An uncountable countable set
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From: MoeBlee on 5 Oct 2006 16:59 mueckenh(a)rz.fh-augsburg.de wrote: > 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? I'm coming into this part of the conversation late, so I hope I have the context correct. We should be clear. Of course we can make various definitions, but you can see that certain of them are conditional definitions. If the conditions fail, then you cannot apply the definition as if the condition holds. In this case, we don't have a definition of subtraction that allows for substracting a finite ordinal k from w (read 'w' as 'omega'). For a finite ordinal k, we don't have w-k as some kind of ordinal such that there is an x such that k+x=w or even as any kind of object (except by some method of assigning a default value for otherwise non-referring terms, which is a whole other subject). MoeBlee
From: Virgil on 5 Oct 2006 17:12 In article <1160079778.871756.325200(a)m73g2000cwd.googlegroups.com>, 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. Seems to be an ingrained habit. You should endeavor to break it. Informally, the Peano axioms may be stated as follows: 1) 0 is a natural number. 2) Every natural number a has a successor, denoted by Sa or a'. 3) No natural number has 0 as its successor. 4) Distinct natural numbers have distinct successors: a = b if and only if Sa = Sb. 5) If a property holds for 0, and holds for the successor of every natural number for which it holds, then the property holds for ALL natural numbers. Alternately at http://en.wikipedia.org/wiki/ZFC#The_axioms 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}). > > 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 your lack of imagination that is the problem. > > It is crazy to hear that covering up to every number is possible but > covering every number shall be impossible. > > > 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. The sets having larger cardinalities are sets OF numbers, and are not necessarily numbers themselves. > Regards, WM
From: MoeBlee on 5 Oct 2006 17:23 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? 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)) MoeBlee
From: Lester Zick on 5 Oct 2006 18:45 On 5 Oct 2006 11:05:06 -0700, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: >mueckenh(a)rz.fh-augsburg.de wrote: [.. .] >That came out with just boxes for the symbols, and unfortunately I >don't read German. > >Would you give us this information in ASCII and English? ASCII and English? Surely symbols must suffice? ~v~~
From: Lester Zick on 5 Oct 2006 18:47
On Thu, 05 Oct 2006 13:29:03 -0600, Virgil <virgil(a)comcast.net> wrote: >In article <45251b3e(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: [. . .] >In Standard reals,"infinitesimal", if it means anything, merely means >very small but not zero. Very small in relation to what precisely? >In The Robinson, or similar, non-standard models, infinitesimals are >different from standard numbers but still non-zero. And what prevents infinitesimals from being very large? ~v~~ |