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On Real Enumeration 
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#1
Nov1012, 02:53 PM

P: 82

Hey, I've been reading up on Cantor's work recently, and was wondering if the following series can be considered an enumeration for the reals between 0 and 1 in binary:
[0] => 0.00000... [1] => 0.10000... [2] => 0.01000... [3] => 0.11000... [4] => 0.00100... [5] => 0.10100... [6] => 0.01100... [7] => 0.11100... [8] => 0.00010... [9] => 0.10010... [10] => 0.01010... [11] => 0.11010... [12] => 0.00110... [13] => 0.10110... [14] => 0.01110... [15] => 0.11110... . . . If not then can you please explain why this isn't an accurate enumeration? Thank You! 


#2
Nov1012, 03:01 PM

P: 772

Cantor clearly shows why in his diagonalization argument. I can easily create an infinite length binary string that isn't on your list.



#3
Nov1012, 03:07 PM

P: 82

I don't get it, the list I made is defined to go through every possible combination of 0's and 1's for a binary expansion of any number of digits. So how could there be a binary string not in the set of every possible binary strings?



#4
Nov1012, 03:12 PM

P: 772

On Real Enumeration
The first digit in your first string is zero, so the first digit in my string is one. The second digit in your second string is one, so the second digit in my string is zero And so on... My string differs from every string on your list. Your enumeration fails. 


#6
Nov1012, 03:19 PM

P: 82

Could you please elaborate? I have just a basic understanding of these ideas, I admit, but I see n=1,5,9,13, which all start with the example you gave. And if you continued with your list, I can keep giving you n's which start with the sequence you give.



#7
Nov1012, 03:22 PM

Mentor
P: 18,246

I'm sure you can find numbers which correspond to 0.1 0.101 0.10101 0.1010101 and so on. But can you find a number corresponding with the entire (infinite) sequence 0.101010101010101... ?? That's what Cantor's theorem is about. 


#8
Nov1012, 03:24 PM

P: 898




#9
Nov1012, 03:38 PM

P: 82




#10
Nov1012, 03:40 PM

Mentor
P: 18,246




#11
Nov1012, 03:51 PM

P: 772




#12
Nov1012, 05:08 PM

Sci Advisor
P: 1,716

you list only has numbers with finitely many zeros



#13
Nov1012, 09:09 PM

P: 82

Thank you all for your responses! It hit me that I had read this idea of finitely many nonzero digits before so I think I understand what you are saying now.
I'm curious though, being outside of academia myself, to what extent are Cantor's ideas criticized? Specifically, I'm speaking about the idea that a onetoone correspondence with the naturals is enough to sensibly conclude that the two sets are of equal cardinality even though the members of one set, in the case of the even numbers for example, might be twice in measure of the members of the other set. And I'm asking this in context to the fact that we are doing the counting without reference to a measurable interval. (Sorry for the nonstandard terminology). I mean, if I generally want to know how many of two marker types there are on an infinitely long road, and you give me that marker type A is spaced X units apart, and marker type B is spaced Y units apart, I couldn't possibly give you a sensible answer without being given some interval to divide the markers into. What I don't get is that if Cantor himself developed the measure of the cardinality of infinite sets, namely aleph0, then why not use alepth0 as an interval measure into which one puts members of two different sets in order to determine their cardinality. In this case, I imagine that one would get a different result for the naturals and the even naturals. 


#14
Nov1012, 09:19 PM

Mentor
P: 18,246

That said: there are of course attempts to make other axiom systems. Other axioms might mean that Cantor's theorems become false. But that doesn't mean that anybody doubts the truth of Cantor's theorems as they are only true in the given axiom system. Furthermore, when you say that the even numbers are "twice in measure", then you first need to define measure. Are you talking about the distance between different natural numbers?? And you're saying that the distance between the even numbers is bigger than the distance between natural numbers?? That makes sense and it is true. But Cantor doesn't talk about distances. A distance is another type of structure. Cantor just talks about sets and nothing else. If you want to talk about distances, then you shouldn't go to Cantor but to the theory of topology or metric spaces. 


#15
Nov1112, 10:44 AM

P: 82




#16
Nov1212, 07:17 AM

P: 932

Define "asymptotic density" for a subset S of the naturals by dividing (how many elements of S are less than n) by n and taking the limit as n increases without bound. The even naturals have an "asymptotic density" of 0.5 If you choose to intuit this as "half of all naturals are even", we can't stop you. A problem is that this notion of "how many" only works for subsets of the natural numbers and does not work for all subsets of the naturals. For instance: { n : 2^k <= n < 2^k+1 for some even k } does not have an asymptotic density. This set is { 1, 4, 5, 6, 7, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 64, ... }. Intuitively, the density of this subset of the naturals ranges between 1/3 and 2/3 depending on how far out you count. 


#17
Nov1212, 11:35 AM

P: 82




#18
Nov1212, 12:41 PM

P: 932

Assuming that density will converge does not make it so. Adding an axiom to make it so won't help  that'll just give you an inconsistent formal system. The asymptotic density of n does not exist  n is not defined. The asymptotic density of S does exist. But it does not fluctuate. The density of S intersect {1 .. n} could be defined as a function of n. That could be seen as fluctuating. Because it fluctuates it would not converge. Perhaps you are saying that an intuitively satisfying notion of density should exist for any subset and should have a numeric value that converges as you "count through" the parent set, regardless of how you count through. I fear that such a metric is impossible to find. 


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