An uncountable countable set
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From: Virgil on 7 Sep 2006 22:57 In article <1157683628.865441.326900(a)e3g2000cwe.googlegroups.com>, imaginatorium(a)despammed.com wrote: > Virgil wrote: > > In article <45001e94(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > > > Virgil wrote: > > > > In article <44fe1d38(a)news2.lightlink.com>, > > > > Tony Orlow <tony(a)lightlink.com> wrote: > > > > > > So that if TO wants justifications, they are available, but they are > > > > often too technical for someone of TO's level of understanding to > > > > comprehend, and even if not, tend to be buried in papers that are > > > > otherwise highly technical in precisely the ways TO object to all the > > > > time. > > > > > > > > > > In other words, you justify the viability of the axioms by the fact that > > > they have gone through a long evolutionary process and survived in their > > > current form, rather than having died out. The same can be said for the > > > platypus. Quack! ;) > > > > The platypus, despite having what looks much like a duck's bill, does > > not quack. > > What noise does it make, then? The call of the platypus is a low growl, at least according to http://library.thinkquest.org/11922/mammals/platypus.htm I have never heard one myself. > http://imaginatorium.org
From: imaginatorium on 7 Sep 2006 23:09 Virgil wrote: > imaginatorium(a)despammed.com wrote: > > Virgil wrote: > > > Tony Orlow <tony(a)lightlink.com> wrote: > > > > Virgil wrote: > > > > > Tony Orlow <tony(a)lightlink.com> wrote: <my goodness, don't they go on...> > > > > In other words, you justify the viability of the axioms by the fact that > > > > they have gone through a long evolutionary process and survived in their > > > > current form, rather than having died out. The same can be said for the > > > > platypus. Quack! ;) > > > > > > The platypus, despite having what looks much like a duck's bill, does > > > not quack. > > > > What noise does it make, then? > > The call of the platypus is a low growl, at least according to > http://library.thinkquest.org/11922/mammals/platypus.htm > I have never heard one myself. How can you say that to the ears of another platypus it doesn't sound like a quack? After all, I believe that all animals have roughly the same number of heartbeats in their lives. So while an elephant appears to you or me to go <<kerrrrrrrrrrrrrrrrr-ttthhukunkkkkkkkkkkkkk kerrrrrrrrrrrrrrrrr-ttthhukunkkkkkkkkkkkkk>> and a mouse goes <tikitikitikitikitikitikitikitikitiki>, presumably each of them hears themself as <tick-tock-tick-tock>. I wonder whether this applies to Orlow's scheme? I mean, I wonder if elephants, being that bit closer to Big'un, perceive even actually infinite numbers as being juuust within finite reach? Whereas to mice, even Littl'un is of appreciable size? Who knows. Brian Chandler http://imaginatorium.org
From: imaginatorium on 7 Sep 2006 23:11 stephen(a)nomail.com wrote: > Mike Kelly <mk4284(a)bris.ac.uk> wrote: > > > Tony Orlow wrote: > >> Mike Kelly wrote: > >> > > >> > Does it not bother you that nobody else agrees with, or even > >> > understands, your proof? > >> > > >> > >> I find it disappointing, but not surprising, that you don't understand > >> such a simple proof, since it's contradictory to your education. I do > >> find it annoying that you feel the right to disagree with it without > >> understanding it. If you feel there is a problem with the proof, please > >> state the logical error I made. If the string is all finite bits, and > >> none of them ever can possibly achieve an infinite value, then how can > >> the string have an infinite value? There's nowhere in the string where > >> that can occur. It's that simple. Grok it. > > > 1) A finite string of 1s represents a (finite) natural number. > > 2) An infinite string of 1s represents a (finite) natural number. > > > 1) doesn't imply 2). > > Tony's argument seems to be a upside-down inside-out version > of one of Zeno's paradoxes. > > ...111111 (in binary) = 1 + 2 + 4 + 8 + ... > > All the numbers on the right are finite, so the sum must > be finite. Apparently a sum can only be infinite if one of the > terms in the sum is infinite. An infinite sum of finite terms > cannot possibly be infinite in Tony's mind. Not entirely sure about this. The problem above is that we are restricting the string ....11111 so that every 1 is in a finite position. The position numbers never achieve true infinity. Whereas, if we had the _different_ string (has Tony invented notation for this? possibly involves colons?).. ***[super-dot] *** *** ...11111 where going left the digit position count goes through the twilight zone, and achieves infinite values, then indeed the sum _would_ be infinite. Uh, I think. Brian Chandler http://imaginatorium.org
From: stephen on 8 Sep 2006 00:17 imaginatorium(a)despammed.com wrote: > stephen(a)nomail.com wrote: >> Mike Kelly <mk4284(a)bris.ac.uk> wrote: >> >> > Tony Orlow wrote: >> >> Mike Kelly wrote: >> >> > >> >> > Does it not bother you that nobody else agrees with, or even >> >> > understands, your proof? >> >> > >> >> >> >> I find it disappointing, but not surprising, that you don't understand >> >> such a simple proof, since it's contradictory to your education. I do >> >> find it annoying that you feel the right to disagree with it without >> >> understanding it. If you feel there is a problem with the proof, please >> >> state the logical error I made. If the string is all finite bits, and >> >> none of them ever can possibly achieve an infinite value, then how can >> >> the string have an infinite value? There's nowhere in the string where >> >> that can occur. It's that simple. Grok it. >> >> > 1) A finite string of 1s represents a (finite) natural number. >> > 2) An infinite string of 1s represents a (finite) natural number. >> >> > 1) doesn't imply 2). >> >> Tony's argument seems to be a upside-down inside-out version >> of one of Zeno's paradoxes. >> >> ...111111 (in binary) = 1 + 2 + 4 + 8 + ... >> >> All the numbers on the right are finite, so the sum must >> be finite. Apparently a sum can only be infinite if one of the >> terms in the sum is infinite. An infinite sum of finite terms >> cannot possibly be infinite in Tony's mind. > Not entirely sure about this. The problem above is that we are > restricting the string > ...11111 > so that every 1 is in a finite position. The position numbers never > achieve true infinity. Yes, which means that each term in the sum 1 + 2 + 4 + 8 + ... never achieves true infinity. Apparently because no single term in the sum is infinite, the sum itself cannot be infinite. If you had a 1 in an infinite position p, then the sum would contain the term 2^p, which would be infinite, because p is infinite, and the sum would be infinite. Stephen
From: imaginatorium on 8 Sep 2006 01:22
stephen(a)nomail.com wrote: > imaginatorium(a)despammed.com wrote: > > stephen(a)nomail.com wrote: > >> Mike Kelly <mk4284(a)bris.ac.uk> wrote: > >> > >> > Tony Orlow wrote: > >> >> Mike Kelly wrote: > >> >> > > >> >> > Does it not bother you that nobody else agrees with, or even > >> >> > understands, your proof? > >> >> > > >> >> > >> >> I find it disappointing, but not surprising, that you don't understand > >> >> such a simple proof, since it's contradictory to your education. I do > >> >> find it annoying that you feel the right to disagree with it without > >> >> understanding it. If you feel there is a problem with the proof, please > >> >> state the logical error I made. If the string is all finite bits, and > >> >> none of them ever can possibly achieve an infinite value, then how can > >> >> the string have an infinite value? There's nowhere in the string where > >> >> that can occur. It's that simple. Grok it. > >> > >> > 1) A finite string of 1s represents a (finite) natural number. > >> > 2) An infinite string of 1s represents a (finite) natural number. > >> > >> > 1) doesn't imply 2). > >> > >> Tony's argument seems to be a upside-down inside-out version > >> of one of Zeno's paradoxes. > >> > >> ...111111 (in binary) = 1 + 2 + 4 + 8 + ... > >> > >> All the numbers on the right are finite, so the sum must > >> be finite. Apparently a sum can only be infinite if one of the > >> terms in the sum is infinite. An infinite sum of finite terms > >> cannot possibly be infinite in Tony's mind. > > > Not entirely sure about this. The problem above is that we are > > restricting the string > > ...11111 > > so that every 1 is in a finite position. The position numbers never > > achieve true infinity. > > Yes, which means that each term in the sum > 1 + 2 + 4 + 8 + ... > never achieves true infinity. Apparently because no single > term in the sum is infinite, the sum itself cannot be infinite. > If you had a 1 in an infinite position p, then the sum would > contain the term 2^p, which would be infinite, because p > is infinite, and the sum would be infinite. Hmm, I don't think you're quite getting into the spirit of this. In Tony's scheme (insofar as I understand it, and of course it appears to me to be as contradictory as it does to you), I believe you can count numbers 1, 2, 3, ... and so on, and go on for ever. But at some foggy point (perhaps after about ever has elapsed) you find that you have arrived in the infinite zone, and are counting numbers which, in an infinite binary notation, have nonzero digits at least close to the (nonexistent) left end. One of these numbers is called Big'un, and if you write the sum 1 + 2 + 4 + 8 + ... + 2^Big'un + 2^(Big'un+1) + ... then it _is_ infinite (in Tony's scheme), because at least one of the numbers in the sum has achieved genuine infinity. So in Tonyspeak, you have to quantify 1+2+4+... by saying what sort of zone the dots extend to. I think. Brian Chandler <<<Advert>>> Understand the mysteries of infinity! Contemplate the cosmos! Learn if Avalokiteshvara was a woman! Buy a 2000-piece mandala puzzle, and keep the Imaginatorium afloat! Guaranteed to remain fun for more hours than reading loony drivel on sci.math! http://imaginatorium.org/shop/mandala.htm |