An uncountable countable set
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From: Tony Orlow on 28 Sep 2006 11:39 Randy Poe wrote: > Tony Orlow wrote: >> Virgil wrote: >>> In article <451b3296(a)news2.lightlink.com>, >>> Tony Orlow <tony(a)lightlink.com> wrote: >>> >>>> Randy Poe wrote: >>>>> Tony Orlow wrote: >>>> You must have been a strange 10 year old, like that kid >>>> down the block that used to pull the legs off of roaches. >>> Only those that looked like TO. >>> >>>>>>> So the reason I don't say it's full "an infinitesimal time >>>>>>> before noon" or "some other time before noon" is that >>>>>>> I don't say it's full. >>>>>> But, you do say it's full or empty, right? >>> One can easily say that it is empty at any time at which every ball >>> that was put in has been taken out again. >>> >>> Does TO suggest that at any time after noon there is any ball that was >>> put in that was not also taken out? >> Yes, at any given time 9/10 of the balls inserted remain. > > Which ball does not have a definite time at which it > is removed? > Any ball which does not have a definite time at which it is inserted. >>>>> So your conclusion from my statement that I would never >>>>> say it's full is that sometimes I would say it's full? >>>> Uh, you would say it contains an infinite number of balls in some >>>> circumstances, as I understand it. >>> Then you misunderstand it. >> No, your labels misconstrue the problem with your silly fixation on >> omega. Do I "misunderstand" that if you remove balls 1, then 11, then >> 21, etc, that the vase will NOT be empty? > > We have different variants of the problem setup. Before > discussing too many details, we need to agree on > what EXACTLY are the starting assumptions. The subject is whether that makes any difference or not. It doesn't. Your dual gedankens imply that changing the labeling scheme after noon makes the balls all disappear. That's ridiculous. > > But in general if: > (a) Every ball has a label n which is a finite natural number. > (b) Every ball has a time t_n at which it is removed. > (c) There exists a supremum T of the set {t_n, n in N} > then for any time t >= T, the vase is empty. What is this "supremum", in terms of iterations? Zeno machines are fine for paradoxes, if that's what you WANT to produce. The greatest lovers of math want to discover truths, not invent fictions. So, what is this greatest finite number of iterations you claim to have? Here's one iteration: (a) 10 balls added AND (b) 1 ball removed IMPLIES (c) net 9 balls added How many iterations? n? Fine. 9n balls remain. > >>>> If you say it empties, then you would agree that it either fills or it >>>> empties. When does it empty? You say, not before noon. You also say >>>> this does not occur at noon, but after noon there are no balls left. So >>>> when does this occur? >>> When every ball that was put in has also been taken out again. >> At noon or before noon? You're skirting the issue. > > In some of our setups, noon is a supremum, and no time > before noon is a supremum. Therefore there is no time > before noon when the vase is empty, and for every > time at noon or after, the vase is empty. Uh, therefore, it happens AT noon? Not before, but from that time afterwards? Like, starting then? What do you think it means for something to occur at a given time? > > - Randy > (sigh) Tony
From: MoeBlee on 28 Sep 2006 12:05 Han de Bruijn wrote: > It's a priorities issue. Do axioms have to dictate what constructivism > should be like? Should constructivism be tailored to the objectives of > axiomatics? I think not. Fine, but if you don't give a formal system, then your mathematical arguments are not subject to the objectivity of evaluation that arguments backed up by formal systems are subject to. MoeBlee
From: stephen on 28 Sep 2006 12:05 Tony Orlow <tony(a)lightlink.com> wrote: > stephen(a)nomail.com wrote: >> Randy Poe <poespam-trap(a)yahoo.com> wrote: >> >> <snip> >> >>> What is the number of the ball which, when removed, >>> makes the vase empty? >> >>> I know the kind of nonsense you will spout in answer to >>> those questions, but the answers within our axiom system >>> are: (1) there is no t<noon which is the moment just >>> before noon. For any t<noon, there is t < t' < noon. >>> (2) There is no such ball. >> >>> Here are the Tony gobbledgook answers: >>> (1) noon - 1/oo >>> (2) Ball number omega >> >>> In TO-matics, one can confidently give an answer like >>> number 2 despite the fact that one can also agree >>> that no ball numbered omega is ever put into the >>> vase. >> >> In TO-matics, it is also possible to end up with >> an empty vase by simply adding balls. According to TO-matics >> >> ..1111111111 = 1 + 1 + 1 + 1 + ... >> >> and >> ..1111111111 + 1 = 0 >> >> So if you just keep on adding balls one at a time, >> at some point, the number of balls becomes zero. >> You have to add just the right number of balls. It is not >> clear what that number is, but it is clear that it >> exists in TO-matics. >> >>> But in mathematics and logic, we don't get to >>> keep a set of self-contradictory assumptions around, >>> only using the ones we want as needed. >> >>> - Randy >> >> Where's the fun in that? :) >> >> Stephen > You drew that from my suggestion of the number circle, and that ...11111 > could be considered equal to -1. Since then, I looked it up. I'm not the > first to think that. It's one of two perspectives on the number line. > It's either really straight, or circular with infinite radius, making it > infinitesimally straight. The latter describes the finite universe, and > the former, the limit. But, you knew that, and are just trying to have fun. > Tony I am just pointing out that according to your mathematics that if you keep adding balls to the vase, you can end up with an empty vase. The fact that other people may have considered a number circle does not change the fact that the number circle implies that if you keep on adding balls, eventually you will have zero balls. So why is it okay to end up with zero balls, when you never remove any at all, but it is not okay to end up with zero balls when each ball is clearly removed at a definite time? Stephen
From: MoeBlee on 28 Sep 2006 12:11 Tony Orlow wrote: > You might want to expand your reading. That's rich coming from a guy who hasn't read a single book on mathematical logic or set theory. MoeBlee
From: Tony Orlow on 28 Sep 2006 12:33
stephen(a)nomail.com wrote: > Tony Orlow <tony(a)lightlink.com> wrote: >> stephen(a)nomail.com wrote: >>> Randy Poe <poespam-trap(a)yahoo.com> wrote: >>> >>> <snip> >>> >>>> What is the number of the ball which, when removed, >>>> makes the vase empty? >>>> I know the kind of nonsense you will spout in answer to >>>> those questions, but the answers within our axiom system >>>> are: (1) there is no t<noon which is the moment just >>>> before noon. For any t<noon, there is t < t' < noon. >>>> (2) There is no such ball. >>>> Here are the Tony gobbledgook answers: >>>> (1) noon - 1/oo >>>> (2) Ball number omega >>>> In TO-matics, one can confidently give an answer like >>>> number 2 despite the fact that one can also agree >>>> that no ball numbered omega is ever put into the >>>> vase. >>> In TO-matics, it is also possible to end up with >>> an empty vase by simply adding balls. According to TO-matics >>> >>> ..1111111111 = 1 + 1 + 1 + 1 + ... >>> >>> and >>> ..1111111111 + 1 = 0 >>> >>> So if you just keep on adding balls one at a time, >>> at some point, the number of balls becomes zero. >>> You have to add just the right number of balls. It is not >>> clear what that number is, but it is clear that it >>> exists in TO-matics. >>> >>>> But in mathematics and logic, we don't get to >>>> keep a set of self-contradictory assumptions around, >>>> only using the ones we want as needed. >>>> - Randy >>> Where's the fun in that? :) >>> >>> Stephen > >> You drew that from my suggestion of the number circle, and that ...11111 >> could be considered equal to -1. Since then, I looked it up. I'm not the >> first to think that. It's one of two perspectives on the number line. >> It's either really straight, or circular with infinite radius, making it >> infinitesimally straight. The latter describes the finite universe, and >> the former, the limit. But, you knew that, and are just trying to have fun. > >> Tony > > I am just pointing out that according to your mathematics > that if you keep adding balls to the vase, you can end up > with an empty vase. The fact that other people may have > considered a number circle does not change the fact that the > number circle implies that if you keep on adding balls, eventually > you will have zero balls. That's a bastardization of the concept. There are two ways to look at the number circle, and you are combining them in mutually contradictory ways. > > So why is it okay to end up with zero balls, when you never remove > any at all, but it is not okay to end up with zero balls when > each ball is clearly removed at a definite time? Because the model of the number circle where all strings are positive is incompatible with the number circle where any string with a left-unending string of 1's (in binary) is negative. Duh. > > Stephen > > > |