An uncountable countable set
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From: mueckenh on 18 Jul 2006 07:40 Virgil schrieb: > WM's version of logic is quite different from the sort used in ZF of NBG > or any other part of mathematics. No the logic used in mathematics is the same as mine. It is only in modern set theory that such silly things as the rejection of the binary tree occur. > > > > Consider the columns spanned by the digit positions of 0.111... Either > > there is a column with only zeros, or there is at least one 1 in each > > column, or ? > > If you mean to list 0.0, 0.1, 0.11,0,111,... so that the 0's line up > vertically, then every "column" has a least one 1. And each column to > the right of the '.' column has the same "number" of 1's as all the > others. Here the existing number 0.111... with its aleph_0 digits can be realized by its not existing natural predecessors in the list? Regards, WM
From: mueckenh on 18 Jul 2006 07:44 Virgil schrieb: > The difference is that our "traditions and folklore", which we chose to > call axioms and definitions, are logically consistent, Only if you decide *arbitrarily* which infinite set can be exhausted and which cannot. Regards, WM
From: mueckenh on 18 Jul 2006 07:51 Virgil schrieb: > In article <1153148551.942037.97110(a)s13g2000cwa.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > This is the deep dilemma of > > set theory: There is no actually infinite set of finite numbers. > > But the existence of this "dilemma" can only be established by assuming > it. > > So for those who do not chose to assume it, it does not exist. Those who choose to close their eyes enjoy always the mercy of not being forced to see the sheer misery of mathematics. Regards, WM
From: mueckenh on 18 Jul 2006 08:04 Franziska Neugebauer schrieb: > aleph_0 def= | omega | > 0.111... =def (a_i) having a_i = 1 A i e omega > a_ij means the well known matrix of figures > > > IF aleph_0 does exist, THEN 0.111... covers aleph_0 columns. > > This is as meaningful as > If i exists then sqrt(-1) is i. > > > IF 0.111... covers aleph_0 columns, THEN aleph_0 columns do exist. > > This is as meaningful as > If sqrt(-1) is i then i exists. > > > IF aleph_0 columns do exist THEN we can consider their contents. > > This is as meaningful as > If i exists then we can consider its value. > > > IF we can consider the contents of each column, THEN we can ask how > > many 1's are therein. > > Lotta questions. Unknown word. > > > IF we can ask how many 1's are in each one, THEN the answer can be > > "zero 1's" or "not zero 1's". > > We can. > > > 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. > > 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. 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. Regards, WM
From: mueckenh on 18 Jul 2006 08:10
Virgil schrieb: > In article <1153168957.805313.57460(a)p79g2000cwp.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > Dik T. Winter schrieb: > > > > > > Now you use an entirely new term: "can be exhausted". I think you mean > > > that you can take out elements one by one and doing this at some stage > > > the infinite set becomes empty. Howver, I think, that if that can be > > > done, that there is a last element you can take out. And, according > > > to the axiom of infinity, that is not possible, so infinite sets can > > > not be exhausted in this sense. > > > > But in another sense? > > In the sense of having a set of all of them, as per the axiom of > infinity, an axiom does it. The axiom is not responsible for the well-order of *all* elements of the set, because well-order is a stepwise process, you see? You take one element and then define its successor and then you define the successor of this one and so on. Of course you can give a compact expression (as I did with my transpositions, for instance) but that doesn't change the principle character of the process. Regards, WM |