An uncountable countable set
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From: mueckenh on 8 Jul 2006 14:29 Virgil schrieb: > In article <1152191614.643204.79420(a)s26g2000cwa.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > Dik T. Winter schrieb: > > > > > In article <1152134107.848676.131670(a)75g2000cwc.googlegroups.com> > > > mueckenh(a)rz.fh-augsburg.de writes: > > > > David Hartley schrieb: > > > ... > > > > > So you acknowledge that your process doesn't give, in the limit, a > > > > > bijection N -> Q+ ? > > > > > > > > Such a bijection does not exist, because there is no smallest positive > > > > rational. > > > > > > There is no need of such for a bijection. Consider the set of positive > > > numbers P and the set of negative numbers N. Clearly there is a bijection > > > from P to N: f(x) = -x. But N has no smallest element. A bijection is > > > not necessarily order preserving. > > > > You are right. But the transpositions which I defined are order > > preserving. > > Then none of them disrupt the ordering. Therefore we have a well-ordered set simultaneously ordered by size, "at the end" of finity. Regards, WM
From: mueckenh on 8 Jul 2006 14:30 Virgil schrieb: > In article <1152191768.157605.15960(a)b68g2000cwa.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > Now, the complete > > set of transpositions is of order type omega. And it maintains the > > well-order by index while it achieves well-order by size. > > If this is the set of rationals "mueckenh" claims is being reordered, > "mueckenh" is claiming to have found a smallest rational number!!! > > But to see why his scheme fails: > Suppose we have a well-ordering of the rationals, > "mueckenh"'s scheme of reordering by size the first two, then the first > 3, then the first 4 and so on overlook the fact that the "number" which > are left unreordered after each stage never diminishes, so his > iterations make no progress at all. They all have been ordered at once by writing down the definition. Like Cantor's list. Alternatively: If we construt Cantor's diagonal, there is always almost all left unconstructed, so this construction makes no progress at all. You can choose one alternative, but the same for me and Cantor, please! Regards, WM
From: mueckenh on 8 Jul 2006 14:31 Virgil schrieb: > In article <1152191944.782522.136800(a)j8g2000cwa.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > Virgil schrieb: > > > > > > > Since any line and the following line both act on the same elements, > > > > > they cannot be applied simultaneously, and they do not "commute", but > > > > > must be applied sequentially. > > > > > > > > What has commutation to do with this proof? > > > > > > Absence of commutativity, which is the case with certain transpositions > > > and sequences of transpostions, means that they must be applied in > > > sequence and not simultaneously as "mueckenh"'s theory requires. > > > > They are not "applied" at all but are given in zero time. > > > If transpositions are not applied sequentially, then their effect is > often undefined, since altering that sequence can alter their effect. > The transpositions are not applied sequentially, but they are *defined* sequentially, i.e. in well-ordered form in zero time. So their effect is never undefined, since altering that sequence cannot happen by definiton. Regards, WM
From: mueckenh on 8 Jul 2006 14:34 Virgil schrieb: > > Are there in fact people who can believe that anybody was unable to > > understand Cantor's arguing? Was it such a problem for you to > > comprehend it? > > I have no problem with Cantor's first proof, nor his second, of the > theorem that the reals are uncountable, nor his proof that there are no > surjections from any set to its power set. > > I can easily believe that there are people who have trouble with each of > these proofs, as I have seen evidence to that effect. > > Mueckenheim has not addressed Cantor's first proof, so I do not know if > he understands it, though considering the way he misunderstands the > second, I would doubt it. Cantor's first proof argues that the limiting point of a sequence of nested intervals does not belong to the sequence. The limiting point can be a rational number as well as an irrational number. Therefore the proof concerns rationals and irrationals equally well. Only by knowing that the rationals are countable the proof is focused on the irrationals. As the algebraic numbers turned out countable too, the proof finally concerns transcendental numbers. But would we know that the transcendentals were countable too, the proof would make us believe in another sort of real numbers. But meanwhile we know that the reals in total are not more than a countable set, by the obvious proof: > | > o > / \ > o o > / \ / \ Therefore Cantor's first proof is a piece of history as a proof which was believed correct for 130 years. Regards, WM
From: mueckenh on 8 Jul 2006 14:40
Dik T. Winter schrieb: > > > > And what is the consequence of this? > > > > > > That the calculation of the digits can be done in parallel? > > > > It is done in zero time. It is determined from the beginning. Why > > should something "be done"? > > It was you who remarked on the calculations... The result of a calculation does not depend on the time you need to obtain it. > > > > > > It is Cauchy's theorem that proves that the limit exists, using the > > > > > epsilon argument. > > > > > > > > But Cantor's arguing is that without epsilon. > > > > > > I do not know Cantor's argument exactly. I think that he implicitly > > > uses that result. In my formulation it was abundantly clear that I did > > > use it. > > > > But he needs to consider every digit with equal weight. That is not a > > limit process. > > What did he *mean* when he wrote "with equal weight"? He did not write that. I used this description to express that all digits must be distinguishable from the exchanged digits and that the resulting number must be distinguishable from the number with one digit left unexchanged, be it the first or the last one (which can be recognized). By the way, Virgil's arguing leads to the "paradox": There is no last number and not that one next to the last and so on. Which one is the first one that does not exist? > > > I define limit by: *Using all finite natural numbers* just as like as > > Cantor does. > > No. You do *not* and he does *not*. In the Cantor diagonal the number > obtained is a real number using limits (in the Cauchy sense) with the > definition of real number by many others (that can be proven to all be > equivalent with each other). The limit used by Cantor is precisely the > epsilon argument, together with majoration and minoration on the ordered > set of rational numbers. By Cauchy any sequence of decimal digits has a > limit, but it is not certain whether that limit is in the defined set of > numbers. By Cantor, Dedekind, Weierstrass and a host of others (using > various formulations), such a limit (when starting with rational numbers) > is defined as a real number. That is the place where Cantor uses the > limit (although he may not have expressed it as such), i.e. showing that > the resulting number is a real. > Cantor does not at all use real numbers but merely infinite sequences with only two different symbols w and m (which might be interpreted as binary representations but were not). But that there is no satisfactory limit consideration becomes clear from the following: We know that 0.999... = 1.000... This leads to the result that a change of 1 in the limit where the digit number goes to oo does not have the effect which would be required in order to distinguish the diagonal number from the list numbers. But in order to concern all lines of the list, n has to go to oo (though never reach it) just as in case of my example. Regards, WM |