An uncountable countable set
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From: Virgil on 5 Oct 2006 15:41 In article <1160066411.999062.265360(a)e3g2000cwe.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1159799981.773875.190290(a)b28g2000cwb.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Han de Bruijn schrieb: > > > > > > > > > > What does your "mathematics" say the answer to this > > > > > question is, in the "limit" as n approaches infinity? > > > > > > > > My mathematics says that it is an ill-posed question. And it doesn't > > > > give an answer to ill-posed questions. > > > > > > You are right, but the illness does not begin with the vase, it beginns > > > already with the assumption that meaningful results could be obtained > > > under the premise that infinie sets like |N did actually exist. > > > > That opinion is a minority opinion. > > Good taste has always been a matter of the minority (August Everding, > the late director of the Bavarian theatres, translation by me). Every particular taste, including the worse, is in a minority. And in mathematics, the minority declaring artificial limits on where mathematics may go is that worst minority.
From: Virgil on 5 Oct 2006 15:43 In article <1160066594.051638.4940(a)i3g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > There are no complete infinite sets. Perhaps not in "Mueckenh"-land, but mathematicians will go blithely on dealing with them despite "Mueckenh" 's self-handicapping.
From: Virgil on 5 Oct 2006 15:44 In article <1160066751.825020.117740(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > MoeBlee schrieb: > > > A CONTRADICTION? From what AXIOIMS? Please show a derivation of a > > sentence P and ~P from the axioms. Oh, that's right, by "contradiction" > > you don't mean a contradiction in the sense of a sentence and its > > negation; you mean something that doesn't sit with your personal > > intuition. > > > > + 1 Gedankenexperiment: Put 10 balls in A and remove two, one of which > is put in B and the other one is put in C. > > P: At noon all balls are in B. > ~P: At noon all balls are in C. > Q: some are in each.
From: mueckenh on 5 Oct 2006 16:22 Dik T. Winter schrieb: > > The axiom of infinity does only state n+1 exists if n is given. It is > > realized by he numbers of my list. > > That is *not* what the axiom of infinity states. The axiom states that > there exists a set that contains all the successors of its elements. Oh I must have read always wrong texts. I never came across the word "all" in connection with this axiom. > > > > > Instead of "to index position" we can also say "to cover up to position > > > > n". Hence you assert that it is possible to cover 0.111... up to every > > > > position but it is impossible to cover every position. > > > > > > Yes. > > > > That is obviously a false claim, because "up to every (including this)" > > means "every". At this point a further discussion is really fruitless. > > Yes, because you still do not understand that there is no "specific > position" such that "every position" is the same as "upto that specific > position". In my list there are all specific positions (if all natural numbers do exist). If the required number is not in the list then it is nowhere. Then your imagination is an impossible and self-contradictory dream. It is crazy to hear that covering up to every number is possible but covering every number shall be impossible. > > > > > Cantor's argument was about reals. He strived for generality but did > > > > not see that two symbols are not enough. > > > > > > You are seriously wrong. > > > > Read his first paper. He treats numbers and rational functions. Nothing > > more. > > Read the two first paragraph of his second paper more thoroughly. Or are > you of the opinion that when I write: > "In paper A I did prove the theorem that there are sets that have larger > cardinality than the natural numbers. In this paper I will give a > simpler proof of that theorem." > you do not know for what theorem I will give a simpler proof in the > current paper, but that you need first to read paper A to be able to > state that? Of course, if there is any doubt, one looks it up and finds that the first paper and the proof and the theorem are only about rational and real numbers. You need only look at the title: On a property of the set of all real numbers. The theorem proven reads essentially: "Wenn eine nach irgendeinem Gesetze gegebene unendliche Reihe voneinander verschiedener reeller ZahlgröÃ?enï? vorliegt, so läÃ?t sich in jedem vorgegeben Intervalle eine Zahl (und folglich unendlich viele solcher Zahlen) bestimmen, welche in der Reihe nicht vorkommt; dies soll nun bewiesen werden." Do you see something else than numbers here? And the generalizations concern numbers and numbers and nothing else. > And you assert that it is not likely that in the current > paper the theorem for which a simpler proof is given is the theorem > "that there are sets that have larger cardinality than the natural > numbers", but something else, unstated in the current paper? Of course there are such sets, but these sets are numbers, namely the real numbers, the irrational numbers and the transcendental numbers. Regards, WM
From: mueckenh on 5 Oct 2006 16:26
Dik T. Winter schrieb: > In article <1160045362.894290.321140(a)c28g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > > Dik T. Winter schrieb: > > > > > In article <1159978513.826507.125470(a)m73g2000cwd.googlegroups.com> muecke= > > nh(a)rz.fh-augsburg.de writes: > > > > Dik T. Winter schrieb: > > > ... > > > > > > You can read it above. There is the order preserving union of a s= > > et of > > > > > > k negative numbers and a set of omega natural numbers including z= > > ero. > > > > > > > > > > But that has ordinal k + omega, > > > > > > > > that is what I said! > > > > > > No. > > > > I used "k + omega " as the ordinal number of {-k, -k+1, ..., 0, > > 1,2,3,...}. > > Sorry, I misread. > > > > > > not -k + omega. A set of k negative numbers > > > > > has ordinal k; not -k. > > > > > > > > But subtraction of a set of positive numbers from the set omega is > > > > expressed, as I did, by -k + omega. > > > > > > Ah, apparently you are defining something new here. > > > > New for you probably. As omega + k is different from k + omega, we > > should not write omea - k for -k + omega. > > Cantor did. And he wrote omega_(-k) for the other. Cantor later changed notation. Perhaps you don't know that Cantor changed notation between 1883 and 1895. In 1883 Cantor had not yet done it. Compare the remark [3] by Zermelo on page 208 of his collected works: "Hier und im folgenden stellt Cantor den Multiplikator voran und schreibt 2 omega für omega + omega; in der späteren systematischen Darstellung III 9 stellt er umgekehrt den Multiplikandus voran und schreibt omega * 2, was aus Gründen der Analogie entschieden vorzuziehen ist, weil auch bei der Addition nur der zweite Summand (der Addendus), wenn er endlich ist, die transfinite Summe modifiziert, vergrößert. Vgl. S. 302, 322." In order to use this analogy which Zermelo mentiones I prefer -k + omega, because so every nutcake sees that the sum is not modified but remains omega while omega - k could be misunderstood as modifying the sum. So I do in addition what, according to Zermelo, has to be preferred in multiplication because of the analogy to addition and which Cantor executed in 1895 or somewhat earlier. > > > > No? Who decides that? You see, I knew already that, according to > > > > Cantor, subtraction is possible. If I express this as addition of > > > > negative, what do you think did I meant? > > > > > > I did not know because there is no definition presented nor available. > > > Addition is defined between ordinal numbers. Ordinal numbers are (by > > > their very definition) larger than or equal to 0. But you want to > > > define addition between ordinal numbers and non-ordinal numbers. > > > > Wrong again. k and omega are ordinal numbers. > > But '-k' is *not* an ordinal number. "k" is an ordinal number and "-" is the advice to subtract it. > > In the meantime I have read it. You may note that in his notation > the solution for x + a = b is b - a (a < b). And the solution for > a + x = b (if that exists, and if there is more than one solution, > the smallest one) as b_(-a). > -- So you have learned that subtraction is possible, which was the main point that you originally doubted. But why should I stick to an old-fashioned and misleading notation? Regards, WM |