An uncountable countable set
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From: mueckenh on 21 Jul 2006 07:51 Dik T. Winter schrieb: > > The "first" successor. Tere is only one "first" successor! > > Conveniently snipping your own text again? You wrote in > <"news:1153052131.852951.273540(a)b28g2000cwb.googlegroups.com">: > > 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. > > Now, if you had written "the first successor" I would not have made any > remarks. You would have done better. > It is not *the* successor. That depends on the definition of "successor". 3 is the first successor of 2 if you consider 4 as another successor. It is usual, however, to define only the first successor at al. Only so we get the completely defined succession as required by Cantor. He says "völlig bestimmte Sukzession" and " zu jedem Elemente E' - mit Ausnahme des letzten, wenn ein letztes vorhanden ist - ein ihm _nächst_ folgendes E'' vorhanden ist."(italics by Cantor).Therefore omega ist "das Nächstfolgende", *the* successor. > > > > Are you again arguing that the statement > > > For all p, there is an n such that An[p] = K[p] > > > > That statement is false. (Without K including the index p = aleph_0, k > > cannt be different from all natural numbers.) > > What has your statement in parentheses to do with the statement I gave? > There is no argument at all about K being different from all natural > numbers. It is just a plain statement: > For all p, there is an n such that An[p] = K[p] > you claim that is false, and again without direct proof. Do you disagree > with the statement: > For all p, Ap[p] = K[p]? > I know that Ap[p] = 1 = K[p]. Now chose in the above n = p and you get > the new statement. Why would this second statement be false? You said "for all p". If p is a natural number, then the segment K[p] is the unary representation of a natural number and, hence, is covered by a natural number. Remember: For each natural (i.e. finite) number p we know: to index and to cover are equivalent. p covers K[p] <==> p indexes p. Would K have only natural indexes p, then it would be covered by at least one natural number. If you deny then please give an example of a *finite* natural number which indexes the n-th digit but does not cover the digits number 1 to n. Regards, WM
From: mueckenh on 21 Jul 2006 08:02 Dik T. Winter schrieb: > > You cannot exhaust the naturals. Therefore what you write as 0.111... > > is not the same as the unary representation which also is denoted by > > 0.111... but means that N is exhausted and there is at least one 1 > > which cannot be indexed by a natural number. > > Makes no sense to me. I see no exhaustion, see, I did use the symbol > "lim" there, and I had the hope that there would be a proper definition. > Apparently not. No. You intermingle two different things, all n e N and infinity. Again: For any *finite* natural n we have a logical equivalence, even better an inclusion: indexing the n-th digit includes covering at least all digits from 1 to n. If each digit of 0.111... is indexed by one natural, then each segment of 0.111... is covered by at least one natural too. As long as only natural numbers are concerned there is no outcome. > > > The reason is that > > > > 0.1 > > 0.11 > > 0.111 > > ... > > gives allegedly the same sum as > > > > 0.111... (as representation of aleph_0) > > 0.1 > > 0.11 > > 0.111 > > ... > > Yes, with a proper definition of "...". But that is not acceptable if aleph_0 is a number. > What was the reason again? If aleph_0 is a number larger than any natural, then it cannot be the *+ sum of a list containing only naturals excluding itself and simultaneously the sum of a list including itself. > You apparently have another definition in mind. Pray give *your* > definition. There are two definitions conceivable. Both lead to a contradiction. 1) If "..." means "forall" n, e N then 0.111... is not different from all n. (Because for naturals indexing includes covering.) But we know that 0.111... is not ín the list of naturals. Contradiction. 2) If "..." means --> oo including aleph_0 as an index, then the digit with this index is not defined. > > > But > > > what you are now meaning is that > > > lim{n -> oo} SUM{i = 1 .. n} An = 0.111... . > > > > I said above: It is different from n really reaching infinity. But for > > 1/9 or aleph_0 infinity is reached. > > With a proper definition of the limit, the sum will *never* reach 1/9 > (or aleph_0). But as you still refrain from giving definitions of the > notations you are using this discussion is less than fruitfull. > Bijection: Every position of a decimal number can be indexed by a natural number. The set of naturals has cardinality aleph_0. The set of positions has cardinality aleph_0. > I ask questions. Again you refrain from giving answers. Are you unwilling > to give answers? > I ask questions too which you do not answer. The most important one: How can a finite natural number index digit number n but not cover all digits from 1 to n? > > 0.1 > > 0.11 > > 0.111 > > ... > > As my interpretation of your "..." was apparently wrong above, pray, finally > supply a definition of that notation. See above. 1) and 2). > > How many digits has 1/9 in decimal representation? > > That is not a natural number. No, but the set of digits has a cardinality, namely aleph_0. > There is no such column. And 0.111... is not a natural. That is a contradiction. And I told you why: You said: all digits of 0.111... could be indexed by natural numbers, but not all could be covered by natural numbers. This is wrong as one can easily prove: Every natural which indexes a digit covers all digits up to that one. If all digits of 0.111... can be indexed than all digits up to every (and that is all digits) can be covered. What is in *your* > opinion the result of *+-ing all natural numbers togther? As we have seen it is simply impossible to *+ sum all naturals. Every result yields a contradiction. The set of all naturals does not exist. Regards, WM
From: mueckenh on 21 Jul 2006 12:17 Dik T. Winter schrieb: > Your transpositions are *conditional* transpostions. That is when you > write (a, b) you mean that the a-th element and the b-th element are > interchanged when in standard order the b-th element is smaller than the > a-th element. I would like to notate that as R(a, b), as the notation > (a, b) in general means: interchange the a-th element and the b-th > element without looking at the magnitude. Now when I encounter in your > list of conditional transpositions at some place R(1, 2) I need to know > what previous conditional transpositions have done to be able to determine > what this conditional transposition is going to do. You may weed out from > your sequence of conditional transpositions those that do nothing. But in > order to know whether you can weed out the n-th transposition you need to > know what earlier transpositions have done. Really a nice example: You may weed out from the well-ordering of the rationals those that have already been put in order like 1/2 before 2/4 comes, which is easy to see, or 3874384778/2374234092384032948 which is difficult. But in order to know whether you can weed out a fraction you need to know what already has been put in order. Therefore this is a sequential process. Regards, WM
From: mueckenh on 21 Jul 2006 12:28 Franziska Neugebauer schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > > Franziska Neugebauer schrieb: > > > >> Dear Wolfgang, > >> > >> I would appreciate if you not cut away that parts of my previous > >> posts you are trying to argue against. It may also be helpful if you > >> read before cutting. > > > > Dear Franziska, I cut what I regard as unnecessary and not helpful in > > order to make reading more comfortable. > > lame excuse Polite explanation. > In set theory one usually counts from zero. I know. That's why travelling set theorists always believe that one piece of their luggage is missing. But we shall not miss a number for that reason. > > >> 2 The *-sum of the sequence does not necessarily "inherit" the > >> properties from their individual sequence members. If this > >> "inherentance" is kind of a "general principle" you should refer to a > >> recognized source. > >> > >> But there is even a finite counterexample: > >> > >> 101 > >> 011 > >> 110 > >> ---- *-sum > >> 111 > >> > >> The property "does not contain at least one 0" is not inherited by > >> the *-sum. So the "inheritance principle" is not so obvious. > > > > But it is obvious for *all* *finite* *unary* number. > > Can't see that. If it is so obvious then please show (prove!) it. For unary representations of finite naturals n we have the theorem: If n indexes the digit number n then n covers all digits from number 1 to number n. Proof: The unary representation (0.a_1,a_2, a_3, ..., a_k, ...) of n is defined by: / 1 for 0 < k =< n a_k = | \ nothing for k > n. We can complete that to / 1 for 0 < k =< n a_k = | \ 0 for k > n in order to simulate the decimal representation. Now you should be able to see the following about *+ sums: If a 1 is in column n, then in columns m < n no zeros are possible. If there is a largest number n, then no column in m < n can be zero, but all columns with m > n are zero. If there is no largest number n, but all columns are indexed, then no column can be zero. If all columns are indexed then at least one number covers all columns. > > I consider my 3-item list a valid counterexample to an "inheritance > principle". It isn't - because of the missing covering. > > > And only finite unary numbers are involved in the second list below: > > > > The *- sum of > > The following notation I will call "A". Good idea! > > > 0.1 > > 0.11 > > 0.111 > > ... > > 0.111... > > > > is > > > > 0.111... > > > > i.e. the result is a number of the list. > > You want to put 0.111... into the sequence. On which position j e N? At any position you can identify, the first one, the second one, ..., the last one. > > What does the sequence-'...' in your context > (0.1; 0.11; 0.111; ...; 0.111...) > ^^^ > mean? In my first context it means a Cauchy sequence with limit 1/9 noted together with its limit. In my second context it means the sequence of finite cardinals 1, 2, 3, .... noted together with the first transfinite cardinal aleph_0. Both are ordered sets of order type omega. > > > The *- sum of > > > > I will call this "B". > > > 0.1 > > 0.11 > > 0.111 > > ... > > > > is > > > > 0.111... > > > > The result is allegedly not in the list. > > We did not put it into the list, so it's not there. Call this finite list "C": 0.1 0.111... Here the *+ sum is 0.111... > > > This is a contradiction, because by tertium non datur either the first > > case or the second must be true (if aleph_0 is a number with respect > > to trichotomy with natural numbers). > > First of all you produced two notations > > A def= (0.1; 0.11; 0.111; ...; 0.111...) > and > B def= (0.1; 0.11; 0.111; ...) > > Notation A is (not yet) defined since the first occurence of '...' has > (not yet) a defined meaning. The first "..." denotes the remaining terms of the sequence. >Fomally different is notation B which > simply denotes the original "list". > > *Assumed* that A are B denote different sequences the proposition > > 0.111... is in *A* & 0.111... is not in *B* > > is not a contradiction and does not violate the tertium non datur > principle. A contradiction is a proposition of the form > > p & ~p > That has been proved by means of the binary tree: |{edges}| >= |{paths}| || || N << R But let us first consider the present result which we obtained under great efforts and pains. Either the *+ sum is lager than the largest list number or it is not. We have for cases of different character: For a finite list of finite numbers we have: The *+ sum of list numbers is the largest list number. For an infinite list of finite numbers (B) we have: The *+ sum of list numbers is larger than the largest list number For a finite list of infinite numbers (C) we have: The *+ sum of list numbers is the largest list number. For an infinite list of infinite numbers (A) we have: The *+ sum of list numbers is the largest list number. > Hence you have (not yet) shown a contradiction as has been pointed out > all too often. > We have the "experimental" result that for all kinds of lists the *+ sum is the largest number of the list. If we remove the largest number in case there is one, the *+ sum decreases*. If we remove another number, the sum remains unchanged. There is only one exception, namely case B, where no largest number is present. But this exception does appear only by definition. Instead of the usual definition we could apply a consistent definition: If there is no largest number, then there is no *+ sum. Regards, WM
From: Virgil on 21 Jul 2006 13:25
In article <1153478099.434931.273560(a)m79g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > Where in the guidelines, and where in the > > consitution, is it layed down that zero is considered a natural number? > > I have access to both, so pray give pointers. > > I do not remember the source, but it was a trustworthy one. It is the von Neumann model for the naturals, which starts with 0 = {} and proceeds by defining the successor of x as (x union {x}) > > > > When you can derive, > > within that system, contradicting conclusions, your axioms are inconsistent. > > In principle, no other knowledge than the axioms and the definitions is > > needed. > > Within this system I defined a *+ sum and showed that > > The *- sum of > > 0.1 > 0.11 > 0.111 > ... > 0.111... > > is > > 0.111... > > i.e. the result is a number of the list. But your ellipsis is malformed as its "last" member does not occur in the indicated sequence, so any "proof" based on it is also flawed. > > The *- sum of > > 0.1 > 0.11 > 0.111 > ... > > is > > 0.111... > > The result is not a number of the list. This one is at least arguable as a limit, but not as a member of the list itself. > > This is a contradiction, because by tertium non datur only one case can > apply. One of these results is wrong if 0.111... represents a number. The first result is wrong for the reason cited. And "0.111...", if it is to represent anything at all, may as well represent the cardinality of the sequence which allegedly generates it. |