An uncountable countable set
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From: Tony Orlow on 9 Oct 2006 16:45 David Marcus wrote: > Tony Orlow wrote: >> Dik T. Winter wrote: >>> In article <1160310643.181133.6720(a)b28g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: >>> > Dik T. Winter schrieb: >>> ... >>> > AXIOM OF INFINITY Vla There exists at least one set Z with the >>> > following properties: >>> > (i) O eps Z >>> > (ii) if x eps Z, also {x} eps Z. >>> > >>> > There are several verbal formulations dispersed over the literature >>> > without any "all". >>> >>> Perhaps. Does this mean that there are some x in Z such that {x} not in >>> Z? Or, what do you mean? >> He means that the axioms can be stated without using universal >> quantifiers or existential quantifiers, as simple logical implications >> between propositions, I believe. WM, am I close? > > If you see "if x is in Z, then {x} is in Z" in a math book (without any > other explicit restriction on what x is), then the convention is that > you are to interpret it as meaning "for all x, if x is in Z, then {x} is > in Z". Otherwise, you have a letter whose meaning you do not know. > You can express that as x e Z -> {x} e Z, as a particular case for every x e Z. It follows, logically, given the infinite chain of logical implication inherent in any recursive definition, that it applies to every member of the set so defined. But, it need not be so stated.
From: Tony Orlow on 9 Oct 2006 16:47 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? > > And as every natural is an ordinal, that scuttles TO's theories. Arrghhh, we scuttles to and fro, M'laddy!
From: Virgil on 9 Oct 2006 16:48 In article <452ab325(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > David Marcus wrote: > > 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. But without the labels, it is a different problem. > > While > > these may be correct, until the problem is stated mathematically, it is > > impossible to tell. Please give the statement of the problem in > > mathematics without also giving any conclusions or derived properties. > > Yeah, I compared and contrasted the two approaches. Please reread. As TO was, as usual, wrong, please rewrite. > > > When discussing mathematics, communication works best one step at a > > time. > > > > So, yes, you can only take a spoonful at a time. TO can't even take that.
From: Tony Orlow on 9 Oct 2006 16:48 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. Tony
From: Tony Orlow on 9 Oct 2006 16:49
imaginatorium(a)despammed.com wrote: > 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 > Yep, dork. Back to your action figures. |