An uncountable countable set
in [Math]
|
Prev: integral problem
Next: Prime numbers
From: Virgil on 15 Aug 2006 16:05 In article <1155664542.356195.210290(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > "Every digit of n can be indexed" > > And can be 'indexed" by using only those digits whose position number > > is a prime, leaving most of them unused as indices but still indexing > > all of them. > > > > > > > This is a *symmetric* relation. > > > > > > Not at all. when one turns it about, only those digits in prime > > positions are indexed leaving infinitely many un-indexed. > > You talk about enumerating the elements of a set and, as usual, you > have not grasped the essential point of indexing. Indexing involves construction a surjective from the set of indices, normally but not necessarily the smallest ordinal number of the same cardinality as the set to be indexed, to the set of objects to be indexed. Ideally, but not necessarily, the mapping will be bijective. N is an ordinal which by the identity function indexes itself.
From: Virgil on 15 Aug 2006 16:14 In article <1155664618.345718.296400(a)i42g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > Because the index number of every row (which equals the number of x's in > > that row) is finite. > > Therefore there exists no number aleph_0 of rows. There are aleph_0 rows ( the cardinality of the set of rows equals the cardinality of N) but no row indexed is aleph_0 if one uses N as an index set. In NBG, in indexing any set with the minimal ordinal which bijects with it, no member of the set is indexed by that minimal ordinal of the set. "Mueckenh" is working without a net in not having any axiom system. Without an axiomatic base, a set of statements accepted as being true, "Mueckenh" cannot prove anything.
From: Virgil on 15 Aug 2006 16:19 In article <1155664719.550846.306150(a)i42g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1155067333.259873.193700(a)p79g2000cwp.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > The v. Neumann model of natural numbers is just the model of segments. > > > n = {0, 1, 2, ..., n-1} > > > Every natural number is finite. > > > Hence every union of finite segments is a finite segment. > > > > Nor quite. Every FINITE union of finite segments is finite, but infinite > > unions need not be. In fact the union of all naturals in NBG equals the > > set of all naturals, and by the axiom of infinity does not have any > > largest member. > > But, by definition, EVERY member is finite. Every member of N is finite, but that does not imply that N itself must be finite. And in NBG, the set we call N is not finite. > > > > > You don't like set theoretic models if they are uncomfortable for your > > > current arguing? > > > > Every model of NBG contradicts "Mueckenh"'s arguing, so he must be > > infinitely uncomfortable with it.. > > > > > > Again, a model, with an unfounded statement. I was talking about sets, > > > > not about specific models of natural numbers. > > > > > > The v. Neumann model of natural numbers is just the model of segments. > > > n = {0, 1, 2, ..., n-1} > > > Every natural number is finite. > > > Hence every union of finite segments is a finite segment. > > > > "Mueckenh" repeats what is false in NBG (the von Neuman-Bernays-Goedel > > model). The union of all von Neumann naturals is not finite. > > Which is the first infinite union? Union of what? First by what criterion?
From: Virgil on 15 Aug 2006 16:22 In article <1155665028.991156.223740(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > In article <1155485913.329014.192160(a)p79g2000cwp.googlegroups.com> > > mueckenh(a)rz.fh-augsburg.de writes: > > > Dik T. Winter schrieb: > > > > In article <1155242105.069297.90260(a)74g2000cwt.googlegroups.com> > > > > mueckenh(a)rz.fh-augsburg.de writes: > > > > > David R Tribble schrieb: > > > > ... > > > > > > What is 10% of Aleph_0? If you start counting your lines, at > > > > > > what > > > > > > point do you know that you've counted the first 10% of them? > > > > > > > > > > At what point do you know that you have all aleph_0 lines? > > > > > > > > When you start at 1, at no point. > > > > > > So Cantor's diagonal is never completed. We are never sure that it is > > > not in the list. > > > > Nobody ever has said that the diagonal is ever completed. There is a > > definition of that number, and it is easy to show, with that definition, > > that that number (when completed) would be different from all numbers on > > the list. > > But if it is impossible to complete it, then the result is void. It is possible to complete the proof for any given number in the list. In ZFC of NBG, one can show by induction that it is then valid for all members of the list. AS "Mueckenh" has no axiom system to work from, what he can do is more limited.
From: mueckenh on 15 Aug 2006 16:33
Dik T. Winter schrieb: > > For a linear problem like the unary numbers of the list, there is no > > quantifier gambling possible. If position n is indexed by a unary > > number, then also all positions m < n are covered by this number n. > > There is no outcome in the set of numbers of the form 0.111...1. > > > > 1) If a digit is indexed then the sequence up to that digit is covered. > > 2) If there are no digits which cannot be indexed, then there are no > > digits, which cannot be covered. > > 3) That is equivalent to the statement, that all digits can be indexed > > and all digits are covered. > > (3) is indeed also true if you reformulate to "all digits can be indexed > and all digits can be covered". OK. All digits can be covered. But all digits are never covered. Hence all digits cannot be covered. Again a new equivalence of set theoretic logic: All digits can be covered <==> All digits cannot be covered. > > You assert that every digit of 0.111... can be covered, but that > > 0.111... cannot be covered. That assertion is void of meaning. It makes > > no sense. > > To you, apparently. I did show a proof, what was wrong with the proof? You showed it for digit positions only which can be indexed by natural numbers. You claimed that all digit positions which are in the true list suffer to index all positions of 0.111... which is not in the true list. You constructed an "equivalence" like that one I wrote down above. Regards, WM |