countability of real numbers? What am I missing?
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From: Transfer Principle on 18 May 2010 13:40 On May 18, 6:27 am, Saijanai <saija...(a)gmail.com> wrote: > OK, all you set/number theorists, what is wrong with this binary > fraction sequence. I assert it lists all real numbers [0,1] (allowing > for duplicates): > {0.0, 0.1}, > {0.00, 0.01, 0.10, 0.11}, > {0.000, 0.001, 0.010, 0.011, 0.100, 0.101, 0.110, 0.111}, > ... Ah, a countability of R thread. This is the perfect opportunity for me to practice new habits of what to say in this situation. I've resolved to acknowledge at least three possibilities to describe the OP, Saijanai: Case 1. Saijanai wishes to describe a theory other than the classical reals (a theory in which only the numbers that are in his list count as reals in the proposed theory). Case 2. Saijanai knows that there exist uncountably many classical reals, but doesn't like this fact. Case 3. Saijanai believes that his list actually contains every classical real. This is the only case for which it would be accurate to call Saijanai wrong. In the past, I've usually assumed Case 1 without considering the other cases. Now I must consider all three cases. This thread is only four hours old. As the thread progresses, we can find out which case is most likely.
From: Tonico on 18 May 2010 14:42 On May 18, 8:40 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > On May 18, 6:27 am, Saijanai <saija...(a)gmail.com> wrote: > > > OK, all you set/number theorists, what is wrong with this binary > > fraction sequence. I assert it lists all real numbers [0,1] (allowing > > for duplicates): > > {0.0, 0.1}, > > {0.00, 0.01, 0.10, 0.11}, > > {0.000, 0.001, 0.010, 0.011, 0.100, 0.101, 0.110, 0.111}, > > ... > > Ah, a countability of R thread. This is the perfect opportunity > for me to practice new habits of what to say in this situation. > > I've resolved to acknowledge at least three possibilities to > describe the OP, Saijanai: > > Case 1. Saijanai wishes to describe a theory other than the > classical reals (a theory in which only the numbers that are > in his list count as reals in the proposed theory). > > Case 2. Saijanai knows that there exist uncountably many > classical reals, but doesn't like this fact. > > Case 3. Saijanai believes that his list actually contains every > classical real. This is the only case for which it would be > accurate to call Saijanai wrong. > > In the past, I've usually assumed Case 1 without considering > the other cases. Now I must consider all three cases. > > This thread is only four hours old. As the thread progresses, > we can find out which case is most likely. Uh? Wasn't enough that he explicitly talked about Cauchy sequences from his list's numbers that can approximate every number in [0,1] INSTEAD of actually having all the real numbers in his list, as he was told by several posters, to conclude that he's wrong? He thought that having Cauche seq. does the trick, and this means, imfho, that he's wrong....what are you waiting for to deduce this? Perhaps during time he'll change in a subtle way his stand so that you'll say he wasn't wrong but...whatever? What has time to do with this at all?? How much will you wait to deduce I'm wrong if I start a thread claiming that 2 + 2 = 5 is true within the usual field axioms of the real numbers? 8 hours, 16 hours, 2 days..? Tonio
From: James Burns on 18 May 2010 15:55 Transfer Principle wrote: > On May 18, 6:27 am, Saijanai <saija...(a)gmail.com> wrote: > >>OK, all you set/number theorists, what is wrong with this binary >>fraction sequence. I assert it lists all real numbers [0,1] (allowing >>for duplicates): >>{0.0, 0.1}, >>{0.00, 0.01, 0.10, 0.11}, >>{0.000, 0.001, 0.010, 0.011, 0.100, 0.101, 0.110, 0.111}, >>... > > > Ah, a countability of R thread. This is the perfect opportunity > for me to practice new habits of what to say in this situation. > > I've resolved to acknowledge at least three possibilities to > describe the OP, Saijanai: > > Case 1. Saijanai wishes to describe a theory other than the > classical reals (a theory in which only the numbers that are > in his list count as reals in the proposed theory). > > Case 2. Saijanai knows that there exist uncountably many > classical reals, but doesn't like this fact. > > Case 3. Saijanai believes that his list actually contains every > classical real. This is the only case for which it would be > accurate to call Saijanai wrong. > > In the past, I've usually assumed Case 1 without considering > the other cases. Now I must consider all three cases. > > This thread is only four hours old. As the thread progresses, > we can find out which case is most likely. Here is my opportunity to demonstrate a technique for distinguishing between the three cases you list, or between other possibilities, uhm, possibly overlooked. A-HEM! Saijanai, which of Transfer Principle's three cases is the case? Or is there some other description of your beliefs about reals and countability that would be more appropriate? This technique doesn't always work. It may be that we will not hear from Saijanai again, soon or ever. Often posters get what they want and go back to real life. While we wait, I will bet that Saijanai knows, or at least strongly suspects, that he(?) is making an error somewhere but wants to understand where specifically the mistake is -- to improve his understanding of cardinality, the reals, etc, etc. See, for example, Saijanai: <I'm not sure if that's cheating or not. I suspect <it is, but I'm only auditing elementary level number <theory/set theory/analysis lectures online right now <and I'm no doubt missing something (or just don't <understand the lectures I've already seen in the <first place). <62540561-1418-4c09-9115-a575cbfc4047(a)31g2000prc.googlegroups.com> It's an interesting question for you what to do, if I am right. Then, Saijanai would be right that he is wrong. Would you disagree and tell him he is right, or agree that he is wrong? (This sort of thing is not a problem for me. I'll tell him which arguments are right or wrong, and let Saijanai worry about what he /wants/ to be right or wrong.) Jim Burns
From: Transfer Principle on 18 May 2010 17:31 On May 18, 12:55 pm, James Burns <burns...(a)osu.edu> wrote: > Here is my opportunity to demonstrate a technique for > distinguishing between the three cases you list, or > between other possibilities, uhm, possibly overlooked. I dis acknowledge that the list isn't necessarily exhaustive, when I wrote: > > I've resolved to acknowledge _at_least_ three possibilities to > > describe the OP, Saijanai: (emphasis added) > A-HEM! Saijanai, which of Transfer Principle's three > cases is the case? Or is there some other description > of your beliefs about reals and countability that would > be more appropriate? Direct asking sometimes works, but I only like to ask when I can avoid making it sound like an _interrogation_ or giving a poster the third degree. In particular, when others are already asking the OP questions, I usually avoid doing so. Asking vs. interrogating... > It's an interesting question for you what to do, > if I am right. Then, Saijanai would be right that > he is wrong. Would you disagree and tell him he > is right, or agree that he is wrong? Burns and Tonio, admittedly, do give evidence suggesting that the OP is in Case 3. Therefore, I have decided to agree with them that Case 3 is the most likely. And so, even though I do find a theory in which the reals with finite binary expansions are the only reals to be an interesting theory, Case 3 tells us that such a theory would have nothing to do with Saijanai. It would be considered wrong ("pandering") of me to attempt to write such a theory in this thread. Case 3 tells us that Saijanai is wrong, but there are already enough posters telling him so in this thread, and so for me to join them would be redundant. I'd rather not post at all than _repeat_ that he is wrong. Therefore, I must say good-bye to this thread. This is my final post in this thread.
From: christian.bau on 18 May 2010 18:53
On May 18, 6:40 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > Case 3. Saijanai believes that his list actually contains every > classical real. This is the only case for which it would be > accurate to call Saijanai wrong. The others fall in the category "this is so bad, it's not even wrong". What the original poster doesn't realise is that he has a sequence with countably many elements, but the number of subsequences that are Cauchy sequences is uncountable. And yes, the (uncountably many) Cauchy sub-sequences _do_ cover all the reals in [0, 1]; but that obviously does nothing to show the reals are countable. |