An uncountable countable set
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From: Tony Orlow on 9 Oct 2006 16:18 Ross A. Finlayson wrote: > Tony Orlow wrote: >> Virgil wrote: >>> In article <4529afa4(a)news2.lightlink.com>, >>> Tony Orlow <tony(a)lightlink.com> wrote: > ... >>>> That is not "the reals strictly between 0 and 1" but a subset thereof. >>> So there is still no element within either set which is its LUB. >> If the Finlayson reals are used, indeed the LUB is the maximal member of >> the set of reals in [0,1). Ross, is that correct? >> >> Tony > > Hi Tony, Hi Ross! :D > > It depends how you use them because they are also the real numbers. Well, no duh! They're just not STANDARD real numbers. Everything on the real line is real, yesno? > > In your sense of use, yes, it's permissible. You just can't then do > other things that would right contradict that statement, you know. Of course not. It's incompatible with transfinitology, so mixing them is crazy. That's the source of most objections to my ideas, as far as I can tell. To > reestablish your discussion about the entire set of real numbers again, > including all the numbers between zero and one where there are only and > everywhere real numbers, Is there anything else ANYWHERE on the real line, outside of 0 and 1? I don't think so. Do you? the segment of the real number line's > continuum, ....the unit interval? involves that claim in context. This is kinda non-committal, it sounds. You should be more direct and willful. It will get you a nice girl, if you're nice about it, and you want one. That's why Virgil loves to hate me. Not that I want "him".... ;) Anyways.... So, what you mean is, and please correct me if I'm wrong, that within the discussion of the interval [0,1), given nilpotent infinitesimals (well, not really nilpotent exactly, but integral, indivisible infinitesimals, with some exact number per unit interval), one can state which is that last real in that interval, and the LUB is the maximal element? That's what I glean of the Finalyson numbers, and it makes sense to me. > > A fellow Spinoza from some time ago is known for introducing the notion > that the set of natural numbers is itself a continuum. Ah yes, Spinoza.... > > Did you know that the closer scientists look at subatomic (well, and > atomic) particles, the smaller the particles appear to be? Are they > infinitesimal? Why is there gauge invariance? > > Ross > I've heard you say that, and heard it disputed once, but can't weigh in on that one. However, I have no personal, intuitive doubt that it all boils down to points, which are numbers, and the underlying "I am that I am" of the Universe. Rock on! TOny
From: imaginatorium on 9 Oct 2006 16:38 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
From: Tony Orlow on 9 Oct 2006 16:39 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? :) TOny
From: imaginatorium on 9 Oct 2006 16:41 Tony Orlow wrote: > David R Tribble wrote: > > David R Tribble wrote: > >>> Hardly. The H-riffics are a simple countable subset of the reals. > >>> Anyone mathematically inclined can come up with such a set. > > > > Tony Orlow wrote: > >> You never paid enough attention to understand them. They cover the reals. > > > > They omit an uncountable number of reals. Any power of 3, for example, > > which you never showed as being a member of them. Show us how 3 fits > > into the set, then we'll talk about "covering the reals". > > > > 3 is an unending string, just like 1/3 is in base-10. Rusin confirmed > that about two years ago. But, you're right, I need to construct a > formal proof of the equivalence between the H-riffics and the reals. Good luck with that. Uh-oh, I suppose you mean a Tproof - now _that_ shouldn't be too difficult at all. Brian Chandler http://imaginatorium.org
From: Virgil on 9 Oct 2006 16:43
In article <452aab46(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > >> It's a 1-D continuum with an origin, a metric space. > > > > But where are the values of that metric if zero is not one of them? > > 0 is the origin, the reference point. > Measure is difference from 0. Which has nothing to do with metric spaces. A metric space is a 2-tuple (X,d) where X is a set and d is a metric on X, that is, a function d: X x X --> R such that 1 d(x, y) ? 0 (non-negativity) 2 d(x, y) = 0 if and only if x = y (identity) 3 d(x, y) = d(y, x) (symmetry) 4 d(x, z) ? d(x, y) + d(y, z) (triangle inequality). > > TO appealing to Ross is the blind asking for a lead from the blind. > >> Tony > > I have better hearing, and Ross knows many smells, Can you hear Ross stink up the place? |