An uncountable countable set
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From: Ross A. Finlayson on 11 Oct 2006 20:35 David Marcus wrote: > 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 Build a set: {x: true}, it's a set, with unrestricted comprehension it's not a set, without it's not either, so there are no true, or provable, predicates in ZF. A lot of the axioms, of ZF(C) set theory, can be seen to enrich or broaden the set-theoretic universe. Regularity is not one of those. There is no universe in ZF. Ross
From: Tony Orlow on 11 Oct 2006 20:38 Lester Zick wrote: > 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. > Circular arcs approach the straight line in the limit as radius->oo, but other than than, no, pi's a quantity, a distance from the origin That's what a real number is. >>>>>>>> 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. In essence, there is no end to subdivision. In the limit, the subsegment is a point, or a pair of identical points. > >>>> 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. I have them. 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. > You and I are specks, Lester. >>>>>>>>>> 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.
From: Ross A. Finlayson on 11 Oct 2006 20:39 David Marcus wrote: > 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 Build a set: {x: true}, it's a set, with unrestricted comprehension it's not a set, without it's not either, so there are no true, or provable, predicates in ZF. A lot of the axioms, of ZF(C) set theory, can be seen to enrich or broaden the set-theoretic universe. Regularity is not one of those. There is no universe in ZF. Ross
From: Tony Orlow on 11 Oct 2006 20:40 Virgil wrote: > 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. The original statement contrasted two situations which both matched this scenario. The difference between them was the label on the ball removed at each iteration, and yet, that's not relevant to how many balls are in the vase at, or before, noon.
From: cbrown on 11 Oct 2006 21:57
Tony Orlow wrote: > Virgil wrote: > > 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. > > The original statement contrasted two situations which both matched this > scenario. The difference between them was the label on the ball removed > at each iteration, and yet, that's not relevant to how many balls are in > the vase at, or before, noon. Do you think that the numbering of the balls is not relevant to determining the answer to the question "Is there a ball labelled 15 in the vase at 1/20 second before midnight?" Cheers - Chas |