An uncountable countable set
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From: MoeBlee on 30 Oct 2006 20:45 Lester Zick wrote: > According to MoeBlee mathematical definitions require a "domain of > discourse" variable such as IN(x) and OUT(x). I did not say that. Regarding a particular definition I gave, I explained to you that the variable ranges over the domain of discourse of any given model. I didn't say, in general, that definitions "require" variables (some defininitonal forms do require variables, but not all definitions do), nor did I suggest using variables in the mindless way you have done in certain examples you've posted that misrepresent the actual definition I gave. MoeBlee
From: David Marcus on 30 Oct 2006 21:29 Virgil wrote: > In article <45463580(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > > David Marcus wrote: > > > > Do you say that g(0) = 0? > > > > No, I say that lim(x->0: g(x))=oo. > > Does that mean that TO can find any ball inserted before noon which is > not removed before noon? Virgil, the post you are replying to presented a problem without balls and time. Please don't put balls and time in where they don't belong! -- David Marcus
From: David Marcus on 30 Oct 2006 21:34 MoeBlee wrote: > Tony Orlow wrote: > > stephen(a)nomail.com wrote: > > > > The point is, there are different types of numbers, and statements > > > that are true of one type of number need not be true of other > > > types of numbers. > > > Well, then, you must be of the opinion that set theory is NOT the > > foundation for all mathematics, but only some particular system of > > numbers and ideas: a theory. That's good. > > Whether he thinks set theory is or is not a foundation, it doesn't > follow that he should not think it is a foundation simply because there > are different kinds of numbers. It certainly appears that Tony doesn't know what the word "foundation" means in this context. -- David Marcus
From: Tony Orlow on 30 Oct 2006 21:53 Randy Poe wrote: > Tony Orlow wrote: >> stephen(a)nomail.com wrote: >>> Tony Orlow <tony(a)lightlink.com> wrote: >>>> stephen(a)nomail.com wrote: >>>>> Tony Orlow <tony(a)lightlink.com> wrote: >>>>>> stephen(a)nomail.com wrote: >>>>> <snip> >>>>> >>>>>>> What does that have to do with the sets IN and OUT? IN and OUT are >>>>>>> the same set. You claimed I was losing the "formulaic relationship" >>>>>>> between the sets. So I still do not know what you meant by that >>>>>>> statement. Once again >>>>>>> IN = { n | -1/(2^(floor(n/10))) < 0 } >>>>>>> OUT = { n | -1/(2^n) < 0 } >>>>>>> >>>>>> I mean the formula relating the number In to the number OUT for any n. >>>>>> That is given by out(in) = in/10. >>>>> What number IN? There is one set named IN, and one set named OUT. >>>>> There is no number IN. I have no idea what you think out(in) is >>>>> supposed to be. OUT and IN are sets, not functions. >>>>> >>>> OH. So, sets don't have sizes which are numbers, at least at particular >>>> moments. I see.... >>> If that is what you meant, then you should have said that. >>> And technically speaking, sets do not have sizes which are numbers, >>> unless by "size" you mean cardinality, and by "number" you include >>> transfinite cardinals. >> So, cardinality is the only definition of set size which you will >> consider.....your loss. >> >>> In any case, it still does not make any sense. I am not sure >>> what |IN| is for any n. IN is a single set. There is only >>> one set, and it does not depend on n. In fact, there isn't >>> an n specified in the problem. Yes I used the letter n in >>> the set description, but that does not define an entity named 'n'. >>> >> There most certainly is an 'n'. The problem describes a repeating >> process, each repetition of which is indexed with a successive n in n, >> and during each repetition of which ball n is removed. What do you mean >> there's no n??? > > The ORIGINAL problem. This is a new one, inspired by > the original, but it is one with no balls, no vases, no > time steps, no iterations. Just a definition of two subsets > of the natural numbers, one called IN and one called > OUT. > > The definition of the set IN does not include a definition > of something called IN(n). > > You are being asked to characterize these two subsets. > > - Randy > Set-theoretically, they are the same set, by the axiom of extensionality. That doesn't mean the axioms of ZFC account for all the information in the problem. There is the combining of +10 and -1 in each iteration n at time t=-1/n, a coupling that is not being addressed by your method. You are considering the two sets statically, outside of time, as completed, but for any n, the max of in is 10 times the max of out. Just because these both have some limit at oo, even though they don't reach it, doesn't mean they reach it at the same time. They DON'T reach it, and if they did, if noon occurred, in would reach it in 1/10 the time as out. But,there is no such ending to the finites, and so the set-theoretic approach using the set N is bogus at its core. - Tony
From: David Marcus on 30 Oct 2006 22:02
Tony Orlow wrote: > Randy Poe wrote: > > Tony Orlow wrote: > >> stephen(a)nomail.com wrote: > >>> Tony Orlow <tony(a)lightlink.com> wrote: > >>>> stephen(a)nomail.com wrote: > >>>>> Tony Orlow <tony(a)lightlink.com> wrote: > >>>>>> stephen(a)nomail.com wrote: > >>>>> <snip> > >>>>> > >>>>>>> What does that have to do with the sets IN and OUT? IN and OUT are > >>>>>>> the same set. You claimed I was losing the "formulaic relationship" > >>>>>>> between the sets. So I still do not know what you meant by that > >>>>>>> statement. Once again > >>>>>>> IN = { n | -1/(2^(floor(n/10))) < 0 } > >>>>>>> OUT = { n | -1/(2^n) < 0 } > >>>>>>> > >>>>>> I mean the formula relating the number In to the number OUT for any n. > >>>>>> That is given by out(in) = in/10. > >>>>> What number IN? There is one set named IN, and one set named OUT. > >>>>> There is no number IN. I have no idea what you think out(in) is > >>>>> supposed to be. OUT and IN are sets, not functions. > >>>>> > >>>> OH. So, sets don't have sizes which are numbers, at least at particular > >>>> moments. I see.... > >>> If that is what you meant, then you should have said that. > >>> And technically speaking, sets do not have sizes which are numbers, > >>> unless by "size" you mean cardinality, and by "number" you include > >>> transfinite cardinals. > >> So, cardinality is the only definition of set size which you will > >> consider.....your loss. > >> > >>> In any case, it still does not make any sense. I am not sure > >>> what |IN| is for any n. IN is a single set. There is only > >>> one set, and it does not depend on n. In fact, there isn't > >>> an n specified in the problem. Yes I used the letter n in > >>> the set description, but that does not define an entity named 'n'. > >>> > >> There most certainly is an 'n'. The problem describes a repeating > >> process, each repetition of which is indexed with a successive n in n, > >> and during each repetition of which ball n is removed. What do you mean > >> there's no n??? > > > > The ORIGINAL problem. This is a new one, inspired by > > the original, but it is one with no balls, no vases, no > > time steps, no iterations. Just a definition of two subsets > > of the natural numbers, one called IN and one called > > OUT. > > > > The definition of the set IN does not include a definition > > of something called IN(n). > > > > You are being asked to characterize these two subsets. > > > > - Randy > > > > Set-theoretically, they are the same set, by the axiom of > extensionality. That doesn't mean the axioms of ZFC account for all the > information in the problem. There is the combining of +10 and -1 in each > iteration n at time t=-1/n, a coupling that is not being addressed by > your method. You are considering the two sets statically, outside of > time, as completed, but for any n, the max of in is 10 times the max of > out. Just because these both have some limit at oo, even though they > don't reach it, doesn't mean they reach it at the same time. They DON'T > reach it, and if they did, if noon occurred, in would reach it in 1/10 > the time as out. But,there is no such ending to the finites, and so the > set-theoretic approach using the set N is bogus at its core. It is remarkable that you seem to be unable to answer any post without mentioning the ball and vase problem. Why is this? Are you afraid that if you do, you will be trapped into an inconsistentcy? Is the following an accurate description of what you are saying? 1. You agree that (given the definitions above) IN = OUT and that |IN \OUT| = 0. 2. You don't agree that, in the ball and vase problem, the number of balls in the vase at noon is |IN\OUT|. -- David Marcus |