An uncountable countable set
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From: mueckenh on 19 Jul 2006 09:33 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
From: Franziska Neugebauer on 19 Jul 2006 09:47 mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: >> mueckenh(a)rz.fh-augsburg.de wrote: >> >> > Franziska Neugebauer schrieb: >> [...] >> >> > IF the answer is in each case is "not zero 1's", THEN in each >> >> > column at least one 1 must be present. >> >> >> >> This is the case, since every a_jj = 1 j e N by definition. >> >> >> >> > However, there is no natural numbers with this property, >> >> >> >> Could you precicely _define_ which /property/ you are talking >> >> about? >> > >> > To contribute a 1 to each column. >> >> Should be a property of which object? > > Should be a property of at least one summand if the *+ sum has this > property. 1. I have shown formally (proved) that there is no single sequence member having the property to "contribute a 1 to each column of the sum". 2. If your assertion is a theorem that claim and 1 form a contradiction. 3. To "complete" it you now have to show that your claim "there exists at least one summand which contributes a 1 to each column" is a theorem. I. E. you have to _prove_ your assertion. >> >> For every column j e N a_mj has the 1 in position m(j) = j, since >> >> a_jj = 1 A j e N. Where exactly lies your problem? >> > >> > I told you recently: In a list like mine a *+ sum is defined. This >> > * sum is equal to the largest number of the list. >> >> The notion of "largest number" is misleading. There is no largest >> number. Neither in the list nor in omega. If it makes sense to talk >> about "a largest number", then please > > 1) That does not mean that your pretension (sum of all naturals is > 0.111...) was correct.2) omega is the largest number less than omega + > 1. 1) I have already proven that the *-sum of all the unary(i) i e N is unary (omega). 2) But omega is not the "largest number" < omega. This is what we are talking about. >> 1. name it (show us its name or position _in_ the list or _in_ >> omega), >> and/or >> 2. present a _proof_ that it exists. > > In > > 0.1 > 0.11 > 0.111 > ... > 0.111... > > this element exists , if it does exist at all. Its position can be > selected arbitraily. This does not imply that (unary(i))_{i e omega} has a member unary(omega). But this is what we are talking about. >> Until then we cannot (meaningfully) talk about "a largest number in >> the list" or "a largest number in omega". Wittgenstein applies here. > > omega is the largest number of he sum above. Which "sum" do you mean? Please state explicitly. > Its existence is proven in set theory. Omega is neither a sum or a *-sum in set theory. It is an axiomatically defined entity. >> > And if a larger number is included, then the sum grows. You pretend >> > that this is not valid for 0.111... because without 0.111... the >> > list has the *+ sum 0.111... and when you include it in the list >> > (as diagonal number or as the first line or anywhere else) the *+ >> > sum does not grow. >> >> Postponed until existential status of "largest number" is clarified >> by > > Is clarified. Not to me. F. N. -- xyz
From: Virgil on 19 Jul 2006 14:47 In article <1153301245.706099.39070(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: > > > >> My understanding of unary represenations is: > > > > > > strange! > > > > > >> n e omega: unary(n) def= (a_i) > > >> a_i = 1 if 0 <= i < n > > > > > > a_i =/= 1 if n = 0. > > > > Of course. unary(0) = 000... > > The unary representation of 0 (the empty set) is the empty string > > (stiputalting only 1's are written). > > correct. If "unary representation" is announced by "0." Then 0. denotes > zero. Unary representation would be better announced by a "U" for unary or even a "b1." for base 1. Starting with a "0." is more like announcing decimal fraction representation. > > > > A *-sum of _infinitely_ _many_ representation of finite numbers does in > > the present case result in a representation of an infinite number. This > > is a fact. > > That is nonsense. Then it is logical nonsense in vivid contrast to your illogical nonsense.
From: Virgil on 19 Jul 2006 14:53 In article <1153301881.393720.57330(a)75g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > 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. > > > 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. Whichever interpretation one puts on the symbols, unary or decimal fraction, in either case one has the ordinality of (N union {N}) in that there is a trivial order isomorphism between the ordered members of your lists and the members of (N union {N})
From: Virgil on 19 Jul 2006 14:58
In article <1153302541.286243.112340(a)s13g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > > > > > > The successor of all naturals is not a natural and, therefore, > > > > > > > must be > > > > > > > larger (because it is not less). > > > > > > > > > > > > There is no such thing as the successor of all naturals any more > > > > > > that > > > > > > there is a single successor common to both 3 a and 6. > > > > > > > > > > This is an expression coined by Cantor: "a number following after all > > > > > natural numbers". > > > > > > > > But he does not call it a 'natural' number any more than he calls it an > > > > even number or a prime number. > > > > > > I said above: The successor of all naturals is not a natural. > > > > No one else even says that such a thing exists. There is a set of all > > naturals but if is not the successor of any of them. > > Cantor said it .. Cantor did not say "successor of all naturals", and if he said anything at all like that he did not mean successor in the same sense as 2 is the successor of 1. > > Perhaps not in your philosophy, but you are not God, to command what is > > true or false. > > But I can see what is contradictory. And you can see things as contradictory which are not as well. Such "visions" are a handicap in logical thinking. > > You are a liar. Therefore I cease my correpondence with you. You are hardly in a position to complain about falsehoods. |