An uncountable countable set
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From: Dik T. Winter on 5 Oct 2006 19:52 In article <1160066594.051638.4940(a)i3g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > > By the only meaningful and consistent definition: A n eps |N : > > > |{1,2,3,...,2n}| = 2*|{2,4,6,...,2n}|. > > > Do you challenge its truth? > > > > No, I never did. But you draw conclusions about it about the set N. Indeed, > > for each finite n, it is true. But this is *not* a proper definition for > > the amounts involved in infinite sets. Given two infinite sets A and B, > > by what method do you determine whether A has more elements than B, or > > the other way around? Are there more Gaussian integers than Eisenberg > > integers, and if so why? And if not, why not? > > There are no complete infinite sets. And I thought you always were talking within the context of the axiom of infinity. At least, that is were I am talking. > > You attack the existence of an infinite sequence, and so also the existence > > of an infinite set. So be it. But that is just a negation of the axiom of > > infinity. With that axioms such things do exist. > > I would like to know whether numbers exist, not what axioms do say > about them. Numbers 1 and 2 and 3 exist without any axioms. Number > omega does not exist, with and without any axiom. Without axiom it is > clear. With the axiom INF you see the results. You must insist on the > silliest ideas (like covering up to every but not covering every). So the axiom of infinity just bothers your sense, but not anything else. So be it. I am not bothered by those ideas. And calling things silly does not make them false. Just try to start mathematics without the axiom of infinity and be done with it. Come back when you have results. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 5 Oct 2006 19:57 In article <1160066751.825020.117740(a)h48g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > 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. Proof? I would say: P: At noon there are countably many balls in both B and C. But let me clarify the experiment (using numbered balls, and you put in A starting with the lowest numbered ones of the remaining balls): Put 10 balls in A and move the two with the lowest order numbers from A to B or C where you put the odd numbered ball in B and the other one in C. At noon: A contains no balls, B contains the odd-numbered balls, C contains the even-numbered balls. Contrast this with: Put 10 balls in A and move the two with the highest order numbers from A to B or C where you put the odd numbered ball in B and the other one in C. At noon: A contains countably many balls, B contains the balls with the numbers 10n - 1, and C contains the balls with the numbers 10n. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 5 Oct 2006 20:12 In article <1160079778.871756.325200(a)m73g2000cwd.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > 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. What texts *have* you read where your version is stated? > > 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. Again, you imply that "every position" is the same as "upto some specific position". There is no such "specific position". > It is crazy to hear that covering up to every number is possible but > covering every number shall be impossible. That is nothing more than opinion and taste. But that is not a disproof of the proof given. > > 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. Yes, but that first article also in essence proves the theorem given in the first sentence of that paragraph. It is simply a corrollary of the theorem actually proven. > Do > you see something else than numbers here? And the generalizations > concern numbers and numbers and nothing else. What does it matter? It was shown that the set of reals has larger cardinality than the set of naturals. An easy consequence of that is the theorem stated in the first sentence: there are sets with larger cardinality than the reals. > > 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. So there are sets. The second paper is not concerned about what the sets contain, it just proves that there are (arbitrary) sets with larger cardinality than the naturals. And shows that with countably infinite sequences of two symbols. Why do you think the annotations state how the proof can be modified to a proof about the reals? -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 5 Oct 2006 20:44 In article <1160079987.659148.278260(a)b28g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: .... > > > 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 f?r omega + omega; in der sp?teren systematischen > Darstellung III 9 stellt er umgekehrt den Multiplikandus voran und > schreibt omega * 2, was aus Gr?nden der Analogie entschieden > vorzuziehen ist, weil auch bei der Addition nur der zweite Summand (der > Addendus), wenn er endlich ist, die transfinite Summe modifiziert, > vergr??ert. Vgl. S. 302, 322." Yes, that change is still current in set theory, and I know the difference. And I even do understand why he used his original notation. What is more natural than asserting that with a.b you combine a copies of set b? But there were more compelling reasons to interchange the operands. > 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. But that notation suggests that there is an ordinal (-k), which is true in most branches of algebra. But whatever. Subtraction and division is nearly nowhere defined or used in set theory. There appears to be not much need. So you should not be surprised that I question what you write, when that was possibly common 120 years ago, but was not taught at all some 40 years ago, when I had my set theory courses. > > > 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. But in the standard ways algebra works, first it is asserted that the negative of a number does exist, and after that subtraction is defined as the addition of the negative... > > 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. It is not unrestricted possible. There are cases where subtraction is not possible at all. Possibly one of the reasons that it is not part of mainstream mathematics. > But why should I stick to an > old-fashioned and misleading notation? No reason at all, but when you invent a new notation, it would be better if you did define the notation before use. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: r.e.s. on 5 Oct 2006 23:38
<mueckenh(a)rz.fh-augsburg.de> wrote ... > There are two equivalent truths: > > (1) (Each) ball number n will come out before noon. > (2) When ball number n comes out, more than n balls remain in the vase. > > Both are absolutely correct. This shows that one can not consistently > calculate with infinity. < ;o) > I don't see why everyone bothers to remove balls from the urn, since the urn can be emptied without removing any balls, all the while adding more and more of them ... At step n (integer n=1,2,3,...) just re-label with n all the balls (if any) that are in the urn, and insert a zillion more balls, each labelled with n. So at the end of step n the urn contains n zillion balls, each of which is labelled by integer n. Obviously, for each integer n, every label less than n permanently ceases to exist in the urn after the nth step; so, after *all* integer-numbered steps have been performed, there can be no label in the urn -- ergo no balls, since every ball is labelled. Hmm ... where'd all those balls go, I wonder? </ ;o) > |