An uncountable countable set
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From: mueckenh on 17 Jul 2006 10:51 Dik T. Winter schrieb: > > > You do not believe in limits? Otherwise, why does the definition not make > > > sense? Quote the definition for precisely that notation: > > > 0.999... = lim{n -> oo} sum{k = 1..n} 9.10^(-k) > > > what part of that definition makes no sense? > > > > "n --> oo" because there is no natural number oo. n cannot become oo > > whether there are infinitely many natural numbers or not. Each is > > finite. Thus, for n e |N, the result is always different from 1.000... > > . > > n -> oo does *not* mean that n will become some oo, because n will not > become oo; it only means that n grows without bound. And as you should > know how the limit given above is defined, you ought to know that. > For each (real) epsilon > 0 there is an n0 such that for n > n0 > |1 - sum{k = 1..n} 9.10^(-k)| < epsilon > at what place does n become oo? In the above, take n0 = log_10(-epsilon) > when epsilon < 1.0, else take n0 = 0. n does not become oo. And therefore we have only the epsilon definition. And therefore we have *always* undefined digits in any irrational number and in the diagonal of Cantor's list. There are *always* most of its digits unknown, i.e., always there are more digits unknown than are known. No matter how small an epsilon you select. Hence you cannot prove that the digits of the diagonal are all different from those of the line numbers. You can prove it for each one but you cannot prove it for all. That is the same problem as with the *+ sum of my list. You can prove the sum is 1 for each column but you cannot prove it for all columns. Because stepwise exhaustion of infinite sets is impossible. Otherwise my (and Cantor's) reordering could be completed. > > > It is not that I fight against the identification of 0.999... with 1 or > > against the identification of the limit of the series of > > Gregory-Leibniz with pi/4 for computational purposes. > > But you fight against the identification for mathematical purposes? > > > But the assertion > > that the digits of these limits could be used to construct a diagonal > > number is simply nonsense. > > Only assertion and meaning. No content. Proven by epsilon. Set theory lives by contradiction. Some exhaustions of nfinite sets are accepted other are not. But except from tradition there are no other reasons. Therefore, set theory is folklore. Regards, WM
From: mueckenh on 17 Jul 2006 11:02 Dik T. Winter schrieb: > In article <1153052131.852951.273540(a)b28g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > In article <1153039371.156834.8960(a)s13g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > > Virgil schrieb: > > > ... > > > > > Every ordinal has successors, whether in ZF or NBG or most other set > > > > > theories, and to index any of them it is sufficient to use at the > > > > > members of any successor. > > > > > > > > But the successor of 4 does not consist of the ingredients of 4 only. > > > > > > Eh? > > > > The successor of 4 is 5, and 5 = 0.11111 which cannot be indexed by 4 = > > 0.1111. > > I think you are meaning that the digits of 5 can not be indexed by the > numbers 1 to 4. That is right. > > > Hence, > > 0.111... as the succesor of all naturals must consist of more 1's, than > > any natural, if it is to be a number. > > (1): 0.111... is not *the* successor of anything, we may sloppily say that > it is *a* successor of all naturals, just like 10 is *a* successor of > 2 It was Cantor who coined this expression saying " daß omega die erste ganze Zahl sein soll, welche auf alle Zahlen nu folgt." (Works, p. 195) So it must be larger anyhow. > > > As stated this makes no sense. But the successor of every natural number > > > is a natural number. > > > > The successor of all naturals is not a natural and, therefore, must be > > larger (because it is not less). > > Yes, it is larger than all naturals, but I would not call it *the* successor, > but *a* successor, or, if you wish, *the smallest* successor. However, not all of its 1's in unary representation can be indexed by natural numbers because they are smaller. This is the deep dilemma of set theory: There is no actually infinite set of finite numbers. Regards, WM
From: Franziska Neugebauer on 17 Jul 2006 11:32 mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: [...] >> Yes, it is larger than all naturals, but I would not call it *the* >> successor, but *a* successor, or, if you wish, *the smallest* >> successor. > > However, not all of its 1's in unary representation can be indexed by > natural numbers because they are smaller. My understanding of unary represenations is: n e omega: unary(n) def= (a_i) a_i = 1 if 0 <= i < n a_i = 0 if n <= i < omega omega: unary(omgea) def= (a_i) a_i = 1 A i e omega > This is the deep dilemma of set theory: There is no actually infinite > set of finite numbers. Non sequitur. Ever considered _your_ representation theory broken? F. N. -- xyz
From: Virgil on 17 Jul 2006 15:07 In article <1153142944.234387.270680(a)s13g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > 0.0... > > > > > > That is not a natural number. > > > > It is to John von Neumann and he know more about it than you do. > > If Newton had eggs called apples, that would have been just as wrong as > the error of John von Neumann. It appears that it is WM's eggs that have become scrambled. JvN made no such errors. And until WM has done anywhere near as many mathematical things right as JvN, he is in no position to criticize someone so obviously his better. > > In Transsylvania horses have four heads and only one leg. That must be WM's native land, then, as his views on mathematics seem to resemble those views on horses.
From: Virgil on 17 Jul 2006 15:09
In article <1153143241.113246.317220(a)i42g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > Obvious is that no transposition of two elements of a well-ordered set > > > can destroy the well-order. > > > Obvious is that, if infinite sets do exist and can be exhausted, the > > > set of my transpositions does exist and can be exhausted. > > > > > > The issue is what would result. > > > > Since at any finite stage one still has infinitely many infinitely > > descending sequences of rationals by magnitude, what evidence is there > > that "exhaustion" would produce anything else? > > Because "exhaustion" is defined by reaching the limit, the order by > size. Unless it is reached, numbers need be transposed, and hence, the > set of necessary transpositions is not yet exhausted. So that WM claims that that which does not and can not end must end. We do not want any of that sort of thing in mathematics. |