My Math Forum  

Go Back   My Math Forum > College Math Forum > Real Analysis

Real Analysis Real Analysis Math Forum


Thanks Tree13Thanks
Reply
 
LinkBack Thread Tools Display Modes
October 17th, 2016, 04:00 AM   #81
Global Moderator
 
Joined: Dec 2006

Posts: 17,919
Thanks: 1383

Quote:
Originally Posted by zylo View Post
As n approaches infinity, the list of n-place decimals approaches the list of real numbers . . .
There's no justification for that wording, as you haven't yet shown that there is such a thing as the list of real numbers. You stated "There is no "list at infinity".

Quote:
Originally Posted by zylo View Post
.... and list count approaches countable infinity.
As there is no "list at infinity", that doesn't imply that the reals are countable.

Suppose instead that you make a list of n identical sequences, all of length 10, each sequence being 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. You can choose a single appropriate digit from these sequences so that your chosen digits are 7, 0, 7, 1, 0, 6, 7, . . . continuing until n digits have been listed. By continuing in this fashion (with increasing values of n), your choices can form the first n places of the decimal expansion of the square root of 1/2, or any other real in [0,1). You need a countable number of steps to do this, so by your logic you could claim that the reals in [0,1) are countable. That would lack proper justification, as being able to take an arbitrarily large, but finite, number of steps towards some goal doesn't imply that the goal is itself countable (in any sense). Even if you took an infinite sequence of steps towards the goal, that still wouldn't prove that the goal is countable.
Thanks from topsquark and Benit13
skipjack is offline  
 
October 17th, 2016, 06:14 AM   #82
Senior Member
 
Joined: Mar 2015
From: New Jersey

Posts: 1,134
Thanks: 88

Quote:
Originally Posted by zylo View Post
List the n-place decimals in [0,1). 4-place shown
.0000
.0001
.0002
.0003
.0004
.0005
.0006
.0007
.0008
.0009
.0010
.0011
.0012
.....
.....
.4999
.5000
.5001
.....
.....
.9999

The count stares you in the face 1,2,3,......,9999
How many numbers in the list between 0 and 1? Hint 9999.
How many numbers in a list of 8 place decimals between 0 and 1? Hint: 99999999
Can you figure out how many numbers in a list of n-place decimals between 0 and 1?

If you've gotten this far, what happens as n $\displaystyle \rightarrow \infty$?
1) each number in the list remains unique.
2) each number in the list approaches a real number.
3) The number of elements in the list becomes 9999999........... without end, otherwise known as countably infinite.

If you didn't get it, put it aside and try again in a few days (if you can still find this post).
That's what I said back on pg 4, now buried, by misrepresentations, delierately ignoring what I said, or irrelevancies based on not understanding what I said.

There is no list of real numbers at infinity (a number), but you can talk about an infinite (unending) list. The list of n-place rational numbers gets closer and closer to the list of real numbers and remains unique as n gets larger and larger. That's the fundamental concept of real analysis.
If a property is true for all (any) n, in this case uniqueness and countability, it is true for countably infinite n.

As for the second part of previous post, I find the example irrelevant and the conclusion unintelligible.

Last edited by zylo; October 17th, 2016 at 06:23 AM.
zylo is offline  
October 17th, 2016, 06:48 AM   #83
Senior Member
 
Joined: Apr 2014
From: Glasgow

Posts: 2,049
Thanks: 680

Math Focus: Physics, mathematical modelling, numerical and computational solutions
Quote:
Originally Posted by zylo View Post
That's what I said back on pg 4, now buried, by misrepresentations, delierately ignoring what I said, or irrelevancies based on not understanding what I said.
No... we are not ignoring what you said, it's just that your argument is flawed and the only responses you are giving to the objections are just reassertions of other flawed things you have said earlier.

Quote:
As for the second part of previous post, I find the example irrelevant and the conclusion unintelligible.
Well, maybe that's where you need to go and study!

Also... your scheme to list all of the real numbers doesn't contain all of the real numbers (e.g. 1/3 is not in your list). It can only contain that number if n is allowed to be infinity (not just approach it), but you already objected to that.

Last edited by Benit13; October 17th, 2016 at 06:59 AM.
Benit13 is offline  
October 17th, 2016, 07:57 AM   #84
Global Moderator
 
Joined: Dec 2006

Posts: 17,919
Thanks: 1383

Quote:
Originally Posted by zylo View Post
There is no list of real numbers at infinity (a number), but you can talk about an infinite (unending) list.
You referred to the specific unending list of all the real numbers in [0,1), and did so before showing that such a list exists.

Quote:
Originally Posted by zylo View Post
If a property is true for all (any) n, in this case uniqueness and countability, it is true for countably infinite n.
That doesn't imply the countability of the irrationals, as all your countable lists contain only decimals corresponding to rational values. You can use a selection process to find listed rationals that approach any irrational real number in [0,1), but that doesn't show that those reals are countable, because each finite selection of rationals gets used for such an approach to infinitely many reals (that you haven't shown are countable).
skipjack is offline  
October 17th, 2016, 08:45 AM   #85
Math Team
 
Joined: Dec 2013
From: Colombia

Posts: 6,938
Thanks: 2265

Math Focus: Mainly analysis and algebra
Quote:
Originally Posted by zylo View Post
The list of n-place rational numbers gets closer and closer to the list of real numbers and remains unique as n gets larger and larger. That's the fundamental concept of real analysis.
You are not doing real analysis here, you are trying to do set theory. Real analysis doesn't care how many reals there are. Neither are lists any part of real analysis except when interpreted as sequences.

Moreover, when $n$ gets larger and larger, there is no suggestion in real analysis that this has anything to do with "infinity". It just means that $n$ can become arbitrarily large.

Related to your discussion, this just means that your $n$-place decimals can become arbitrarily long, but by virtue of having a length they remain finite.

Quote:
Originally Posted by zylo View Post
If a property is true for all (any) n, in this case uniqueness and countability, it is true for countably infinite n.
This is not generally true. The $n$-place decimals are all of finite length and are all rational numbers. Neither of these properties hold for infinite decimals. Your claims (uniqueness and countability) are no more than unproven (and untrue) assertions. Moreover, there is no mechanism by which you can increase $n$ sufficiently that it becomes, or even "approaches" an infinite value.

The concept of limit in real analysis applies to the values of the elements of a sequence. In particular, the sequence converges to the value $L$ only if for any (arbitrarily small) value $\epsilon$ we can always find an $N$ such that the value of every element of the sequence after the $N$th is within $\epsilon$ of $L$. At no point does any concept of "infinity" get involved in this definition. You cannot use this to to tell you anything about set theoretic infinities.
Thanks from topsquark

Last edited by v8archie; October 17th, 2016 at 08:50 AM.
v8archie is offline  
October 19th, 2016, 02:49 PM   #86
Senior Member
 
Joined: Mar 2015
From: New Jersey

Posts: 1,134
Thanks: 88

My posts, buried in this thread, are summarized and generalized, simply and clearly, in:

Decimal Representation and Expansion
zylo is offline  
October 19th, 2016, 03:09 PM   #87
Math Team
 
Joined: Dec 2013
From: Colombia

Posts: 6,938
Thanks: 2265

Math Focus: Mainly analysis and algebra
So the sum total of your posts here is that you believe, on no evidence whatsoever, that whatever is true for finite cases must also be true for infinite cases.

Moreover, you prefer to ignore the evidence presented to the contrary, instead repeating your bogus claim as if nobody had commented at all.

It doesn't matter how many times you repeat nonsense, it remains nonsense.
v8archie is offline  
October 19th, 2016, 08:55 PM   #88
Global Moderator
 
Joined: Dec 2006

Posts: 17,919
Thanks: 1383

Quote:
Originally Posted by zylo View Post
My posts are summarized and generalized in: ...
The thread you linked to doesn't give any mathematical argument at all; it does little more than define a term that you haven't used in any of your posts here.

In particular, you don't explain what you mean by "limit" here, and you cannot claim to have proved anything here if you don't explain what you mean by that.
skipjack is offline  
Reply

  My Math Forum > College Math Forum > Real Analysis

Tags
decimal, representation, unique



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Matrix representation Robert Lownds Linear Algebra 2 April 15th, 2013 01:13 AM
representation of permutation bvh Advanced Statistics 1 February 28th, 2013 07:47 AM
Basis Representation guynamedluis Number Theory 2 January 29th, 2012 10:20 PM
Conjecture: Decimal Representation of root John Creighton Number Theory 2 March 14th, 2011 10:03 AM
Decimal To Fraction To Decimal demipaul Linear Algebra 2 November 19th, 2009 05:42 AM





Copyright © 2017 My Math Forum. All rights reserved.