My Math Forum  

Go Back   My Math Forum > College Math Forum > Topology

Topology Topology Math Forum

Thanks Tree3Thanks
  • 1 Post By v8archie
  • 2 Post By greg1313
Closed Thread
LinkBack Thread Tools Display Modes
June 29th, 2016, 07:42 AM   #1
Banned Camp
Joined: Mar 2015
From: New Jersey

Posts: 1,720
Thanks: 126

Natural Numbers and the Continuum

Reverse binary/decimal representation of the natural numbers:
a0x2^0 + a1x2^1 + a2x2^2 + a3x2^3 + ....., an=0,1
a0x10^0 + a1x10^1 + a2x10^2 + a3x10^3 + ....., an=0,1,.....,9
ALL n.
01523 <-> 32510 (reversed <-> standard)

From the definition it follows there is a 1:1 correspondence between the natural numbers and the real numbers (continuum) by adding or removing a "decimal" point.
015 <-> .015 (0x10^0 + 1x10^1 + 5x10^2 <-> 0x10^-1 + 1x10^-2 + 5x10^-3)
1101 <-> .1101
00101.. <-> .00101..
33333..... <-> .33333

Finite and repeat natural numbers correspond to the rational numbers, and non-repeating natural numbers correspond to irrational numbers. Thus there is a 1:1 correspondence between the natural numbers and the continuum.

In decimal notation, corresponding to each point in [0,1) is an integer point on the line. The representation of all the points on the line by natural numbers is obvious.


i) The reals are countable.

ii) Cantor's Diagonal Argument is wrong because any countably infinite binary sequence corresponds to a natural number, by above.
The Argument fails because:
a) The countably infinite sequence of countably infinite binary digits is not square (2^n sequences per n digits, ALL n), or
b) The constructed sequence whose nth digit is the complement of the nth digit of the nth sequence in the list can only be at the end of the list, and so doesn't exist because the list has no end.

[Note by moderator: a natural number is finite (the summation you give doesn't converge unless it contains only a finite number of non-zero terms), so 33333... isn't a natural number. Also, if the "constructed sequence" cannot be in the list used, that confirms that an all-inclusive list is impossible, so Cantor's "not enumerable" conclusion is correct.]

Last edited by skipjack; June 29th, 2016 at 09:10 PM.
zylo is offline  
June 29th, 2016, 10:23 AM   #2
Math Team
Joined: Dec 2013
From: Colombia

Posts: 7,674
Thanks: 2654

Math Focus: Mainly analysis and algebra
Originally Posted by zylo View Post
From the definition it follows there is a 1:1 correspondence between the natural numbers and the real numbers (continuum)
Ye gods, you are stupid.

Just saying "for all $n$" like some magic mantra doesn't make what you say any less incorrect. Instead of repeating your incoherent nonsense over and over again, try learning what that phrase actually means.
Thanks from topsquark

Last edited by skipjack; June 29th, 2016 at 09:12 PM.
v8archie is offline  
June 29th, 2016, 05:57 PM   #3
Global Moderator
greg1313's Avatar
Joined: Oct 2008
From: London, Ontario, Canada - The Forest City

Posts: 7,958
Thanks: 1146

Math Focus: Elementary mathematics and beyond
v8archie, please address other members respectfully, regardless of how impertinent their ideas may be.

As this thread is redundant I am closing it.
Thanks from topsquark and Azzajazz
greg1313 is offline  
Closed Thread

  My Math Forum > College Math Forum > Topology

continuum, natural, numbers

Thread Tools
Display Modes

Similar Threads
Thread Thread Starter Forum Replies Last Post
Natural numbers yo79 Math Events 2 April 7th, 2013 02:11 AM
Natural numbers EulerRules Number Theory 9 March 5th, 2013 09:39 AM
The paradox between prime numbers and natural numbers. Eureka Number Theory 4 November 3rd, 2012 03:51 AM
natural numbers ! rose3 Number Theory 1 January 13th, 2010 08:41 AM
natural numbers from sets....not very natural jinjouk Number Theory 12 June 3rd, 2008 06:11 AM

Copyright © 2019 My Math Forum. All rights reserved.