An uncountable countable set
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From: Virgil on 20 Oct 2006 14:50 In article <45390ce5(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > > The state of affairs with regard to the VASE. Ball 15 is obviously out > of the vase. All finitely numbered balls are out. But not before noon. > And if the experiment continues UNTIL noon, infinitely-numbered balls > are added. Where do these phantom balls appear from? Spontaneous generation? > > In light of my observations, why won't you directly address the > > validity of your argument as you /actually stated it/ in your previous > > post, instead of haring off on some new and unrelated argument? > > It's all the same argument, with many different disproofs of the > standard conclusion. But no valid "disproofs". Every alleged disproof requires some assumption not justified by the statement of the problem. > > All of this is logical. The question is which ideas are logically > consistent with each other. I don't think I'm being illogical, or that > many here are. It's just that we're working with incompatible assumptions. We who see the vase as empty at noon assume nothing other than ZF and the statement of the problem. > > > > > For example, how does your response above /in any way/ address my > > statement "because t=0 is /not/ a time such that t=-1/n for some > > natural n"? > > All events happen before noon. At all times before noon, there are balls > in the vase. At noon they are gone. Either something happened to the > balls at noon, or it happened sometime between then and every moment > before noon, which doesn't even make sense. Neither do any of TO's analyses, but a discontinuity at noon is at least mathematically possible within the constraints of the problem , whereas all of TO's solutions contravene the constraints of the problem. > > Now, say something does happen to the balls in the vase at noon, and you > get to remove all of the naturals completely. At noon, 1/n=0, so n is > infinite, and to continue the process until noon, you need to insert > infinitely-numbered balls in the vase. TO may need to but no one who goes by the statement of the problem needs to.
From: Virgil on 20 Oct 2006 14:52 In article <45390fcc(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Either the experiment goes until noon or it doesn't. If all moments are > naturally indexed, then it doesn't. So that natural indexing won't work. But time indexing with discontinuous values at each transition and at noon does.
From: MoeBlee on 20 Oct 2006 14:55 Han de Bruijn wrote: > > Anyway, what is confusing about the theorem that there exist finite > > ordinals and that there exist infinite ordinals? What do you find confusing about the following statement: There exists an x such that the following hold: x is well ordered by membership; x is transitive by membership; there is no natural number that is bijectable with x. MoeBlee
From: MoeBlee on 20 Oct 2006 14:57 Han de Bruijn wrote: > I've formulated more than one, _without_ set theory. What is the language, primitives, axioms, and rules of inference? MoeBlee
From: Virgil on 20 Oct 2006 15:54
In article <4539117c(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > I'm reading Non-Standard Analysis instead. Robinson agrees there's no > smallest infinity, and that there are an uncountable number of countable > neighborhoods with what he calls the '~' relation. I'm very encouraged > to see essentially the same ideas as mine, put in technical terms. If TO thinks that is what he is reading, he is not reading very carefully. |