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Bob Kolker schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > I have understood the concept and its failure. (That is the parallel > > > between those who have not yet arrived and those who have already left: > > > Both are not there.) > > > > What failure? > > One of many examples: The set {2,4,6,...,2n} has a cardinal number less > than some numbers in the set. This does not change when n grows (yes, > it can grow!) over all upper bounds. Therefore the assertion that the > set of all even natural numbers has a cardinal number gretaer than any > even number is false. 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." If I am misunderstanding, WM, let him state his own understanding of what that negations wold be, but without merely appending "is false". > > >Te concept of infinite sets has lead to the theory of > > real and complex numbers, of integration and ultimately to theoretical > > physics which as made possible the computer on which you spew your > > nonsense internationally. > > > > 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. |