February 20th, 2018, 10:41 AM  #11 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,431 Thanks: 105 
"You are talking the same old unmitigated nonsense again."* is not a refutation of Cantor's Diagonal Argument. * v8archie, last post Last edited by zylo; February 20th, 2018 at 10:46 AM. 
February 20th, 2018, 11:02 AM  #12 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,355 Thanks: 2469 Math Focus: Mainly analysis and algebra 
No, but then I'm not trying to refute it.

February 20th, 2018, 11:57 AM  #13 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,431 Thanks: 105  Would someone else be justified in using your argument to refute it? What amazes me is that a nonsensical statement passes unnoticed, and buries a serious contribution, but when I attempt to respond to a serious question to my thread in "Real Analysis," Real Numbers and Limits, the thread is closed. 
February 20th, 2018, 01:15 PM  #14  
Senior Member Joined: May 2015 From: Arlington, VA Posts: 374 Thanks: 26 Math Focus: Number theory  Quote:
Maschke, thank you for taking the time to answer my questions directly and thoroughly. You would do well teaching undergraduate math majors. You essentially answered my questions, written or surmised, and brought forth new lines of thought. I can largely understand your interpretations. You got it. Keep on topic. You all see most of my intent although without formal descriptions from me, and contribute what is condensed by Maschke in most helpful form. Btw, what significance does "47" have?  
February 20th, 2018, 01:25 PM  #15 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 374 Thanks: 26 Math Focus: Number theory 
Is "interleaving" anything like the diagonal argument?

February 20th, 2018, 01:31 PM  #16  
Senior Member Joined: Aug 2012 Posts: 1,999 Thanks: 572  Quote:
Interleaving works like this. Say you have a point $(x,y)$ in the unit square, and say that the decimal expression of $x$ is $.x_1 x_2 x_3 \dots$ and the decimal expression of $y$ is $.y_1 y_2 y_3 \dots$. Then you can map the pair $(x,y)$ to the real number $.x_1 y_1 x_2 y_2 x_3 y_3 \dots$. Likewise given the decimal expression for some real number in the unit interval, you can reverse the process to get a pair of reals representing the coordinates of some point in the unit square. You can do the trick for any $n$dimensional Euclidean space. Cantor was shocked to discover this. He thought each dimension up gave you a higher cardinality. Turns out to be false. The line, the plane, 3space, 4space, etc. all have exactly the same cardinality. Last edited by Maschke; February 20th, 2018 at 01:47 PM.  
February 20th, 2018, 02:11 PM  #17 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 374 Thanks: 26 Math Focus: Number theory 
Does the uncountability over the set of epsilons correspond to a particular cardinality? Can the set of cardinal numbers (ordinal numbers?) itself belong to a set with its own cardinality (ordinality?), or is this "circular" reasoning? Do we know somethings about "2 to the Alef" by being able to define a "greater" cardinal set? Where can I find examples of cardinal number sets (like that of the set of all curves in space)? 
February 20th, 2018, 05:11 PM  #18 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,852 Thanks: 1077 Math Focus: Elementary mathematics and beyond  So we're back to this are we? There's no point in discussing it, as you have consistently shown.
Last edited by greg1313; February 21st, 2018 at 11:34 AM. 
February 20th, 2018, 08:42 PM  #19  
Senior Member Joined: Oct 2009 Posts: 439 Thanks: 147  Quote:
Quote:
 
February 21st, 2018, 07:52 AM  #20  
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,431 Thanks: 105  Quote:
Why can't I express my opinion like everyone else in this thread? What are you afraid of? Part of what I have said I said about 1000 posts ago. Has nothing in this thread ever been said before? Quote:
Real Numbers and Limits I wrote "REAL NUMBERs are defined uniquely by "infinite" (unending) sequences of natural numbers. The sequence IS the real number. LIMIT is a defined property of real numbers." which is a simple, clear, logical, transparent foundation of real analysis, which is why I posted it in the Real Analysis forum. This thread was closed when I tried to post an answer to a question.  

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