An uncountable countable set
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From: Tony Orlow on 11 Oct 2006 11:43 Ross A. Finlayson wrote: > David Marcus wrote: > > ... > >>> the null axiom theory. >> So, you aren't saying that ZF is inconsistent. You are saying that you >> prefer to use a different system of axioms. Is that correct? >> >> -- >> David Marcus > > Hi David, > > As you said, it's good to take these things one at a time, so, I am > happy to explain my opinion about set theory to you. > > Having been explaining set theory for some time, I was able to to take > part in some of these discussions here for example with everybody. > > No, David, Dave, I say ZF is inconsistent. That doesn't mean all its > results are false. It just lets me say whatever I want about a less > inconsistent set of "axioms". > > Ross > Can you state axioms which you use which contradict set theory?
From: Tony Orlow on 11 Oct 2006 11:46 David Marcus wrote: > Tony Orlow wrote: >> David Marcus wrote: >>> Tony Orlow wrote: >>>> David Marcus wrote: >>>>> Tony Orlow wrote: >>>>>> David Marcus wrote: >>>>>>> Tony Orlow wrote: >>>>>>>> David R Tribble wrote: >>>>>>>>> Virgil wrote: >>>>>>>>>>> Except for the first 10 balls, each insertion follow a removal and with >>>>>>>>>>> no exceptions each removal follows an insertion. >>>>>>>>> Tony Orlow wrote: >>>>>>>>>>> Which is why you have to have -9 balls at some point, so you can add 10, >>>>>>>>>>> remove 1, and have an empty vase. >>>>>>>>> David R Tribble wrote: >>>>>>>>>>> "At some point". Is that at the last moment before noon, when the >>>>>>>>>>> last 10 balls are added to the vase? >>>>>>>>>>> >>>>>>>>> Tony Orlow wrote: >>>>>>>>>> Yes, at the end of the previous iteration. If the vase is to become >>>>>>>>>> empty, it must be according to the rules of the gedanken. >>>>>>>>> The rules don't mention a last moment. >>>>>>>>> >>>>>>>> The conclusion you come to is that the vase empties. As balls are >>>>>>>> removed one at a time, that implies there is a last ball removed, does >>>>>>>> it not? >>>>>>> Please state the problem in English ("vase", "balls", "time", "remove") >>>>>>> and also state your translation of the problem into Mathematics (sets, >>>>>>> functions, numbers). >>>>>> Given an unfillable vase and an infinite set of balls, we are to insert >>>>>> 10 balls in the vase, remove 1, and repeat indefinitely. In order to >>>>>> have a definite conclusion to this experiment in infinity, we will >>>>>> perform the first iteration at a minute before noon, the next at a half >>>>>> minute before noon, etc, so that iteration n (starting at 0) occurs at >>>>>> noon-1/2^n) minutes, and the infinite sequence is done at noon. The >>>>>> question is, what will we find in the vase at noon? >>>>> OK. That is the English version. Now, what is the translation into >>>>> Mathematics? >>>> Can you only eat a crumb at a time? I gave you the infinite series >>>> interpretation of the problem in that paragraph, right after you >>>> snipped. Perhaps you should comment after each entire paragraph, or >>>> after reading the entire post. I'm not much into answering the same >>>> question multiple times per person. >>> I snipped it because it wasn't a statement of the problem, as far as I >>> could see, but rather various conclusions that one might draw. >> I drew those conclusions from the statement of the problem, with and >> without the labels. > > I'm sorry, but I can't separate your statement of the problem from your > conclusions. Please give just the statement. > The sequence of events consists of adding 10 and removing 1, an infinite number of times. In other words, it's an infinite series of (+10-1).
From: Lester Zick on 11 Oct 2006 12:29 On Wed, 11 Oct 2006 11:41:00 -0400, Tony Orlow <tony(a)lightlink.com> wrote: >Lester Zick wrote: >> On Tue, 10 Oct 2006 13:56:43 -0400, Tony Orlow <tony(a)lightlink.com> >> wrote: >> >>> Lester Zick wrote: >>>> Tony, I'm going to strip as much as seems unessential. >>>> >>>> On Mon, 09 Oct 2006 15:56:32 -0400, Tony Orlow <tony(a)lightlink.com> >>>> wrote: >>>> >>>>> Lester Zick wrote: >>>>>> On Sun, 08 Oct 2006 22:07:23 -0400, Tony Orlow <tony(a)lightlink.com> >>>>>> wrote: >>>> [. . .] >>>> >>>>>> As far as transcendentals are concerned, Tony, the only thing that can >>>>>> lie on a real number line in common with rationals/irrationals are >>>>>> straight line segment approximations. That's the only linear order >>>>>> possible. So either you give up transcendentals or a real number line. >>>>> The trichotomy or real quantity itself defines a linear order. Each such >>>>> value is greater than or less than every different value. Pi is >>>>> transcendental - is it less than or greater than 3? Is there any doubt >>>>> about that? >>>> No but that only applies to linear approximations for pi, Tony. It has >>>> nothing to do with the actual value of pi which lies off to the side >>>> of any real number line. Imagine if you will the linear order you >>>> speak of superimposed on a real number plane instead of a straight >>>> line (which isn't even possible either). Then pi has to lie off to one >>>> side of the straight line. So the metric for the straight line becomes >>>> a non linear variable instead of linear. Thus the fact that the value >>>> of pi lies between 3 and 4 doesn't show any value for pi and never >>>> will. All you wind uyp with is a linear metric which approximates pi >>>> which doesn't provide us with any real number line running along the >>>> lines of 1, 2, e, 3, pi. So the real number line metric is variable. >>>> >>> Pinpointing particular points on the line may require varying degrees of >>> computation/construction. A natural requires only a finite number of >>> iterations of increment, a rational may require a division operation, >>> and an irrational or a transcendental like pi may require an infinite >>> computation to exactly pinpoint the value. That doesn't mean that value >>> doesn't exist as a point on the line. It just mans specifying it >>> requires an infinite process. >> >> It requires an infinite process only because it isn't on the line. The >> only thing an infinite process determines is how close the curve comes >> to a straight line without ever getting there. Every other rational/ >> irrational on the line can be located with finite processes because >> they are on the line. >> > >If the line is defined by trichotomy, and 3<pi<4, isn't pi on that line? If by "trichotomy" you mean <=> on a straight line then no pi isn't on that line, Tony. Pi lies on a circular curve not on a straight line. Approximations for pi such as 3<pi<4 do lie on a straight line but only indicate how close circular arcs come to straight line approximations. >>>>>>> Perhaps not, but if, say, y=1/x is both oo and -oo at x=0, and oo=-oo, >>>>>>> then the function is continuous in every respect, which is what we might >>>>>>> desire in such a fundamental algebraic relation. >>>>>> But for the division operation x never becomes zero. Which indicates >>>>>> that there can be no plus or minus infinity and no continuity. >>>>> Is 0 part of the continuum, or just another arbitrary "limit" >>>>> discontinuity? When you ask yourself, "If I divide this finite space >>>>> into individual points, how many will I have?", what answer do you get? >>>>> How much of the space between 0 and 1 does each real in that interval >>>>> occupy, and how many are there? >>>> Well points don't occupy any space at all, Tony. >>> Right, their measure is 0. >>> >>> Intersection ends in >>>> points but subdivision never ends in points. >>> In the limit, subdivision results in infinitesimal segments, where the >>> values of the endpoints cannot be distinguished on any finite scale. >> >> Oh I don't know about that, Tony. Certainly the first bisection in a >> series is exactly half the length of the starting finite segment. > >Yes, and every finitely-indexed subdivision results in finite segments, >but that cannot be truly when the segment is infinitely subdivided. As far as that goes it never is. >>> That's the difference >>>> between intersection and differentiation. Although not a natural zero >>>> certainly exists but the question is how it's used. Zero used as a >>>> limit is approachable but not reachable. Same with infinity. Zero used >>>> as a divisor is undefinable. Maybe you're trying to combine arithmetic >>>> and infinitesimal calculus operations in ways that are incompatible? >>>> >>> It's true that absolute 0 has an indeterminate reciprocal of oo, and >>> that regular arithmetic won't work on those. But, a specific unit >>> infinity lends itself to arithmetic quite nicely. >> >> So does zero. It just doesn't have any meaning when used for division. >> Nor does infinity when used to multiply. >> > >But specific infinite values and specific infinitesimals do. But those are things you never have and can never have. The idea of infinitesimals is only a conceptual device or limit used to denote the end of an ongoing process which never actually ends and can never end. >>>>>>>>> When it comes down to this argument, Wolfgang's argument, I agree with >>>>>>>>> his logic concerning the naturals and the identity function between >>>>>>>>> element count and value. >>>>>>>> To me "element count" and the number of commas are the same. >>>>>>> Sure, that sounds okay to me. In the naturals, the first is 1, the >>>>>>> second 2, etc. What is the aleph_0th? >>>>>> I don't know what you're asking here, Tony. If there is no real number >>>>>> line aleph there are no aleph ordinals either. There can be aleph >>>>>> infinitesimals but that represents a continual process of subdivision >>>>>> and not one of division in which case the ordinality would be one of >>>>>> relation between various curvatures where straight lines would be >>>>>> first or minimal and the ordinality of others judged in relation to >>>>>> it. I think the question you really need to be asking in this context >>>>>> is whether there can be such a thing as an open set. If not then the >
From: Virgil on 11 Oct 2006 14:21 In article <452d11ca(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > > I'm sorry, but I can't separate your statement of the problem from your > > conclusions. Please give just the statement. > > > > The sequence of events consists of adding 10 and removing 1, an infinite > number of times. In other words, it's an infinite series of (+10-1). That deliberately and specifically omits the requirement of identifying and tracking each ball individually as required in the originally stated problem, in which each ball is uniquely identified and tracked.
From: David Marcus on 11 Oct 2006 18:42
Ross A. Finlayson wrote: > David Marcus wrote: > > > the null axiom theory. > > > > So, you aren't saying that ZF is inconsistent. You are saying that you > > prefer to use a different system of axioms. Is that correct? > > Hi David, > > As you said, it's good to take these things one at a time, so, I am > happy to explain my opinion about set theory to you. > > Having been explaining set theory for some time, I was able to to take > part in some of these discussions here for example with everybody. > > No, David, Dave, I say ZF is inconsistent. That doesn't mean all its > results are false. It just lets me say whatever I want about a less > inconsistent set of "axioms". So, you are saying that ZF is inconsistent. By "inconsistent", do you mean that ZF proves both P and not P, for some statement P? If so, please be explicit: what is the statement and what is the proof in ZF of it? -- David Marcus |