An uncountable countable set
in [Math]
|
Prev: integral problem
Next: Prime numbers
From: Virgil on 8 Jul 2006 15:48 In article <1152383427.970423.325990(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > 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. Not so. Cantor's rule works for each n independently of all other members of N, so all can be done simultaneously by a single rule. Yours does not allow simultaneity because of the non commutativity of permutations. > Alternatively: If we construt Cantor's diagonal, there is always almost > all left unconstructed If you do it Cantor's way, all steps are done at once.
From: Virgil on 8 Jul 2006 16:09 In article <1152383519.271656.179220(a)p79g2000cwp.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > 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. Since the sequence in which the tranpositions are applied effects the result obtained (unlike what happens with Cantor's "diagonal"), one must retain the sequential application. After each step in such a sequence, there remains a countable set not in natural order. So the process is never complete. The distinction resembles that between getting N by the axiom of infinity (Cantor, ends in one step) or trying to get N by sequencially appending successors one a time ("mueckenh", never ends).
From: Virgil on 8 Jul 2006 16:15 In article <1152383645.198338.92290(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > But meanwhile we know that the reals in total are not more than a > countable set "It ain't what you don't know that hurts most, it's what you 'know' that ain't so" If "mueckenh" claims that the set of reals are countable, he must equally be claiming that the power set of the naturals is countable, since those two sets may easily be shown to be of the same cardinality.
From: Virgil on 8 Jul 2006 16:29 In article <1152384019.621477.88880(a)b28g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > 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. What if, as "mueckenh"'s reordering of the well ordering of the rationals, it is impossible to finish it at all. > > > > 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). How is there a "last" digit to a decimal expansion which does not have any last digit? > > By the way, Virgil's arguing leads to the "paradox" Only the "paradox" of "mueckenh"'s failure to comprehend simple logic. > 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. Since no version of Cantor's rule allows 0 or 9 to appear as a digit in the "diagonal" number for any list, that number cannot be equal to any of those having dual representation. Thus the problem claimed has long since been totally solved. > > 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. Any version of the Cantor "diagonal" rule establishes that set of digit positions for which the "diagonal" differs from the list member ( and differs from both 0 and 9) is N. There are at least as many variations to that rule as there are members in P(N).
From: Virgil on 8 Jul 2006 16:39
In article <1152384241.144334.238730(a)p79g2000cwp.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > > > > > > Either the diagonal number 0.111... is not distinguished from > > > > > > > all > > > > > > > finitely large numbers of the list > > > > > > > 0. > > > > > > > 0.1 > > > > > > > 0.11 > > > > > > > 0.111 > > > > > > > ... > > > > > > > then Cantor's proof fails. > > > > > > > > If we assume that that list contains natural numbers in some unary > > > > notation > > > > (as I think you do) than: > > > > > > > > > > > Or 0.111... is distinguished from all finitely large numbers > > > > > > > of the > > > > > > > > 0.111... is not a natural number in unary notation. So it is > > > > inherently > > > > different from all elements of the list. > > > > > > OK. it is the unary representaton of omega. > > > > Interesting. What is the unary representation of w+1? > > As we will see that the unary representation of w does not exist, it > will not be necessary to look for the unary representation of w + 1. > > > But more clear, you > > admit that it is not a natural number, and so inherently different from all > > elements of the list. > > This would be necessary, yes. > > > > > > > "To be different" means for all unary representations of n > > > > > An : 0.111... - n =/= 0 > > > > > > > > How do you propose to define that subtraction when 0.111... is not a > > > > natural number? > > > > > > I chose just this number because it is also the decimal representation > > > of 1/9. > > > > Interesting, although it makes no sense. > > It allows us to subtract: 0.111... - 0.111...1. Only if the same numeral is allowed simultaneoulsy to have several different values (in several different bases). Notice also that in unary, 0.1 and 0.11, and 0.111..., really have no meaning at all, as zero digits and radix points don't work. So "mueckenh" is essentially saying that the base being used makes no difference to the meaning of the numeral. So that 10 base two means the same as 10 base ten? |