An uncountable countable set
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From: Virgil on 22 Oct 2006 18:59 In article <453bc7c9(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <453b326d(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Virgil wrote: > >>> In article <4539000e(a)news2.lightlink.com>, > >>> Tony Orlow <tony(a)lightlink.com> wrote: > > > >>> Claimed but not justified. TO's usual technique! > >> You didn't justify yours. It's clearly nonsensical. It pretends there's > >> a time between noon and all times before noon. > > > > I only claim there is a time between any time before noon and noon. > > > > When does the vase become empty? It is empty at noon and is not empty at any time before noon, but I have no idea what TO means by "When does the vase become empty?", as it seems to imply a continuity at 0 that does not exist. > >> The iterations do not occur in linear time. > > > > They occur in linear time but not at equally spaced intervals in that > > time. Time being linear merely means that all times can be lined up in > > order. > > > > Yeah, and a linear function just means it looks like some kind of a > line. Sure, Virgil. There is a difference between a set being linearly ordered, as the reals are, and as time is in this problem, and a function being linear, which the number-of-balls-as-a-function-of-time is not. So TO is off base again, as usual.
From: Lester Zick on 22 Oct 2006 19:34 On Sun, 22 Oct 2006 05:38:11 -0400, Tony Orlow <tony(a)lightlink.com> wrote: >Lester Zick wrote: >> On Fri, 20 Oct 2006 14:12:27 -0400, Tony Orlow <tony(a)lightlink.com> >> wrote: >> >>> MoeBlee wrote: >>>> Tony Orlow wrote: >>>>> Also, upon which axioms is the definition of cardinality based? >>>> The usual definition is: >>>> >>>> card(x) = the least ordinal equinumerous with x >>>> >>>> The definition ultimately reverts to the 1-place predicate symbol 'e' >>>> (and the 1-place predicate symbol '=', if equality is taken as >>>> primitive). For the definition to "work out" ('work out' is informal >>>> here) in Z set theory, we usually suppose the axioims of Z set theory >>>> plus the axiom of schema of replacement (thus we're in ZF) and the >>>> axiom of choice (thus we're in ZFC). However, there is a way to avoid >>>> the axiom of choice by using the axiom of regularity instead with a >>>> somewhat different definition from just 'least ordinal equinumerous >>>> with'. Also, we could adopt a "midpoint" between the axiom schema of >>>> replacement and the axiom of choice by adopting the numeration theorem >>>> (AxEy y is an ordinal equinumerous with x) instead, which would be a >>>> method stronger than adopting the axiom of choice, but weaker than >>>> adopting both the axiom of choice and the axiom schema of replacement. >>>> As to the more basic axioms of Z, for the definition to "work out", I'm >>>> pretty sure we need extensionality, schema of separation (or schema of >>>> replacement if we go that way), union, and pairing (pairing is not >>>> needed if we have the schema of replacement). I'm not 100% sure, but my >>>> strong guess is that we don't need the power set axiom for this >>>> purpose. And we don't need the axiom of infinity. >>>> >>>> Why don't you just a set theory textbook? >>>> >>>> MoeBlee >>>> >>> I'm reading Non-Standard Analysis instead. Robinson agrees there's no >>> smallest infinity, >> >> Just out of curiosity, Tony, what's his rationale? > >I quoted it for MoeBlee, and you can see, but basically, he extends N to >*N by applying the basic truths about finite numbers to the infinite. >Since it is true for all n in N that, except for 0, every element has a >predecessor n-1, then this must also be true for any infinite n. We >assume some smallest infinite n. Since n-1<n, and since n-1 is infinite >if n is infinite, we have n-1 being a smaller infinity than n, which is >a contradiction. For any smallest n we can assume, there is a smaller >n-1, so there is no smallest infinite, just like there is no greatest >finite. Thanks for the explanation, Tony. I can see what the argument amounts to and basically I agree. But I've become extremely skeptical of the combination of finites and infinites in arithmetic operations in general. I'm beginning to suspect there is no such thing as trans finite arithmetic. I think arithmetic works with finites and calculus with infinites. And the rest is just so much mathematical pretense. Yet I've also been considering what it looks like you're trying to do with trans finite arithmetic.In particular it occurs to me that if one takes +00 to be larger than any positive finite -00 correspondingly must be smaller than any negative finite such that your concept of circularity among arithmetic numbers might be combined in the following way: [-00, . . . 3, 2, 1, 0, 1, 2, 3 . . . +00]. The only difference would be that whereas +00 represents the number of infinitesimals, -00 would represent the size of infinitesimals. Thus we'd have a positive axis with the number of infinitesimals and a negative axis with the size of infinitesimals. At least that's the best I can make of the situation. >>> and that there are an uncountable number of countable >>> neighborhoods with what he calls the '~' relation. I'm very encouraged >>> to see essentially the same ideas as mine, put in technical terms. Maybe >>> when I'm through with that, although I'm also reading Boole's treatise >>> on logic where he eventually talks about probabilistic logic. >> >> ~v~~ ~v~~
From: Ross A. Finlayson on 22 Oct 2006 21:06 Ross A. Finlayson wrote: > Naturals: countable AND uncountable. > > It's like, where the sets are Cantor-Bernstein, but not > Cantor-Schroeder-Bernstein. > > Infinite sets are equivalent. > > Where you can put any number on the list, the transfer principle can be > applied. Where you can't, the reals aren't a set. > > I wonder how the anti-anti-Cantorians will deny this one. Maybe they > won't and become post-Cantorians. > > Did you know the halting problem is not a problem? If there's an > infinite program there are infinite programs. I guess that's > negateable. > > Let's see, foundations of mathematics, check. > So, I got to thinking about, and got to thinking that I was probably wrong about the uncountable lists and the countable lists and so on. Where I might be right anyways I realized I was incorrect. The set of lists has in ZF most directly cardinality |P(R)|. A countable collection of countable lists? If there are 2^x lists, or functions from the reals to the naturals, in ZF none are injections, or else. (Or else ZF is most directly inconsistent.) The universal set would already contain all sets and be it own powerset in a set theory, and it is. I derive humor from the fact that my previous post invoked notions along the lines of "Oh my God, he killed Kenny. You bastards." You can make a list that has as the first element each real, or the second, and etcetera. Oh wait, yeah that's right, each item in the sequence can be a natural, so each item on the list can be a real. That's coconsistent with that Dedekind/Cauchy as used in standard analysis is sufficient to represent any real number, which for example I say is not so, as it's known that it posseses no cardinal where in ZFC it's known that every set has a cardinal that is an ordinal. For example, it's consistent in ZF that the reals are not even a set. Nor would be any other large cardinal. Similarly, in ZF it's required that there exists no universe. Maybe it would help if you had a well-founded universe. A well-founded universe in a set theory is still its own powerset. N E N. So, now I'm trying to recall the logical progression that enabled the above description of a resolution of inadequcy of description of a function between the infinite set of natural numbers and reals. Trust me, I do. Ross
From: David Marcus on 23 Oct 2006 01:01 cbrown(a)cbrownsystems.com wrote: > Tony Orlow wrote: > > Well, allow me to repeat them here (with two minor changes): > > In order to interpret the problem > > "At each time t = -1/n where n is a (strictly positive) natural number, > we place the balls labelled 10*(n-1)+1 through 10*n inclusive in the > vase, and remove the ball labelled n from the vase. What is the number > of balls in the vase at time t=0?" > > I make the following simple (and I would claim, fairly uncontroversial > and natural) assumptions: > > --- (object permanence) > > (1) When we speak of a time t, we mean some real number t. > > (2) If a ball is in the vase at any time t0, there is a time t <= t0 > for which we can say "that ball was placed in the vase at time t". > > (3) If a ball is placed in the vase at time t1 and it is not removed > from the vase at some time t where t1 <= t <= t2, then that ball is in > the vase at time t2. > > (4) If a ball is removed from the vase at time t1, and there is no time > t such that t1 < t <= t2 when that ball is placed in the vase, then > that ball is not in the vase at time t2. > > ---- (obedience to the problem constraints) > > (5) If a ball is placed in the vase at some time t, it must be in > accordance with the description given in the problem: it must be a ball > with a natural number n on it, and the time t at which it is placed in > the vase must be -1/floor(n/10). > > (6) If a ball is removed from the vase at some time t, it must be in > accordance with the description given in the problem: it must be a ball > with a natural number n on it, and the time t at which it is removed > from the vase must be -1/n. > > (7) If n is a natural number with n > 0, then the ball labelled n is > placed in the vase at some time t1; and it is removed from the vase at > some time t2. > > --- (very general definition of "the vase is empty at noon") > > (8) the number of balls in the vase at time t=0 is 0 if, and only if, > the statement "there is a ball in the vase at time t=0" is false. > > --- > > Perhaps you would add other assumptions (9), (10), etc.; but my > question is: > > Given the problem statement, do you agree that /each/ of these > assumptions, /on its own/, is reasonable and not just some arbitrary > statement plucked out of thin air? > > If not, which assumption(s) is(are) not reasonable or is(are) > unneccessarily arbitrary? Your assumptions seem consistent with the following formulation of the problem. For n = 1,2,..., define A_n = -1/floor((n+9)/10), R_n = -1/n. For n = 1,2,..., define a function B_n by B_n(t) = 1 if A_n < t < R_n, 0 if t < A_n or t > R_n, undefined if t = A_n or t = R_n. Let V(t) = sum{n=1}^infty B_n(t). What is V(0)? > > I said that any specific ball was obviously out of the vase at noon. > > That's good: we at least agree that it logically follows from (1) - (8) > that there are no labelled balls in the vase at t=0. > > What I honestly find baffling is your repeated claim that it doesn't > then logically follow from assumptions (2) and (5), that if a ball is > in the vase at /any/ time, it is a ball which is labelled with a > natural number; and so therefore the above statement is logically > equivalent to "there are no balls in the vase at t=0". It is rather amazing. The logic seems to be that the limit of the number of balls in the vase as we approach noon is infinity, so the number of balls in the vase at noon must be infinity, but all numbered balls have been removed, therefore the infinity of balls in the vase at noon aren't numbered. It does have a sort of surreal appeal. -- David Marcus
From: David Marcus on 23 Oct 2006 01:07
Tony Orlow wrote: > cbrown(a)cbrownsystems.com wrote: > > Tony Orlow wrote: > >> cbrown(a)cbrownsystems.com wrote: > >>> Tony Orlow wrote: > >>>> You have agreed with everything so far. At every point before noon balls > >>>> remain. > >>> To be precise, the assertions above all imply that at every time t = > >>> -1/n, where n is a natural number, there are balls in the vase. > >>> > >>> But that *alone* does not even include every time t before noon; let > >>> alone every time t. For example, notice that nowhere above do you or I > >>> /explicitly/ assert: "at t=-2/3, the number of balls in the vase is a > >>> positive finite number". > >>> > >>> We assert something specific about t = -2/4, and something specific > >>> about t = -1/3, but nowhere do we directly state somthing about t = > >>> -2/3. > >>> > >>> On the other hand, given the problem statement, I think we would both > >>> /agree/ that there "should be" an obvious (perhaps even unique) > >>> well-defined answer to the question : "what is the number of balls in > >>> the vase at time t = -1/pi?" > >>> > >>> Assuming in the remaining statements that one agrees with the previous > >>> statement, this leads us to the question: what are the unstated > >>> assumptons that allow to agree that this must be the case? > >>> > >>> I attempted to describe those assumptions in my previoius post. Did you > >>> read those assumptions? If so, do you agree with those assumptions? > >> At this point I don't recall your previous post. I've been off a bit. > > > > Well, allow me to repeat them here (with two minor changes): > > > > In order to interpret the problem > > > > "At each time t = -1/n where n is a (strictly positive) natural number, > > we place the balls labelled 10*(n-1)+1 through 10*n inclusive in the > > vase, and remove the ball labelled n from the vase. What is the number > > of balls in the vase at time t=0?" > > > > I make the following simple (and I would claim, fairly uncontroversial > > and natural) assumptions: > > > > --- (object permanence) > > > > (1) When we speak of a time t, we mean some real number t. > > > > (2) If a ball is in the vase at any time t0, there is a time t <= t0 > > for which we can say "that ball was placed in the vase at time t". > > > > (3) If a ball is placed in the vase at time t1 and it is not removed > > from the vase at some time t where t1 <= t <= t2, then that ball is in > > the vase at time t2. > > > > (4) If a ball is removed from the vase at time t1, and there is no time > > t such that t1 < t <= t2 when that ball is placed in the vase, then > > that ball is not in the vase at time t2. > > > > ---- (obedience to the problem constraints) > > > > (5) If a ball is placed in the vase at some time t, it must be in > > accordance with the description given in the problem: it must be a ball > > with a natural number n on it, and the time t at which it is placed in > > the vase must be -1/floor(n/10). > > > > (6) If a ball is removed from the vase at some time t, it must be in > > accordance with the description given in the problem: it must be a ball > > with a natural number n on it, and the time t at which it is removed > > from the vase must be -1/n. > > > > (7) If n is a natural number with n > 0, then the ball labelled n is > > placed in the vase at some time t1; and it is removed from the vase at > > some time t2. > > > > --- (very general definition of "the vase is empty at noon") > > > > (8) the number of balls in the vase at time t=0 is 0 if, and only if, > > the statement "there is a ball in the vase at time t=0" is false. > > > > --- > > > > Perhaps you would add other assumptions (9), (10), etc.; but my > > question is: > > > > Given the problem statement, do you agree that /each/ of these > > assumptions, /on its own/, is reasonable and not just some arbitrary > > statement plucked out of thin air? > > > > If not, which assumption(s) is(are) not reasonable or is(are) > > unneccessarily arbitrary? > > > > <snip> > > Those all look reasonable to me as I read them. I don't see any > statement regarding the fact that ten balls are added for every one > removed, though that can be surmised from the insertion and removal > schedule. That's the salient fact here. You never remove as many as you > add, so you can't end up empty. What about #5? It says that every ball in the vase has a natural number on it. Do you agree with that? > Either something happens an noon, or it doesn't. Where do you stand on > the matter? What does "something happens" mean, please? I really don't know what you mean. -- David Marcus |