An uncountable countable set
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From: Virgil on 19 Jul 2006 15:00 In article <1153316030.583020.125150(a)p79g2000cwp.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > Sorry, I have 150 examinations these days, so I cannot answer as yet. > But I will come back. (And you meanwhile have time to ponder about the > inconsistencies of set theory.) > > Regards, WM I hope that your exams are on set theory. As that might make you see what it is all about.
From: Dik T. Winter on 19 Jul 2006 20:39 In article <1153301881.393720.57330(a)75g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > > > If the *+ sum of the list is a unary representation of some 1's, then, > > > by the construction of the unary numbers of the list, this sum must be > > > a unary numer of the list. > > > > As long as your list is finite. > > As long as it contains *numbers* which can be distinguised from each > other and which obey trichotomy. No, as long as your list is finite. > > Suppose the list is infinite, and the > > *+ sum of all members of the list is a member of the list. That would > > mean that the *+ sum of the list is the last element of the list, > > contradicting that the list is infinite. > > If we assume that non-terminating fractions have aleph_0 digits, then > this case is realized by > > 0.1 > 0.11 > 0.111 > ... > 0.111... > > Here the last number of an infinite list is in this list. I thought we were talking about natural numbers. I have not yet seen a definition that calls the last one a natural number. You should *not* switch between representations during the process. Either you have a list of natural numbers (in that case 0.111... does not belong to it) or you have a list of rational numbers in decimal notation. The latter is easier to reason about because it can be tackled easier. In that case your list consists of: An = sum{i = 1..n} 10^(-n) = (1 - 10^(-n))/9. Also we can easily show what repeated *+ is (notated here as SUM): SUM{i = 1 .. k} Ai = Ak. Now we can also get a proper definition about what your notation ... means in the list: lim{k -> oo} SUM{i = 1 .. k} Ak = 1/9 = K. And K is (by convention) decimally notated as 0.111..., which makes sense, in my opinion. K clearly is not equal to any of the An, unless there is a natural number for which 10^(-n) = 0. > Here the last number of an infinite list is in this list. It is not a list when you use the proper definition of list (a mapping from N to the items of the list). An infinite list does not have a last element. Because, suppose that list had a last element. By the definition of list there should be a natural number n that maps to that element. But n+1 maps to another element. > > Hence the assumption is false. > > Yes, the assumption is clearly false that omega or aleph_0 is a number > larger than any natural but counting all the naturals. This is insidious misquoting. You should start working with the definitions. When you want to show an inconsistency in a system you have to do: (a) show that within that system you can prove proposition P (b) show that within that system you can disprove proposition P But *all* proves should be within the system. So you should also use the definitions from the system. Otherwise the only thing you show is that the system is inconsistent with *your* definition when used within the system. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 19 Jul 2006 20:45 A bit late, but I think it requires response: In article <1153168957.805313.57460(a)p79g2000cwp.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: .... > > > One should have seen that earlier, then Bourbaki would not have > > > succedeed to define 0 as natural number, even in political decisions. > > > > Pray explain the last part "even in political decisions". > > It is laid down in the guide lines of the European Community that zero > was a natural number. I am indebted to your compatriots that they have > dismissed the constitution of this disastrous association. You are talking about two different things. First the guide lines and second the constitution. 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. But you fail to understand mathematics. Within mathematics you start with axioms, use definitions and derive conclusions. 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. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: mueckenh on 21 Jul 2006 06:34 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. > > 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. The *- sum of 0.1 0.11 0.111 .... is 0.111... The result is not a number of the list. 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. Regards, WM
From: mueckenh on 21 Jul 2006 06:39
Dik T. Winter schrieb: > In article <1153301881.393720.57330(a)75g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > > > > If the *+ sum of the list is a unary representation of some 1's, then, > > > > by the construction of the unary numbers of the list, this sum must be > > > > a unary numer of the list. > > > > > > As long as your list is finite. > > > > As long as it contains *numbers* which can be distinguised from each > > other and which obey trichotomy. > > No, as long as your list is finite. The numbers count the lines. As long as there are only finite numbers, the list is finite. But as you have not the understanding, consider the related case: A finite number covers what it indexes. This does not depend on how much numbers are in the list. >From this discovery you may obtain the result for what you call an infinity list. > > > Suppose the list is infinite, and the > > > *+ sum of all members of the list is a member of the list. That would > > > mean that the *+ sum of the list is the last element of the list, > > > contradicting that the list is infinite. > > > > If we assume that non-terminating fractions have aleph_0 digits, then > > this case is realized by > > > > 0.1 > > 0.11 > > 0.111 > > ... > > 0.111... > > > > Here the last number of an infinite list is in this list. > > I thought we were talking about natural numbers. I have not yet seen a > definition that calls the last one a natural number. I never said so. > You should *not* > switch between representations during the process. It is important for the discovery. > Either you have a > list of natural numbers (in that case 0.111... does not belong to it) > or you have a list of rational numbers in decimal notation. The latter > is easier to reason about because it can be tackled easier. In that > case your list consists of: > An = sum{i = 1..n} 10^(-n) = (1 - 10^(-n))/9. > Also we can easily show what repeated *+ is (notated here as SUM): > SUM{i = 1 .. k} Ai = Ak. > Now we can also get a proper definition about what your notation ... means > in the list: > lim{k -> oo} SUM{i = 1 .. k} Ak = 1/9 = K. > And K is (by convention) decimally notated as 0.111..., which makes sense, > in my opinion. K clearly is not equal to any of the An, unless there is > a natural number for which 10^(-n) = 0. Nevertheless we have the *+ sum 0.111... from 0.1 0.11 0.111 ... Now we can insert this in the list and we find that there is no change in the *+ sum. This is only possible, if it was in the list before already. > > > Here the last number of an infinite list is in this list. > > It is not a list when you use the proper definition of list (a mapping > from N to the items of the list). An infinite list does not have a last > element. Because, suppose that list had a last element. By the definition > of list there should be a natural number n that maps to that element. But > n+1 maps to another element. Then call 0.1 0.11 0.111 ... 0.111... an extended list or an ordered set or what you like. The proof should not fail by lack of names. Regards, WM |