Cantor Confusion
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From: Poker Joker on 30 Sep 2006 09:27 "Tonico" <Tonicopm(a)yahoo.com> wrote in message news:1159605340.208078.275580(a)c28g2000cwb.googlegroups.com... > > Poker Joker wrote: >> "Alan Morgan" <amorgan(a)xenon.Stanford.EDU> wrote in message >> news:efkegr$6d9$1(a)xenon.Stanford.EDU... >> >> >>if its true for ANY list, then it must be >> >>true for a specific list. So if considering a single specific list >> >>shows a flaw, then looking at ANY (ALL of them) list doesn't >> >>help. >> > >> > But if it's true for ANY list then it must be true for a specific >> > list. So if considering a single specific list shows a flaw then >> > perhaps that list doesn't really exist. >> >> That's true, but that's not the entire story. >> >> Suppose I claim that I have a list that contains all the reals. >> You claim you can take that list and construct a real not >> on the list. You procede to show the construction. I would >> claim that your construction is flawed because it is >> self-referential, which it must be if I truly gave you a list of >> all the reals. So in that *SPECIAL CASE*, unlike the >> general case, your construction isn't valid. The only way >> to eliminate that special case is to use what the >> conclusion of the proof would be if you neglected the >> special case. > ************************************************************************************ > Hollie Mollie and Holy Moses!! Poker, son: not only you beliittle and > "scoff" (or so you seem to believe you do) people that is WAY more > prepared and educated than you to deal with these things, just as good > old well-reknown cranks, crackpots and trolls usually do, but you also > have the logic of a raving TV preacher trying to convince people that > his god is a better > player of golf than his devil. > Common, do you REALLY believe the stupidity that you just wrote above?? > Not that it is your first in this thread, but the above one....dude! > You wrote: "Suppose I claim that I have a list that contains all the > reals. You claim you can take that list and construct a real not > on the list. You procede to show the construction. I would claim that > your construction is flawed because it is self-referential, which it > must be if I truly gave you a list of > all the reals...", being "if" the key word here. Dudeman, the word *IS* "if". So why must reality be involved? Look what you've written. > Iff you gave a complete list of the reals then the other part would NOT > be able to construct a real that is not in it...kid, DAH! And any construction of such a number would be flawed. This statement is untrue. Perfectly grammatically correct. Just like the process that takes all reals and produces on that wasn't in its input. The output LOOKS good. No proof that if you feed it ALL the reals that it actually produces as advertised. It can't. > how hard is this > to for you to understand? Are you trying to make the competence to > James Harris or what? > So the construction is flawed because YOU think it is self-referential So you think a process that takes all reals can produce one that isn't in the set of all reals is possible? Wait, I get it. You want to use the consequences of having such a process to prove that there is such a process. Now I understand your logic. > IF you actually gave > a complete "list" of the reals, uh?? By your """logic""" then, it is > IMPOSIBLE to rebuke > anything you say, because IF it is true then the rebuttal will have, OF > COURSE!, to be flawed....great! It took you this long to figure that out? Of course, you can't stop thinking that the conclusion stands on its own. So how could you see the flaw?
From: Peter Webb on 30 Sep 2006 09:53 <mueckenh(a)rz.fh-augsburg.de> wrote in message news:1159610608.002214.150810(a)i3g2000cwc.googlegroups.com... > > William Hughes schrieb: > > >> For every real number x, there exists a list, L_x such >> that x is a member of L_x > > Where do they exist? It is impossible for most real numbers even to > name them. You cannot name or construct more than aleph_0 real numbers. > Nevertheless you insist, that also the other ones, which have no names > and no other identification properties, should have complete lists? This is a complete red herring. There is no question that the Real generated by Cantor's proof is computable (r. e,) if the original list is, as he gives an explicit construction. >> >> There exists a list L, such that every real number x is >> a member of L. >> >> The first is true, the second is false. There is no way to put >> all the L_x together to get a "countable set of entries" >> (the list L). > > But if they existed, then we could put them together. Why can't we > connect in our thoughts all these thought lists such that there is only > one thought list, i.e., the thought list of all thought lists? At least > a square of all thought lists should be possible. We can connect them. We know that aleph_0 squared is still aleph_0, because we pair the elements off as follows: 1 <-> (1,1) 2 <-> (1,2) 3 <-> (2,1) 4 <-> (2,2) 5 <-> (3,1) 6 <-> (3,2) etc. The same will work for aleph_0 to any power; it is not until we get to aleph_0 to the aleph_0 power (which is the same as 2 to the aleph_0 power) that it changes. > > But if you like, you can schematically consider all real numbers by the > infinite binary tree which contains them all represented by a countable > set of nodes and edges. I have shown by a rational relation that the > set of branches (corresponding to real numbers) is not larger than the > set of edges. > Then the set of edges isn't countable either. > Regards, WM >
From: Peter Webb on 30 Sep 2006 10:05 "Poker Joker" <Poker(a)wi.rr.com> wrote in message news:3HmTg.25601$QT.205(a)tornado.rdc-kc.rr.com... > "MoeBlee" <jazzmobe(a)hotmail.com> wrote in message > news:1159581865.120392.117490(a)h48g2000cwc.googlegroups.com... > >> Like I said, you've not refuted the uncountability of the reals nor >> Arturo's point to which you originally responded. > > I never tried to refute the uncountability of the reals. Too bad > you've never been able to understand that. > No, but you introduced a specious point that seemed to support this argument though. You said that for any real x exists y such that x/y=0. This statement is false, at least within standard arithmetic, as you point out. Cantor is different in that the statement is true, and easily proved.
From: Virgil on 30 Sep 2006 14:11 In article <1159611066.767146.101490(a)e3g2000cwe.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > cbrown(a)cbrownsystems.com schrieb: > > > > Therefore, the assertion "M is a complete list of reals" is only true > > if the assertion "M is complete, and M is not complete" is true. > > > > (A and ~A) = false. > > A system has the property W, if it can be proved that the reals can be > well-ordered. A system has the property ~W if it can be proved that the > reals cannot be well-ordered. A system is self-contradictive, if W and > ~W can be proved. Therefore the system does not exist. > So "Mueckenh" concludes that there is no system in which there is a complete list of reals? So what else is new?
From: Virgil on 30 Sep 2006 14:23
In article <1159611187.639749.195260(a)i3g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Unfortunately, his method fails in some cases which is fatal for an > impossibility proof. The list, for instance, > > 0.0 > 0.1 > 0.11 > 0.111 > ... > > with the prescription that the diagonal digit 0 is replaced by 1, > delivers a number which is not different from any list number, except > in the last digit, which, however, does not exist. Except that that list is really the list 0.000... 0.1000... 0.11000... 0.111000... which is the same as the the rational list 0 1/2 3/4 7/8 .... or the base 4 list 0.000... 0.1000... 0.3000... 0.31000... 0.33000... .... And in this last form, it is easy to construct a number to in the list: 0.222..., for example |