An uncountable countable set
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From: mueckenh on 25 Jun 2006 04:46 Daryl McCullough schrieb: > mueckenh(a)rz.fh-augsburg.de says... > > >Such a function exists from |N --> |R for all real numbers which can be > >individualized, i.e. which are really real numbers. > > If such a function exists, then demonstrate it. Give us a function > f from N to R such that every real number that is "individualized" > is in the image of f. Cantor's general proof breaks down by one counter example like this one: Construct the diagonal number of the following list by digit replacement 0 --> 1 (only the digits behind the point are to be considered): 0.0 0.1 0.11 0.111 .... It is D = 0.111... Do you really believe that actually infinitely many ones (1) can be contained in 0,111... without exactly as many appearing in the list numbers? *All* sequences of ones are present in the aleph_0 numbers of the list. (aleph_0 - 1 = aleph_0) Hence the list contains its D. On the other hand, the limit 1/9 = 0.111... is not contained in the monotonic sequence of the list numbers. Therefore you may insist that 0.111... differs from each number of the list. But then the representation 0.111... of 1/9 is not well defined. Why? The list 0.1 0.11 0.111 .... contains all digits in places which can be indexed and identified by natural numbers. The third digit behind the point, for instance, can be indexed by 3 or, equivalently, by 0.111. The seventh digit by 0,1111111 and so on. Without the possibility of being indexed, a place cannot be identified and does not exist. So, if you insist that 0.111... has more digits than any one of the list numbers, then there must be some digits (at least one) at places which cannot be indexed and hence have no well-defined positions. So, whatever is your choice: There is a contradiction: Either 0.111... differs from any list number and, therefore, does not exist at all, or it does not differ from any list number; then the diagonal proof fails. And don't talk about quantifiers. Either D exists actually with all its digits and the list numbers do exist completely too, then my reasoning holds, or they do not, then Cantor's proof fails anyhow. Regards, WM
From: mueckenh on 25 Jun 2006 04:52 Virgil schrieb: > In article <1151135731.658456.259240(a)g10g2000cwb.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > Daryl McCullough schrieb: > > > > > mueckenh(a)rz.fh-augsburg.de says... > > > > > > >Cantor said: A well-ordered set remains well-ordered, if finitely many > > > >or infinitely many transpositions are executed. > > > > > > I don't believe that Cantor ever said that. But whether he did or not, > > > it's a false statement. > > > > As false as yours about a well-ordering of *all* rationals. > > In that case, " mueckenh" is declaring it true, as there are many > explicit well-orderings of the set of all rationals. > > And, in fact, if any set, such as the rationals, can be demonstrated to > be countable it is immediately well-orderable by that very demonstration > of countability. > > So either "mueckenh" must be declaring the set of rationals uncountable > or he must admit that they can be well-ordered. They are neither well-orderable nor "countable". Because they just do not exist. I don't want to start a new discussion here. But in case you are interested: Elements of a set must be distinguished from one another. In the universe there are less than 10^100 bits which can be use to distinguish two elements. Therefore there is no set with more than 10^100 elements possible. Regards, WM
From: mueckenh on 25 Jun 2006 05:01 Virgil schrieb: > If an almighty God cold not arrange to have the sun appear to stand > still in the sky without disastrous side effects, such a God would > hardly be almighty. Of course not. He would not even be able to construct a stone so heavy that he was unable to lift it. If such a God first says: It is not good for the man to stay alone, so we will create a women for him. But then curses this very man for having obeyed his wife, this God is hardly knowing everything, not even much about the psychology of women. Cantor stated that infinit is in God, in mathematics, and in nature including physics, chemistry, ..., even sociology. He was wrong in all three points. Physicists have already noticed that. How long will t last until mathematicians get to know it? Regrads, WM
From: mueckenh on 25 Jun 2006 05:06 Virgil schrieb: > In article <1151153501.336865.131980(a)m73g2000cwd.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > Virgil schrieb: > > > > > > > In order to be unambiguous transpositions must be carried out > > > sequentially, meaning that only countably many can be applied. > > > > Not more is required. > > > > > > Mueckenh's analysis is not unambiguously a sequence of transpositions. > > > > > > Also he does not show that any single transposition can re-order a set > > > having no first member into one which does have a first member. > > > > Of course that is impossible. It is just showing a contradiction in set > > theory. > > In "mueckenh"'s set theory , perhaps, but not in anyone else's. > > Until "mueckenh" can show that a SINGLE transposition can switch an > ordered set with a first element with one not having a first element, > his alleged proof fails. Until it can be shown that Cantor's diagonal proof does reach the last digit of the last list number, his proof fails. > > > > > > Similarly in the opposite direction, no single transposition will cause > > > a set with a first element to become one without a first element. > > > > Of course that is impossible. It is just showing that there is nothing > > like an actually infinite set. > > Wrong! It is just showing that "mueckenh"'s "proof" is invalid. It is sowing that it does not deliver the result preferred by you, Virgil. Such proofs must be wrong in general, and even the allmighty could change this fact. Regards, WM
From: mueckenh on 25 Jun 2006 05:12
Virgil schrieb: > In article <1151153602.449685.154410(a)u72g2000cwu.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > Virgil schrieb: > > > By definition, a set M is countable if and only if there SOME surjection > > > g:N --> M exists and is uncountable if no such surjection can exist. > > > > > > This means that, given a set M, one must allow ANY function from N to M > > > to be considered. > > > > > > So that given that M(f) exists at all, one is not constrained to only > > > the function f which defines K(f) and M(f), but can consider whatever > > > functions imaginable from N to M(f). > > > > > > Since it is easy to construct surjections from N to the set of all > > > finite subset os P(N), it is also easy to do it from N to M(F). > > > > Such a function exists from |N --> |R for all real numbers which can be > > individualized, i.e. which are really real numbers. > > > What real numbers are not really real numbers? > In the Dedekind model, every set of rationals that is bounded below > determines a real number (or bounded above, for that matter). > In the Cauchy sequence model, every Cauchy seqence of rationals > determines a real number. > Which of the real numbers so determined are not really real numbers? Irrationals, for instance, are not numbers at all, because Cauchy's epsilon cannot be made arbitrarily small. But we will keep this disussion at the reordering of the well-ordered set of rationals. If you have understood that there are not all rationals, you will understand easily, that there is no irrational. Regards, WM |