countability of real numbers? What am I missing?
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From: Tim Little on 19 May 2010 00:24 On 2010-05-18, Saijanai <saijanai(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}, > ... Every number in this list is of the form k/2^N for some natural numbers k and N. That doesn't even include all the rationals, let alone the reals. For example: if 1/3 is somewhere in the list, you should be able to tell me at least one pair of (k,N) values it has. - Tim
From: Tim Little on 19 May 2010 00:44 On 2010-05-18, Saijanai <saijanai(a)gmail.com> wrote: > My assertion is that the sequence doesn't miss any real and by > extension, you can get arbitrarily close to any real by going out > far enough. Getting arbitrarily close is not enough. Definition: X is "countable" iff there exists f:N->X such that for all x in X, there exists n in N with f(n) = x. For each x there must exist *specific* n where f(n) is *equal to* x. > The relevant issue is countability, at least to me. This gives a > countable sequence of numbers that doesn't miss any reals. The range of the sequence is "dense" in the interval (in the sense of the usual topology), but fails to include most of them. > Again, I think its an issue with conflating different definitions of > ordering, No, it's an issue with what it means for a sequence to include a real number. - Tim
From: Mike Terry on 19 May 2010 13:11 "Transfer Principle" <lwalke3(a)lausd.net> wrote in message news:0df6bdc8-51e2-4390-bf33-6bb42af73978(a)t14g2000prm.googlegroups.com... > 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. It seems to me that nearly all of these threads fall into Case 3, and can be subclassified as Case 3a. OP does not understand the standard meaning of "list" (or "sequence") Case 3b. OP does not understand what it means for a number "to be in" a list/sequence. Case 3c. Other error. (Mostly I think 3a or 3b is the issue. Hard to tell which since they're obviously related. No doubt you would say that in these cases the OP is intending to be using some non-standard definition of list/sequence? :-) Mike.
From: cwldoc on 19 May 2010 13:09 > 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. case 4. Saijanai simply ignores everything posters have said, repeating the same nonsense over and over. He probably had no intention from the outset of being enlightened in any way! Posters seem naive about his intentions, thinking that he just needs more instruction, or perhaps they suspect the truth but just give him the benefit of the doubt. Then they keep posting more clarifications, apparently still being surprised that their insights are not heeded! I don't know whether to be amused or upset at having wasted my time reading this post!
From: Ronald Bruck on 21 May 2010 14:12
In article <6b027f1d-3ba8-4f9f-9a12-7fb28e08fafd(a)y18g2000prn.googlegroups.com>, Saijanai <saijanai(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}, > ... By "listing" every real number, we mean it must ACTUALLY APPEAR somewhere in the list--not just that it can be constructed as some sort of limit of items in the list. If the list is L, then it can be enumerated L1, L2, L3, ... Thus every real number must appear EXACTLY ONCE. So, for example, I don't see the number 1/3 in your list. In binary, that would be 0.010101..., which has infinitely digits and so is at NO level in your list. Your list only enumerates the dyadic fractions m/2^n, where 0 <= m < 2^n. But I don't expect you to understand this. You're either a troll, deliberately provoking responses, or you don't understand standard nomenclature. --Ron Bruck If one wishes to patent a number--some have actually done this--I claim it would be best to patent 0 or 1, or better yet, 0 AND 1. For no other number can be written in binary without using at least one of these. The net effect of this observation is that we probably all owe a lot in royalties to either AT&T or IBM. You object, But people have been using these for many years already. I reply, That doesn't seem to stop companies from ignoring prior art in many other patent applications. |