An uncountable countable set
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From: Ross A. Finlayson on 29 Oct 2006 12:42 Virgil wrote: > In article <4543931c(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > > > David Marcus wrote: > > > > Please answer "yes" or "no". > > > > > > > By the way, yes and no. > > TO gets it wrong again. '"Yes" or "no"' is a barely defensible answer, > but 'yes and no' is not. About these marbles/balls in a jar/vase, that Wikipedia "supertask" page is good reading. http://en.wikipedia.org/wiki/Supertask Described is Laraudogotia's "beautiful" supertask, which appears to be similar to one of those executive desk toys with bearings hung from a frame where a bearing on one side is lifted and dropped and the momentum carries through thus that the bearing on the opposite side is deflected as much as the initial bearing was deflected. Instead, Laraudogotia has infinitely many balls on, say, [0,1) measured in meters, at 0 and 1/2^n for n E N. (Where N E N, that interval is closed.) Accelerate the ball at 1 m/s, it collides with the next imparting all of its momentum, as does it to its successor ad infinitum. It is argued that since there is no last ball, after exactly one second there is no output, instead of the parallel situation where a ball by itself was imparted 1 m/s and evacuates at t = 1 second. In physics, well, there are various considerations as to what comprises matter, basically, without a point at infinity, there is no motion, which is why zero, one, and infinity are the identity constants. Consider Maxwell's demon and why it can't very simply alter the laws of thermodynamics. Similarly with this balls in a vase problem, the Ross-Littlewood "paradox" as is described on the web page reference above, there is a consideration that without there being a ball labelled infinity, the process doesn't complete. Consider a possible line betwen zero and one, a pen and eraser. It seems that the various times of insertions of the ten balls can be drawn, and the erasure of the balls in order can be accomplished by erasing the line from the same beginning and along the same track at a correspondingly lower rate, else the erasure would overtake the drawn line. Then, for the eraser to reach point 1 at exactly the same time as the pencil, it becomes spork, both spoon and fork, the pencil and eraser coincide, their meanings are conflated _and_ dual. If you're a sophisiticate you might understand why that gets into the vague fugue of the real numbers, and why how probabilistically the pencil or eraser moves in a straight line, the point at the pencil tip covers multiple intervals. Well, I hope you and Virgil will forgive my rudeness to him and chalk it up to his, I am sincere, not a troll, etcetera, and if we're talking poetry, we're still talking mathematics. Hey, tell Conway we're talking about him. If there's a theory of everything, there's only one of them. There is no set of all numbers in ZF, so, you can't know all their properties. Ross
From: Lester Zick on 29 Oct 2006 14:02 On Fri, 27 Oct 2006 20:08:18 -0400, Tony Orlow <tony(a)lightlink.com> wrote: >Lester Zick wrote: >> On Thu, 26 Oct 2006 23:28:04 -0400, Tony Orlow <tony(a)lightlink.com> >> wrote: >> >>> MoeBlee wrote: >>>> Tony Orlow wrote: >> >> [. . .] >> >>>> We JUST agreed that 'smallest infinity' means two different things when >>>> referring to ordinals and when referring to certain kinds of other >>>> orderings! It is AMAZING to me that even though I took special care to >>>> make sure this was clear, and then you agreeed, you NOW come back to >>>> conflate the two ANYWAY! >>> Ahem. I said that Robinson's analysis seems to have nothing to do with >>> transfinitology. They appear to be unrelated. However, they cme to two >>> very different conclusions regarding a basic question: is there a >>> smallest infinite number? It seems clear to me there is not, for the >>> very same reason that Robinson uses: if there is an infinite number, you >>> can subtract 1 and get a different, smaller infinite number. It's the >>> same logic y'all use to argue that there's no largest finite. It's >>> correct. The Twilight Zone between finite and infinite CANNOT really be >>> pinpointed that way. >> >> Hey, Tony. You know this is an interesting problem but I think you're >> wasting your time here arguing the issue with the Holy Order of Self >> Righteous Mathematikers. Let me outline my own thinking for you. > >Alright, but keep in mind that changing the way the world works involves >changing people's minds. :) Sure, Tony. Just don't expect academics to change their minds. >> I think there is a smallest infinity but that subtracting finites from >> infinites isn't the way to get at the problem because as far as I can >> tell arithmetic operations cannot be defined between infinites and >> finites any more than finite division by zero can be. > >Why do you think there is a smallest infinity, when removing a finite >portion thereof leaves a, smaller, infinite portion? Well part of my contention is that arithmetic operations between finites and infinites cannot be defined and in effect aren't possible. So you have to attack the problem of "least" infinites some other way. >> Instead you need a different approach altogether and I suspect the way >> to get at the problem is to assess the kind of infinity according to >> the number of infinitesimals in various intervals. And in this manner >> I suspect you'll find the infinity associated with straight line >> segments is the smallest and various kinds of curves larger. >> >> ~v~~ > >You are intuiting in the derivative sense. As segments get smaller, on >whatever curve, the angle between them decreases. Well the angle with respect to any radius would certainly get smaller but I don't intuit what you're trying to suggest as a result. > The trick here is to >declare, as Ross aludes, to a "universe", a complete range of values for >the set or sets, and measure according to that "range". The problem here is that with respect to sets there is no complete range of values which includes both finites and infinites. There is no closed set of naturals in this sense. Yet every value in naturals is finite. In other words there is no [1, 2, 3 . . . 00].Mathematikers seem to want to call the set of naturals infinite but that can only refer to application of the generating mechanism "1+1". It doesn't mean the contents generated are or can be infinite. And as far as I'm concerned there is no way to generate infinities except through infinitesimal subdivision because mathematikers can't define their idea of infinity through "1+1" because that only produces finites. > I have a feeling >maybe you'll get it soon. It fits, I'm sure, with some of your ideas. >You do have ideas, don't you? ;) Sure. I haven't had any cause to change any of them. As far as I've been able to tell they remain true. ~v~~
From: stephen on 29 Oct 2006 14:01 stephen(a)nomail.com wrote: > Tony Orlow <tony(a)lightlink.com> wrote: >> stephen(a)nomail.com wrote: >>> >>> Do you think there exists a positive integer n such that >>> -1/(2^(floor(n/10))) < 0 >>> and >>> -1/(2^n) >= 0 >>> >>> Stephen >> Hell no! > So you must believe that IN is a subset of OUT, as every > integer n that satisfies > -1/(2^(floor(n/10))) < 0 > also satisfies > -1/(2^n) >= 0 oops, that should be -1/(2^n) < 0 Stephen > and if IN is subset of OUT then > | IN - OUT | = 0
From: Lester Zick on 29 Oct 2006 14:05 On Sat, 28 Oct 2006 15:46:26 -0400, Tony Orlow <tony(a)lightlink.com> wrote: >Lester Zick wrote: >> On Fri, 27 Oct 2006 21:04:07 -0400, Tony Orlow <tony(a)lightlink.com> >> wrote: >> >>> MoeBlee wrote: >>>> Lester Zick wrote: >>>>> Ah, Moe, truth is often a nuisance. >>>> Then you're as much of a nuisance as is a cool breeze on a sunny spring >>>> day, as a cleansing and quenching rain that ends a drought, as a >>>> magnificent symphony orchestra heard in an amphitheatre of impeccable >>>> acoustics. >>>> >>>> Moe Blee >>>> >>> As a kitty cat on your lap, on a chilly night... :) >>> >>> Lesterrrrrrrr...... >>> >>> >>> Actually, I named my last cat Lester, before I met Lester here online. I >>> had to scrape his pieces off the road.... >> >> LOL, Tony! Did you try to put the set of pieces back together? >> >> ~v~~ > >Uh, no Lester, I buried him out back, in a pillow case, next to Nelie >and Freddy. I was going to make a whole talk show around him, too..... But you see, Tony, modern math thinks you can put the pieces back together again. Ah well RIP, Namesake. ~v~~
From: Lester Zick on 29 Oct 2006 14:06
On Sat, 28 Oct 2006 15:48:58 -0400, Tony Orlow <tony(a)lightlink.com> wrote: >Lester Zick wrote: >> On Fri, 27 Oct 2006 21:23:44 -0600, Virgil <virgil(a)comcast.net> wrote: >> >>> In article <4542aa5c(a)news2.lightlink.com>, >>> Tony Orlow <tony(a)lightlink.com> wrote: >>> >>> >>>> Stop confusing me with facts, Lester. >>> TO is easily confused by facts. >>> >>> But Lester rarely provides any. >>> >>> A marriage made in Heaven? Or Hell? >> >> You're too fickle for me, Virgil. Aatu calls you "pathetic" for your >> dogged faithfulness to me so you just jump over to Tony instead. >> >> ~v~~ > >It's okay. Aatu says he's pathetic for wasting his time with me too. >Kerberos gots some snarlin' to do. Woof! Then it must be a compulsive disorder. ~v~~ |