An uncountable countable set
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From: Tony Orlow on 9 Oct 2006 17:29 Virgil wrote: > In article <452ab540(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: >>> In article <452aa5c9(a)news2.lightlink.com>, >>> Tony Orlow <tony(a)lightlink.com> wrote: >>> >>> >>>> If you increment a natural n times, you have added n to it. If successor >>>> is increment, and there are an infinite !number! of such increments, you >>>> have added this infinite number to your starting value. Adding an >>>> infinite number to a finite yields an infinite. Therefore, the infinite >>>> set includes infinite values. >>> Without a limit ordinals in between or as the larger, one never can have >>> two ordinals separated by infinitely many successors of the smaller. >> Proof, please? > > Why? TO never proves any of his claims. "Why, to show me how it done properly; that is, if it would please you, Sir, to pass this invaluable knowledge along, as is your calling as an educator, or are you not? This is the finest apple my trees have produced, and I polished it all the way here. So, if you please, Sir, let's exchange gifts...." - Phlebotomy
From: Lester Zick on 9 Oct 2006 17:50 On 9 Oct 2006 13:38:18 -0700, imaginatorium(a)despammed.com wrote: >Lester Zick wrote: >> On Sun, 08 Oct 2006 22:07:23 -0400, Tony Orlow <tony(a)lightlink.com> >> wrote: >> >> >Lester Zick wrote: >> >> But there is no single real number line. There's a single rational/ >> >> irrational line but not even a single transcendental line, Tony. So on >> >> the surface I don't see what this speculation has to recommend it. And >> >> more to the point I don't see any way to effect a crossover in >> >> mechanical terms. >> > >> >I've certainly heard you discuss your views on the number line, and how >> >pi lies on a curve and rationals lie on straight lines, etc. To me, it >> >sounds like a matter of construction, or meaning of the number, but not >> >one of raw quantity. In terms of raw quantity on the real number line, >> >they all obey the law of trichotomy, for any a and b, either a=b, a>b, >> >or a<b. So, it's a linear order. >> >> 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. > >Ah, Lester, the words flow from your pen as the limpid waters of the >Olchon Brook babble their way down the hillside. (That's poetry, by the >way.) > >Skip a bit here, I think. We have to draw the line somewhere. (That's a >figurative line, not the real line.) The figurative line is considerably realer than the real line, Brian. >> 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 > >Oh, would the relation between subdivision and division be akin to that >between subtraction and traction? Yes. >> .... 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 >> question becomes how to close open sets and whether there can be >> anything besides finites in closed sets. > >More poetry. What's it mean, if anything? Not sure. Still working out the meaning. ~v~~
From: Lester Zick on 9 Oct 2006 17:51 On Mon, 09 Oct 2006 16:48:38 -0400, Tony Orlow <tony(a)lightlink.com> wrote: >imaginatorium(a)despammed.com wrote: >> Lester Zick wrote: >>> On Sun, 08 Oct 2006 22:07:23 -0400, Tony Orlow <tony(a)lightlink.com> >>> wrote: >>> >>>> Lester Zick wrote: >>>>> But there is no single real number line. There's a single rational/ >>>>> irrational line but not even a single transcendental line, Tony. So on >>>>> the surface I don't see what this speculation has to recommend it. And >>>>> more to the point I don't see any way to effect a crossover in >>>>> mechanical terms. >>>> I've certainly heard you discuss your views on the number line, and how >>>> pi lies on a curve and rationals lie on straight lines, etc. To me, it >>>> sounds like a matter of construction, or meaning of the number, but not >>>> one of raw quantity. In terms of raw quantity on the real number line, >>>> they all obey the law of trichotomy, for any a and b, either a=b, a>b, >>>> or a<b. So, it's a linear order. >>> 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. >> >> Ah, Lester, the words flow from your pen as the limpid waters of the >> Olchon Brook babble their way down the hillside. (That's poetry, by the >> way.) >> >> Skip a bit here, I think. We have to draw the line somewhere. (That's a >> figurative line, not the real line.) >> >>> 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 >> >> Oh, would the relation between subdivision and division be akin to that >> between subtraction and traction? >> >>> .... 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 >>> question becomes how to close open sets and whether there can be >>> anything besides finites in closed sets. >> >> More poetry. What's it mean, if anything? >> >> Brian Chandler >> http://imaginatorium.org >> > >Brian, you're a dork. > >Have a nice day. Don't worry about Brian, Tony. He's just confused. ~v~~
From: Tony Orlow on 9 Oct 2006 17:52 David Marcus wrote: > Tony Orlow wrote: >> David Marcus wrote: >>> Ross A. Finlayson wrote: >>>> David Marcus wrote: >>>>> I thought you said there was a contradiction in ZF. In the context of >>>>> ZF, the Burali-Forti argument shows that there is no set of all >>>>> ordinals, but does not lead to a contradiction. So, do you still say >>>>> there is a contradiction in ZF? If so, what is it? >>>> {For any x: x is a set} = emptyset <=/=> {For any x: x is a set} = U >>>> (V, L) >>>> >>>> That says, for any x, that's the empty set, and, for any x, that's the >>>> universal set, it seems sufficient to show the universe non-empty. >>>> >>>> There is no set of ordinals nor cardinals in ZF. Yet, because there's >>>> the axiom of infinity, those infinite ordinals/cardinals as there ever >>>> would be are claimed to exist, basically where they're all hereditarily >>>> finite, those ordinals of the cumulative hierarchy. >>> Sorry, but I don't follow. Are you saying this is a contradiction within >>> ZF? By "within" I mean that ZF proves this contradiction. >>> >> Hi David - >> >> This part of Ross I don't quite follow, but perhaps we should discuss >> it. What is your background, if you don't mind my asking? > > Is that relevant? Or, are you just curious? > Curious, and probably, therefore, relevant... ....Tony
From: Lester Zick on 9 Oct 2006 18:39
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. >>> 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. Intersection ends in points but subdivision never ends in 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? >>>>> 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 >> question becomes how to close open sets and whether there can be >> anything besides finites in closed sets. >> > >Well, yes, that is the question, but I would pose it in a different way. > >Let's say there CAN be open sets, by which I assume you mean boundless, >without definite range, either element-wise, or value-wise given a >formula defining membership and any kind of measure (is that what you >mean?). Such a set can be declared as "existing" simply by defining the >rules of the object. I don't see that it *can't* exist. Well the question I have then is whether such a set can be infinite or contain only finites without limit? >The question then becomes, can we tell anything about the "size" of the >set, when the elements themselves are not associated with any measure, >and there is no bound to the count of the set? The only way I can see to >measure an unending set is to establish a standard unending set, and >formulaically compare the two. Unfortunately, the "standard" unedning >set is unending due to there being no largest finite, and not due to >infinite measure persay. And I would agree. But the elements in the set never become non finite. So "infinite" in that sense only means successive finite elements "without limit". But there are different kinds of infinities in this sense which unfortunately only make sense in relation to one another in terms of the infinitesimal calculus used to calculate various forms and relate them to one another. L'Hospital's rule. And once again it looks like a lot of mathematkers are inappropriately trying to transmogrify arithmetic into infinitesimal calculus. >>>>> He chooses then to reject infinities, while I >>>>> choose to retool them. I think the problem he and I both see clearly is >>>>> that using finiteness as a "bound" on the set simply doesn't tell you >>>>> anything, but rather clouds the entire subject. >>>>> >>>>> Did that answer your question? >>>> Kinda, Tony, although I really don't get all the arguing over vases >>>> and balls. And I don't really see finiteness as an issue if you view >>>> infinity as the number of infinitesimals between limits. Then >>>> infinities are contained whatever their magnitude. >>>> >>>> ~v~~ >>> Yes, that's pretty much how I see them, the workable infinities, anyway. >>> Pure infinity isn't really a number, any more than 0 is really a >>> quantity. The ball and vase show comes from an example showing how >>> ludicrous the standard take is on the question, given set theory. >> >> Yeah my problem with arguing examples is that the issues themselves >> are submerged in the examples and disappear never to be seen again. >> >> ~v~~ > >Oh, Lester, bullshit. The argument |