October 9th, 2018, 06:34 AM  #181 
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 124 
The natural numbers don't end.

October 9th, 2018, 06:39 AM  #182 
Global Moderator Joined: Dec 2006 Posts: 20,375 Thanks: 2010  Each string in that list is, by definition, identified by a finite integer, whereas the nonterminating string 333....... isn't. As zylo has just posted, the list of natural numbers doesn't end, and that's true even if nonterminating strings are explicitly excluded, so there's nowhere where nonterminating numbers can exist in that list, and there is no "end" of that list after which they could be added.

October 9th, 2018, 07:13 AM  #183  
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 124 
It/s really quite simple Quote:
I can calculate (specify) sqrt2 to any number of digits. That's the definition of a real number. The end of the calculation doesn't exist. The real numbers only exist to arbitary n digital places. To the extent they exist they can be counted. Hence my remark that you can't count a set of things that don't exist. EDIT This thread exhibits the futility of dealing with the real numbers beyond the above quote. Last edited by zylo; October 9th, 2018 at 07:33 AM.  
October 9th, 2018, 08:46 AM  #184 
Senior Member Joined: Jun 2014 From: USA Posts: 493 Thanks: 36 
So Zylo is a finitist and Cantor’s argument doesn’t apply. Glad we cleared that up.

October 9th, 2018, 09:00 AM  #185 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,621 Thanks: 2609 Math Focus: Mainly analysis and algebra  
October 9th, 2018, 09:26 AM  #186  
Global Moderator Joined: Dec 2006 Posts: 20,375 Thanks: 2010  Quote:
Most nonterminating sequences (or reals) don't have any convenient definition, but that doesn't mean that they don't exist. It's not required that there is a known method of determining their digits. Any list of strings can be diagonalized if each string has a position in the list that can be specified by a finite integer. Without that restriction, it isn't possible to prove that the list is countable. That restriction is therefore crucial. It wasn't met by your list, as the nonterminating string 333..... can't have a position identified by a finite integer. Diagonalization allows a string to be specified by reference to the list in such a way that it can't be in the list. This means that knowing the content of the list is irrelevant. It doesn't even matter whether two strings in the list are the same. It doesn't matter if nobody knows what any of the strings are. All that matters is that a finite integer suffices to denote the position of each digit in a string, regardless of whether the value of that digit is known or defined in some way, and that a finite integer suffices to denote the position in the list of each string, regardless of whether the digits in the string are known or defined in some way. That's incorrect. It's also irrelevant in relation to diagonalization.  
October 9th, 2018, 09:40 AM  #187 
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 124 
To 2 places: 0'0 10' 01 11 
October 9th, 2018, 10:16 AM  #188  
Senior Member Joined: Jun 2014 From: USA Posts: 493 Thanks: 36  Quote:
Quote:
Perhaps I'm being bias, but I thought my example was a decent way of contemplating this. You didn't seem interested and just kept repeating the same old stuff as though, if you said it enough, it would be true. I asked you if $\bigcup B = \bigcup B'$. I was basically asking if you think there is an element of $f(k)$ that doesn't exist in $\bigcup f([0,1)_2)$. It's a logical proposition because $f(k)$ is unique and $f([0,1]_2)$ is pairwise almost disjoint. The only problem is that we can't solve for whatever element of $f(k)$ isn't in $\bigcup f([0,1)_2)$ because such an element would be the 'last' element of $f(k)$. I then showed a couple of funky results if we accept that interpretation, such as the reals being countable and the ability to show a difference in cardinality between sets that in fact have equal cardinality. So, either there is a problem with induction and selfreferential arguments, or we can't assume that there is an element of $f(k)$ that isn't in $\bigcup f([0,1)_2)$. Quote:
Quote:
The set of natural numbers doesn't 'exist' outside of the theoretical, so if the very set we're using to do the counting doesn't exist, why can't we use it to count other sets that don't exist? In my example, Cantor's diagonal equated to what would have been the last element of an infinite sequence. You asserted that the sequence didn't end while simultaneously asserting that Cantor's diagonal appeared in the sequence (a contradiction). I tried to expand on this by asking how $f(k)$ could possibly be a subset of the union of $B$ when Cantor's very argument assures that it is different from every element of $B$. I personally could view the proof by induction that $f(k) \subset \bigcup B$ as being flawed, but when I do, the results are inconsistent so, in my view, we are choosing between complete and inconsistent versus incomplete and consistent. What do you think about that last sentence of mine Zylo (anybody)?  
October 9th, 2018, 11:53 AM  #189 
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 124  So you want to see CDA with infinite binary sequences?: 0) 0 0 0 0 0 0 0 0 ........... 1) 1 0 0 0 0 0 0 0 ........... 2) 0 1 0 0 0 0 0 0 ........... 3) 1 1 0 0 0 0 0 0 ........... 4) 0 0 1 0 0 0 0 0 ........... 5) 1 0 1 0 0 0 0 0 ........... 6) 0 1 1 0 0 0 0 0 ........... 7) 1 1 1 0 0 0 0 0 ............ ...................................... Cantor's diagonal string is 1 1 1 1 1 1 1 1 ....... which is the last number in a sequence of natural numbers. So CDA proves that a sequence of natural numbers has no last member. I knew that. 
October 9th, 2018, 12:06 PM  #190 
Global Moderator Joined: Dec 2006 Posts: 20,375 Thanks: 2010  That's incorrect. There's no such thing as a last natural number. If your particular list doesn't contain all natural numbers, the last one that it does contain cannot possibly contain just '1's and no '0's.


Tags 
binaryexpressed, cardinality, continuum hypothesis, diagonal argument, numbers, real, set 
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