An uncountable countable set
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From: mueckenh on 6 Oct 2006 04:49 Virgil schrieb: > That axiom of infinity says, in symbols, there is an x such that {} is a > member of x and FOR ALL y, if y is a member of x so is (y union {y}). That is a quantifyer which is not expressed as a word "all" meaning that all are there. (In text versions we have only: if y is in x then {y} is in x (Zermelo's version).) "FOR ALL" concerns all those y which are in x but it does not state that a set of all y did exist. Because then it would be easy to define this set by: "The set of all y". > Regards, WM
From: mueckenh on 6 Oct 2006 04:53 MoeBlee schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > 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 text tells you that part of the axiom of infinity is that n+1 > exists if n is given? If x belongs to the set then {x} belongs to it. From thia n --> n+1 can be proved. > > And the quantifier 'for all' is part of the axiom of infinity: > > There exists an x such that 0 is a member of x and, for all y, if y is > a member of x then yu{y} is a member of x. > > In symbols: > > Ex(0ex & Ay(yex -> yu{y]ex)) I know these symbols. A means "for all" y which are there, it does not state that "all" y are there. Regards, WM
From: mueckenh on 6 Oct 2006 04:57 Dik T. Winter schrieb: > 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. I do so, when I contradict your position but I cannot do so when I explain the correct position. Regards, WM
From: mueckenh on 6 Oct 2006 05:08 Dik T. Winter schrieb: > 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. That is correct according to set theory but it is obviously unserious because, in a serious theory, the result even of a Gedanken-experiment cannot depend on the labels attached. Therefore set theory is not capable of any useful aplication, even in its own domain, the description of infinite sets. Further the first result is incorrect if the fact is observed that the number of balls in A increases continuously. So we have the result: At noon all balls are out of A, but in A there are not less (even some more) balls than outside. This is exactly what I call a silly (i.e. unserious) result. Regards, WM
From: mueckenh on 6 Oct 2006 05:27
Dik T. Winter schrieb: > 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? I read many texts. (Don't intermingle the quantity "all" with the quanifyer "forall".) > > > > 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". I imply that "every position" is equivalent with "up to evey position" in a linear set. I imply that "every specific position" is equivalent with "up to evey specific position" in a linear set. Not more. > > > 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. Of course. There cannot be a disproof, because a disproof would not be a disproof. > > > > 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 It proves in essence and out of essence that there are set and subsets of real numbers which are not countable. >proves the theorem given in > the first sentence of that paragraph. It is simply a corrollary of the > theorem actually proven. Your reaction shows me why set theory can never be conradicted. It is a religion. > > > 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. Of course. And all these sets imaginable at that time were subsets of 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? Because in 1933 it had become clear that the proof without modification was incorrect for the reals. Regards, WM |