Cantor Confusion
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From: Eckard Blumschein on 7 Dec 2006 05:48 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.
From: mueckenh on 7 Dec 2006 05:52 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. The set of all natural numbers contains only natural numbers. These number count themselves by |{1, 2, ... , n}| = 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. > 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? Incorrect is not my reasoning but your assertion that the complete set N exists and that a bijection n <--> 2n can be complete. This is incorrect because it leads to a contradiction with the fact that any set of positive even integers contains numbers greater than its cardinal number. This fact remains for {any positive even integers}. It is also correct for any sequence {2,4,6,...,2n}. If you want to refute this obvious truth, then you should be able to name a number 2n for which it is no longer valid. Handwaving hints to "the infinite" are not sufficient, because there are only finite natural numbers. Regards, WM
From: mueckenh on 7 Dec 2006 05:56 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. Regards, WM
From: Eckard Blumschein on 7 Dec 2006 06:08 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. BTW: The meaning of "many" already includes countably. > > 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. even though according to > EB each of those square roots is "uncountable" in some weird > anti-mathematical sense.
From: Franziska Neugebauer on 7 Dec 2006 06:13
mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: >> >> This does not imply that the set of former times "has changed" in >> >> time. Only the naming has changed due to a changed definition. You >> >> may compare this with Gerhard Schr�der who did not undergo a >> >> gender transformation when Angela Merkel became chancellor in >> >> 2005. >> >> >> > But "the chancellor" did. >> >> I would think most people start laughing at the questioner when asked >> whether >> >> In 2005 the chancellor underwent a gender transformation. [*] >> >> is true. > > Many people start laughing in case they fail to understand. Only few > start thinking. The sentence [*] says that the _person_ referred to by "chancellor" has undergone a gender transformation. Which is apparently not true. So what do you mean by stating 'But "the chancellor" did.'? F. N. -- xyz |