Cantor Confusion
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From: Virgil on 7 Dec 2006 14:51 In article <4577F17F.7000005(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/7/2006 5:24 AM, Virgil wrote: > > In article <4576FD8D.60408(a)et.uni-magdeburg.de>, > > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > > > > > >> It started with the question how to deal with the nil in case of > >> splitting IR into IR+ and IR-. I got as many different and definitve > >> answers as there are possibilities. I expected that there is only one > >> correct answer, and I found a reasoning that compellingly yields just > >> one answer in case of rationals and a different one in case of reals. > > > > It might depend on the purpose of performing the split. I can think of > > several ways to deal with it, at least one of which should be > > satisfactory for any given purpose. > > > > If, for example, one is trying to create Dedekind cuts, about the only > > requirement is that if one puts 0 in one of IR+ or IR-, one goes the > > same way with the boundary rational at every rational cut. > > dedekind himself let the question open whether his cut belongs to the > left or the right side. He was also lacking insight. Actually, Dedekind saw that it made no difference which way it went, as long as one one was consistent about it. Those, like EB, who fail to see what Dedekind saw are the ones lacking insight.
From: Virgil on 7 Dec 2006 15:03 In article <1165488738.492139.68600(a)73g2000cwn.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Bob Kolker schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > 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. > > > > The set of natural numbers has a cardinal greater then any set > > > > {1, 2, ... , n} for any integer n. > > Wrong. It is WM who is wrong > The set of all natural numbers contains only natural numbers. > These number count themselves by > |{1, 2, ... , n}| = n. For every {1,2,3,...,n}, there is an n+1 greater than n. > > > Also the set of natural numbers has a > > cardinality greater than the cardinality of any set {2*1, 2*2 ..., 2*n} > > for any integer n. Since the set of integers is equinumerous with the > > set of even integers (by way of the mapping n<->2*n) > > They are not equinumerous by the way of mapping n <--> n. They are not under the mapping n -> sin(n) either, but both are equally irrelevant. The only relevant question is whether there is some bijective mapping, all other mappings are irrelevant. > > it follows that the > > cardinality of the set of even integers is greater than the cardinality > > of the (finite) set {2*1, 2*2, ... ,2*n} for any integer n. > > > > You have reasoned incorrectly. So what else is new? > > An obvious truth should be declared to be an incorrect reasoning, only > in order to protect a dubious assumption from becoming inconsistent? What is an "obvious truth" to such anti-mathematically inclined trolls as WM, is an equally obvious falsehood to pro-mathematically inclined people. > Incorrect is not my reasoning but your assertion that the complete set > N exists Within any axiom system that says a set exists, WM's claims to the contrary are of no effect. And WM does not appear to have an axiom system of his own in which he can dictate what will be allowed. When, if ever, WM presents an axiom system of his own, he can try to make it behave his way, but until then, he is operating in a vacuum.
From: Virgil on 7 Dec 2006 15:06 In article <1165488974.236980.79250(a)73g2000cwn.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Franziska Neugebauer schrieb: > > >> mueckenh(a)rz.fh-augsburg.de wrote: > > >> > Bob Kolker schrieb: > > >> [...] > > >> >> Rational numbers are non-genuine. Nowhere in the physical world > > >> >> outside of our nervouse systems do they exist. > > >> > > > >> > The nervous system is a part of the physical world. > > >> > > >> Hence this issue is to be disussed in a physics or neuro sciences > > >> newsgroup. > > >> > > >> > The integers belong to the nervous system. > > >> > > >> The nervous system is generally not an issue in mathematics. > > > > > > That is why many mathematicians overestimate their capabilities so > > > grossly. > > > > Which mathematician overestimates her or his capabilities? > > Those who pretend to be able of knowing every integer and, therefore, > to imagine the whole actually infinite set. We operate within axioms systems, and those axiom systems determine what we are allowed to do and what we are prohibited from doing, at least while we are in them. What is WM's "system"? A whole lot of inchoate rules of thumb which are often in conflict with one another.
From: Virgil on 7 Dec 2006 15:15 In article <4577F611.6090704(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/7/2006 5:18 AM, Virgil wrote: > > In article <4576FC0A.7000004(a)et.uni-magdeburg.de>, > > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > > > > > >> According to the axiom of extensionality, a set has been determined by > >> its elements. Why do you not admit the possibility that countability of > >> a set requires countable numbers. > > > > The countability of a set requires no more than it have countably MANY > > members, but is totally independent of the nature of those members. > > No. The numbers also have to have an approachable address. Which axiom says that? > > BTW: The meaning of "many" already includes countably. Where is that written in stone? > > > > > > For example: > > > > The set of square roots of prime naturals is as countable as the set of > > prime naturals itself by an obvious bijection, > > In this case you are using addresses approachable via bijection. You are > not really counting the square roots but primarily the prime naturals. In set theory, counting is done by bijection. So those square roots are /really/ counted as much as the primes are counted. > > even though according to > > EB each of those square roots is "uncountable" in some weird > > anti-mathematical sense.
From: Virgil on 7 Dec 2006 15:23
In article <4577F7BE.4060503(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/7/2006 4:41 AM, Virgil wrote: > > 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.. > > Just try and refute: > > Reals according to DA2 are fictitious Reference to any statement, other than by EB or his ilk, to the effect that "Reals according to DA2 are fictitious" > Reals according to DA2 are fictitious Repeating a falsehood does not make it any less false. > With fictitious I meant: They must not have a directly approachable > numerical address. What is this stuff about "addresses"? What rubric of mathematics demands that every "number" have a directly approachable address? That may be so in engineering, but is nowhere writ in mathematics. And what is so special about direct approaches. Lots of mathematical theorems have to be approached quite indirectly, but are still valid and important, despite that. |