An uncountable countable set
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From: David R Tribble on 13 Oct 2006 22:14 Tony Orlow wrote: >> 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). > Virgil wrote: >> 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. > Tony Orlow wrote: > The original statement contrasted two situations which both matched this > scenario. The difference between them was the label on the ball removed > at each iteration, and yet, that's not relevant to how many balls are in > the vase at, or before, noon. How about a slightly different, but equivalent, approach to the problem? At each moment 2^-n sec prior to noon, add 10 balls (10n+1, 10n+2, ...., 10n+10) to the vase. (Tony, the numbering works out if you start with n=0.) Obviously, the insertions stop by noon. Now at each moment 2^-n sec before 1:00pm, remove ball n from the vase. (Tony, the numbering works if you start with n=1.) Obviously, the removals stop by 1:00pm. So how many balls are left in the vase at 1:00pm?
From: Virgil on 13 Oct 2006 22:18 In article <4530430e$1(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > In the axiom of infinity, S(x) is defined as x union {x}. > > In Peano, there is successor. The Axiom of Infinity employs ordinals - a > big mistake. TO is the one making the only big mistake involved here.
From: Virgil on 13 Oct 2006 22:19 In article <4530434f(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > David Marcus wrote: > > Han de Bruijn wrote: > >> Virgil wrote: > >> > >>> In article <9020$452f46c4$82a1e228$31963(a)news2.tudelft.nl>, > >>> Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > >>> > >>>> Virgil wrote about the Balls in a Vase problem: > >>>> > >>>>> Everything takes place before noon, so that by noon, it is all over and > >>>>> done with. > >>>> Noon is never reached, because your concept of time is a fake. > >>> > >>> No one expects the experiment to take place anywhere except in the > >>> imagination, so that everything about it, including its time, is > >>> imaginary, but logic continues to hold even there, at least for > >>> mathematicians. And logic says that a ball removed from a vase is not > >>> later in the vase. > >> Since your logic and the logic of others give contradictory results for > >> the same problem, logic alone is unreliable. > > > > Are you saying that Mathematics gives contradictory results for a > > problem? If so, please state the problem. > > > > The problem, among others, is the vase. If you haven't gotten a clue > about the problem yet, well, get on the bus. Along with TO, HdB, "Mueckenh", Ross, and others of that ilk. I am surprised that JSH has not joined in.
From: Dik T. Winter on 13 Oct 2006 22:13 In article <452fc558$1(a)news2.lightlink.com> Tony Orlow <tony(a)lightlink.com> writes: .... > So, we can interpret the empty set as 0, the origin, and then define > successor any way we want. IF we define the successor of n as n+1, then > we get the naturals. If we define the successor as 1-1/2(1-n), then we > get our Zeno moments. The inductive set produced depends on what the > null set represents and how successor is defined. This, again, is putting the cart before the horse. What do those expressions *mean* when arithmetic is not defined? Again, arithmetic is defined, based on the Peano axioms. Not the other way around. And when you have defined arithmetic on the natural numbers (based on the axioms), have stated what the rationals are, and what the reals are and all that stuff, it is only *then* that you can conclude that there is a bijection between the "natural numbers" and the subset of the reals defined by 1-1/2(1-n). Indeed, the Zeno moments. Which does *not* imply that those real numbers do not have a limit when n goes to infinity (although n does not have a limit). -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Virgil on 13 Oct 2006 22:22
In article <45304389(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > David Marcus wrote: > > Consider this situation: Start with an empty vase. Add a ball at time 5. > > Remove it at time 6. > > > > How you would translate that into Mathematics? > > > > What happens between times 5 and 6? Are there other balls involved? If TO can't do it when it involves only one ball, what makes him think he can juggle infinitely many of them any more successfully? |