An uncountable countable set
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From: David R Tribble on 13 Oct 2006 21:52 Tony Orlow wrote: >> then there are naturals which are >> the result of infinite increments, which must have infinite value. > David R Tribble wrote: >> Where's your proof? >> What is an "infinite increment" (or "infinite successor")? > Tony Orlow wrote: > If there is a !number! n of successors, there exists a successor n steps > ahead. If there are an infinite !number! n of successors, there is a > successor n, an infinite number of steps ahead. > > If you increment a natural n times, you have added n to it. If successor > is increment, and there are an infinite !number! of such increments, you > have added this infinite number to your starting value. Adding an > infinite number to a finite yields an infinite. Therefore, the infinite > set includes infinite values. So how do you know when you've reached the point of adding an infinite number of increments? Is there some way of counting all the increments?
From: Tony Orlow on 13 Oct 2006 21:53 David Marcus wrote: > Tony Orlow wrote: >> Han de Bruijn wrote: >>> Virgil wrote: >>> >>>> http://en.wikipedia.org/wiki/ZFC >>>> Axiom of infinity: There exists a set x such that the empty set is a >>>> member of x and whenever y is in x, so is S(y). >> 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. > > No. Yes. 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. However, > the details of axiomatic set theory aren't really relevant to the > present discussion. > Indeed they are.
From: Tony Orlow on 13 Oct 2006 21:54 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.
From: Tony Orlow on 13 Oct 2006 21:55 David Marcus wrote: > Han de Bruijn wrote: >> Virgil wrote: >> >>> In article <66a8$452f4298$82a1e228$30886(a)news2.tudelft.nl>, >>> Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: >>> >>>> Randy Poe wrote about the Balls in a Vase problem: >>>> >>>>> Tony Orlow wrote: >>>>>> Specifically, that for every ball removed, 10 are inserted. >>>>> All of which are eventually removed. Every single one. >>>> All of which are eventually inserted. Every single one. >>> None are reinserted after being removed but,each is removed after having >>> been inserted, so that leaves them all outside the vase at noon. >> Huh! Then reverse the process: first remove 1, then insert 10. It must >> be no problem in your "counter intuitive" mathematics to start with -1 >> balls in that vase. > > 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?
From: David R Tribble on 13 Oct 2006 21:58
Tony Orlow wrote: >> That doesn't seem "real", and the axiom of choice aside, I don't see >> there being any well ordering of the reals. The closest one can come is >> the H-riffic numbers. :) > David R Tribble wrote: >> Hardly. The H-riffics are a simple countable subset of the reals. >> Anyone mathematically inclined can come up with such a set. > Tony Orlow wrote: >> You never paid enough attention to understand them. They cover the reals. > David R Tribble wrote: >> They omit an uncountable number of reals. Any power of 3, for example, >> which you never showed as being a member of them. Show us how 3 fits >> into the set, then we'll talk about "covering the reals". > Tony Orlow wrote: > 3 is an unending string, just like 1/3 is in base-10. Rusin confirmed > that about two years ago. But, you're right, I need to construct a > formal proof of the equivalence between the H-riffics and the reals. Your definition of your H-riffic numbers excludes unending strings. So 3 can't be a valid H-riffic, and neither can any of its successors. I know you don't get this, but go back and read your own definition. Every H-riffic corresponds to a node in an infinite, but countable, binary tree. The H-riffics is only a countable subset of the reals, and omits an uncountable number of reals. |