An uncountable countable set
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From: imaginatorium on 31 Oct 2006 15:04 Mike Kelly wrote: > Tony Orlow wrote: > > Mike Kelly wrote: > > > Tony Orlow wrote: <snip> > > > 2) How come noon "exists" in this experiment but it didn't exist in the > > > original experiment? Or did you give up on claiming noon doesn't > > > "exist"? What does that mean, anyway? > > > > Nothing is allowed to happen at noon in either experiment. > > Nothing "happens" at noon? I take this to mean that there is no > insertion or removal of balls at noon, yes? Well, I agree with that. Hmm. Yes, there is no ball whose insertion time or removal time is noon. But it seems to me that this "happen" is underdefined in a way that can cause confusion. Does something "happen" to either of these functions at x=0: f(x) = 1 if x<0 ; 0 if x>=0 g(x) = 1 if x<=0 ; 0 if x>0 It seems to me that it is true (within the accuracy of normal communication) to say that both f() and g() "drop from 1 to 0 at x=0" even though the functions are different. Similarly, it seems to me that clearly something "happens" (in any normal sense) at noon in the standard vase problem - what happens is that the frenzy of unending sequences of insertion and removal come to a halt. Good luck with the rest of this anyway... <g> > How about this experiment, does noon "exist" in this experiment : > > Insert a ball labelled "1" into the vase at one minute to noon. > > ? > > >They both end up with countably many balls in the vase at noon. > > For now, I am going to try to restrict myself to discussing this new > experiment, because I want to understand what "noon doesn't exist" is > supposed to mean. And, again, your answer is ambiguous. I asked which > balls are in the vase at noon, not the cardinality of the set of balls > in the vase at noon. I then asked whether "noon exists", not whether > anything "happens" at noon. Please try answering the questions people > actually ask; it aids in communication. Of course, using words with their standard meanings would also be a big help - who knows what "countably" means in Tony's claim above. Brian Chandler http://imaginatorium.org
From: Randy Poe on 31 Oct 2006 15:07 Lester Zick wrote: > On 30 Oct 2006 19:50:42 -0800, "Randy Poe" <poespam-trap(a)yahoo.com> > wrote: > > > > >Lester Zick wrote: > >> It doesn't? My mistake. So there's "no least real" > > > >There's no least real. > > > >> but a "least integer real"? > > > >There's no least integer. > > > >There's a least POSITIVE integer. Is there some reason you > >keep ignoring the critical word POSITIVE? > > I don't ignore it. I considered it as understood You shouldn't consider it understood. That's plain wrong. When we say "the real numbers" we certainly aren't restricting ourselves to positive reals. > but even if you > include it above such that you have "there is a positive least > integer" and "there is no positive least real" you're still stuck with > the implication that there is a distinction between integers and > reals. That's certainly true. There is indeed a distinction between integers and reals. The integers are a subset of the reals. In general, a subset A of set B can have different properties. B could have a least member, and A not. Or A could have a least member, and B not. This difference does not take away the subset relationship. Example: A = {x real : x > 2} B = {x real: x >= 2} A is a proper subset of B. B has a least member. A doesn't. Example: A = {x real : x >= 2} B = {x real: x > 1} A is a proper subset of B. A has a least member. B doesn't. - Randy
From: Lester Zick on 31 Oct 2006 15:09 On 30 Oct 2006 16:52:02 -0800, "MoeBlee" <jazzmobe(a)hotmail.com> 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. Huh? Maybe you could run that by us again, Moe. ~v~~
From: Lester Zick on 31 Oct 2006 15:09 On Mon, 30 Oct 2006 21:34:52 -0500, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >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. Nor does it appear that you know what words mean in combination. ~v~~
From: Virgil on 31 Oct 2006 15:09
In article <1162276209.968986.110750(a)f16g2000cwb.googlegroups.com>, imaginatorium(a)despammed.com wrote: > Virgil wrote: > > In article <1162268163.368326.64650(a)m73g2000cwd.googlegroups.com>, > > imaginatorium(a)despammed.com wrote: > > > > > Virgil wrote: > > > > In article <45462ba0(a)news2.lightlink.com>, > > > > Tony Orlow <tony(a)lightlink.com> 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: > > > > > > <snipola> > > > > > > > > >> 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. > > > > > > > > It is the only definition of set size that is known to produce a valid > > > > partial ordering on sets. > > > > > > Huh? I thought cardinality produced a valid *total* ordering on sets. > > > > The cardinalities are totally ordered, but the sets are not. > > A total order on sets would require that when neither of two sets was > > "larger than" the other that they must be the same set, not merely the > > same size. > > Oh, right. But - and I'm not quite sure how to say this, but the > cardinalities _are_ totally ordered; for any two sets A and B, c(A) < > c(B), or c(A) = c(B), or c(A) > c(B). If you "reduce" the sets by the > cardinality equivalence relation, they are totally ordered. The subset > relation doesn't lead to an equivalence relation, only a partial > ordering: so there is no s(A) = s(B) unless A=B; but for most pairs of > sets, the subset relation simply says nothing at all. (Until His > Master's Voice is heard, telling us something totally arbitrary.) > > Anyway, your claim was clearly wrong, since the subset relation > provides a valid partial ordering on sets. Since there are sets that the subset relation cannot compare for "size", since trichotomy does not hold for the subset relation, I do not regard the subset relation as a valid measure of size in the same sense that cardinality is, at least assuming the axiom of choice. |