An uncountable countable set
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From: MoeBlee on 26 Oct 2006 19:32 David Marcus wrote: > I think it is too much for someone to learn on their own (unless they > have quite a bit of mathematical maturity). Going to school is probably > required. Orlow doesn't want to learn mathematics. He wants the world to learn it from him. MoeBlee
From: cbrown on 26 Oct 2006 21:21 cbrown(a)cbrownsystems.com wrote: > Tony Orlow wrote: > > Mike Kelly wrote: > > <snip> > > > > My question : what do you think is in the vase at noon? > > > > > > > A countable infinity of balls. > > > > This is very simple. Everything that occurs is either an addition of ten > > balls or a removal of 1, and occurs a finite amount of time before noon. > > At the time of each event, balls remain. At noon, no balls are inserted > > or removed. > > No one disagrees with the above statements. > > > The vase can only become empty through the removal of balls, > > Note that this is not identical to saying "the vase can only become > empty /at time t/, if there are balls removed /at time t/"; which is > what it seems you actually mean. > > This doesn't follow from (1)..(8), which lack any explicit mention of > what "becomes empty" means. However, we can easily make it an > assumption: > > (T1) If, for some time t1 < t0, it is the case that the number of balls > in the vase at any time t with t1 <= t < t0 is different than the > number of balls at time t0, then balls are removed at time t0, or balls > are added at time t0. > > > so if no balls are removed, the vase cannot become empty at noon. It was > > not empty before noon, therefore it is not empty at noon. Nothing can > > happen at noon, since that would involve a ball n such that 1/n=0. > I apologize; my comments that follow regard the original problem; not the modified problem describe by mike kelly (which involve no ball removals at all). I'd stil like to hear Tony's opinion on them, in the context of the original problem... Cheers - Chas
From: Ross A. Finlayson on 26 Oct 2006 22:53 Dik T. Winter wrote: > In article <1161829100.130312.44340(a)i42g2000cwa.googlegroups.com> "Ross A. Finlayson" <raf(a)tiki-lounge.com> writes: > > Dik T. Winter wrote: > > > In article <1161825225.141821.66550(a)e3g2000cwe.googlegroups.com> "Ross A. Finlayson" <raf(a)tiki-lounge.com> writes: > > > ... > > > > Cardinals are ordinals and in a set theory: sets. > > > > > > > > Generally it's known that you get cardinals from the previous using the > > > > powerset operation, > > > > > > Through what powerset operation do you get the cardinal 3? > > > > > > > and ordinals from the previous using the successor > > > > operation. > > > > > > Through what successor operation do you get the ordinal omega? > ... > > Please identify something you see as incorrect or don't understand. > > Read above. Cardinals are not *all* obtained from the previous using the > powerset operation. Ordinals are not *all* obtained using the successor > operation. Hence my questions. > -- > dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 > home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ Hi Dik, It seems the most obvious sets in construction are a) the empty set and b) the universal set. Then, uniquification over comprehension of individua in the continuum between those two results in the ordinals. Also, for any finite natural the successor is finite, for all finite naturals the successor is not finite. That's about a reasonable application of the transfer principle. N E N and so on and so forth. The naturals are compact. What do you think about completion of the supertask implying infinity in the naturals? That's interesting reading about the quite oposite available interpretations of the system to the standard here. There is no set of ordinals in ZF. Ross
From: Tony Orlow on 26 Oct 2006 22:55 MoeBlee wrote: > Ross A. Finlayson wrote: >> Please identify something you see as incorrect or don't understand. > > What's the point? People have been pointing out your incorrect > statements for years. You just sail right past every time. > > So really think you are correct and that you make sense? > > It must be nice. > > MoeBlee > For what it's worth, and I know this doesn't add a lot of credibility to Ross in your eyes, coming from me, but I think Ross has a genuine intuition that isn't far off with respect to what's controversial in modern math. Sure, he gets repetitive and I don't agree with everything he says, but his cryptic "Well order the reals", which I actually haven't seen too much of lately, is a direct reference to his EF (Equivalence Function, yes?) between the naturals and the reals in [0,1). The reals viewed as discrete infinitesimals map to the hypernaturals, anyway, and his EF is a special case of my IFR. So, to answer your question, I think Ross makes some sense. But, of course, coming from me, that probably doesn't mean much. :) TOE-Knee
From: Tony Orlow on 26 Oct 2006 23:04
David Marcus wrote: > Tony Orlow wrote: >> MoeBlee wrote: >>> Tony Orlow wrote: >>>>> But none of Robinson's non-standard numbers are cardinalities. >>>> No kidding. They actually make sense. >>> You said you have not properly studied chapter II in the book - the one >>> that includes mathematical logic, model theory, and set theory (does it >>> not? I'll stand corrected if it doesn't). What are you going to say >>> when you find out that what you say makes sense rests on a foundation >>> of set theory that you say doesn't make sense? Or, if I'm incorrect >>> that Robinson's work in non-standard analysis doesn't presuppose basic >>> mathematical logic, model theory, and set theory, then I'll benefit by >>> being corrected in my admittedly cursory understanding of the matter. >>> >>> MoeBlee >>> >> Uh, if Robinson's thesis is built upon transfinite set theory, > > Nonstandard analysis is built on mathematical logic as is set theory. > >> then that >> is evidence right there that it's inconsistent, since you have a >> smallest infinity, omega, but Robinson has no smallest infinity. >> Robinson doesn't use ordinals or cardinals that I've seen. He basically >> defines what a well-formed formula is in his system, which is a little >> more restrictive that some others, it seems, and uses the language to >> extend what can be said about finite n in N to include infinite n in *N. > That doesn't mean that mathematical logic is being properly applied in the transfinite case. There is yet some debate as to which logical constructions are valid and which are not in that situation, as I see it. It's hardly cut and dried. If Euclidean geometry as universal truth can be overturned after more than 2,000 years, what makes one so sure that transfinitology is "correct" after less than 200, especially when it's at odds with all the intuitions we've developed over those thousands of years.? I think the problem is solved most immediately by adopting the method of inductive proof in the infinite case, provided that one is proving either an equality between formulas, or an inequality where the difference establishing it does not have a limit of 0 as n->oo. Given this simple rule, it's easy to order all sorts of infinite sets, when they can be expressed as formulas f and g for which f(x)>g(x) for all x greater than some finite value. Infinite values are greater than any finite value. Therefore, the proof should hold. This leads to a whole other set of conclusions about this and other problems. |