An uncountable countable set
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From: Ross A. Finlayson on 13 Oct 2006 08:31 Han de Bruijn wrote: > Ross A. Finlayson wrote: > > > That helps to explain why I am number one. > > Yeah, it's lonely at the top. > > But I think the best place for you is the stock market, not mathematics. > > Han de Bruijn Han, then I would just run my linear model solver and test making money. You see, you can compute a perfect derivative curve, and, then, when you run it, lose it all, because there is not perfect information. Would you rather I do that? Don't get me wrong: I always feel like I'm done. Still, I wonder: when I closed Virgil down, was that not funny? Han, I'm about done with status quo mathematics. Ross
From: Randy Poe on 13 Oct 2006 08:56 Tony Orlow wrote: > Randy Poe wrote: > > Tony Orlow wrote: > >> Randy Poe wrote: > >>> Tony Orlow wrote: > >>>> David Marcus wrote: > >>>>> Virgil wrote: > >>>>>> In article <452d11ca(a)news2.lightlink.com>, > >>>>>> Tony Orlow <tony(a)lightlink.com> wrote: > >>>>>> > >>>>>>>> I'm sorry, but I can't separate your statement of the problem from your > >>>>>>>> conclusions. Please give just the statement. > >>>>>>> The sequence of events consists of adding 10 and removing 1, an infinite > >>>>>>> number of times. In other words, it's an infinite series of (+10-1). > >>>>>> That deliberately and specifically omits the requirement of identifying > >>>>>> and tracking each ball individually as required in the originally stated > >>>>>> problem, in which each ball is uniquely identified and tracked. > >>>>> It would seem best to include the ball ID numbers in the model. > >>>>> > >>>> Changing the label on a ball does not make it any less of a ball, and > >>>> won't make it disappear. If I put 8 balls in an empty vase, and remove > >>>> 4, you know there are 4 remaining, and it would be insane to claim that > >>>> you could not solve that problem without knowing the names of the balls > >>>> individually. > >>> That's a red herring. It's not the name of the ball that's relevant, > >>> but whether for any particular ball it is or isn't removed. > >> The "name" is the identity. It doesn't matter which ball you remove, > >> only how many at a time. > >> > >>>> Likewise, adding labels to the balls in this infinite case > >>>> does not add any information as far as the quantity of balls. > >>> No, but what the labels do is let us talk about a particular > >>> ball, to answer the question "is this ball removed"? > >> We care about the size of the collection. If replacing the elements with > >> other elements changes the size of the set, then you are doing more than > >> exchanging elements. > >> > >>> If there is a ball which is not removed, whatever label > >>> is applied to it, then it is still in the vase. > >> How convenient that you don't have labels for the balls that transpire > >> arbitrarily close to noon. You don't have the labels necessary to > >> complete this experiment. > >> > >>> If there is a ball which is removed, whatever label is > >>> applied to it, then it is not in the vase. > >> If a ball, any ball, is removed, then there is one fewer balls in the vase. > >> > >>>> That is > >>>> entirely covered by the sequence of insertions and removals, quantitatively. > >>> Specifically, that for each particular ball (whatever you > >>> want to label it), there is a time when it comes out. > >>> > >> Specifically, that for every ball removed, 10 are inserted. > > > > All of which are eventually removed. Every single one. > > > > Every single one, Yes. > each after another ten are inserted, of course. And I can tell you the time that each of those is removed. > Come on! Come on yourself. You *know* there is a removal time associated with every ball. - Randy
From: Dik T. Winter on 13 Oct 2006 10:11 In article <1160669646.780939.326190(a)b28g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: .... > I wrote: > > > The axiom of infinity does only state n+1 exists if n is given. It is > > > realized by he numbers of my list. > You replied: > > 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. > And I reply: > That is wrong. To quote you: > > > AXIOM OF INFINITY Vla There exists at least one set Z with the > > > following properties: > > > (i) O eps Z > > > (ii) if x eps Z, also {x} eps Z. > The axiom says: For all possible n in all possible > cases: If n is a natural number, then n+1 is a natural number too. It > does *not* say that it is meaningful to speak of *all* natural numbers. It states: "there exists at least one set Z". > It does not say that the set N does actually or completely exists. What does the statement "there exists at least one set Z" mean if it does not mean that? > Well: "there is a set". But the meaning of > these three words depends heavily on the interpretation. What other interpretation can you give for "there exists at least one set Z"? -- 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 13 Oct 2006 10:14 In article <1160669820.603144.288450(a)e3g2000cwe.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: .... > > > A set containing all positions "up to position x" is a superset of a > > > set containing "position x". > > > > By what rule? What do you *mean* by "a set containing all positions..."? > > What do you *mean* by "a set containing...". I would state the the > > set {1, 2, 3, 4} contains all positions up to position 4, but that the > > set {4, 5} contains the position 4. But neither is a superset of the > > other. > > A set containing all positions "up to position x" is a superset of a > set containing just "position x". Ok. > > > A set containing "up to every position" defines a superset of set > > > containing "every position". But "every position" cannot be a proper > > > subset. Hence both sets are equivalent. > > > > Please first answer my question above, next, elaborate. > > A set containing all positions "up to every position" is a superset of > a set containing just "every position". What do you mean with 'a set containing all positions "up to every position"'? But I would agree. The set N is such a set. That does not make N a natural number, but the elements can (by it's very definition) be indexed by the natural numbers. -- 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 13 Oct 2006 10:27
In article <1160669936.904325.187140(a)m73g2000cwd.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: .... > > Oh. Whatever. Care to explain? > > 0.111... is the representation of a number, e.g., of 1/9 in decimal > notation. And this representation is not unique, because the indexes > are undefined. I have no idea what you mean here. I thought the indices were 1, 2, 3, 4, ..., i.e. the natural numbers. What is undefined about that? > Therefore there is no unique number 0.111... as > you erroneously stated. With 0.111... this does not matter, but with > 3.1415... we see that this number is not well defined. Well, that is not the way to *define* that number, of course. And, indeed, as a string, 3.1415... has no meaning at all in mathematics. It is commonly understood that pi is intended, but there is no actual definition for that string at all. And to *define* pi, there are quite a few actual definitions (that all can be shown to be equivalent). -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |