February 17th, 2018, 10:34 PM  #1 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 319 Thanks: 26 Math Focus: Number theory  Limits' and reals' cardinality
Does the limit function relate to a maximum cardinality? Does the set of real numbers, as they are an "absolute" continuum? Can one map the set of real numbers onto a finite surface? By bijection? 
February 18th, 2018, 07:48 AM  #2 
Senior Member Joined: Oct 2009 Posts: 275 Thanks: 92 
Please start by defining: 1) Limit function. Do you just mean $\lim_{x\rightarrow a} f(x)$? 2) Maximum cardinality 3) Absolute continuum 4) Finite surface 
February 19th, 2018, 03:54 PM  #3 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 319 Thanks: 26 Math Focus: Number theory 
1) Yes. 2) Does the limit function approach f(a) infinitesimally, e.g. uncountably in decimal form? Might this difference, the limit function minus f(a) itself, be represented by a set of infinite cardinality, since the limit function is actually arbitrary? 3) The set of real numbers, being ["absolute" is redundant and informal, I guess] a continuum, are represented by what cardinality? 4) Can the set of real numbers have a bijection onto any finite surface? Last edited by Loren; February 19th, 2018 at 03:57 PM. 
February 19th, 2018, 07:48 PM  #4 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,764 Thanks: 1012 Math Focus: Elementary mathematics and beyond 
4) How many points are on a finite surface? What is the cardinality of this set?

February 19th, 2018, 08:01 PM  #5 
Senior Member Joined: Sep 2015 From: USA Posts: 1,790 Thanks: 922  
February 19th, 2018, 08:25 PM  #6  
Senior Member Joined: Aug 2012 Posts: 1,777 Thanks: 482  Quote:
Perhaps you can give a specific example of some function you have in mind. You use the phrase, "this difference, the limit function minus f(a) itself ..." and asked if this is a cardinality. But it's a NUMBER, not a set. [Numbers are ultimately sets, but not in this context!] For example let $f(x) = 0$ if $x \neq 0$, and $f(0) = 47$. In other words when you input anything other than $0$ into the function, you get back $0$. But when you put in $0$, you get $47$. Now in this case the limit of $f$ as $x \to 0$ is $0$. But $f(0) = 47$. So the difference of the limit of the function at that point, and the value of the function at that point, is $47$. It's a number. Does that make sense? Remember, a function doesn't even need to have a value at a given point in order to have a limit there. Quote:
These are very deep waters. Nobody has any idea what is the cardinality of the continuum. We know it's $2^{\aleph_0}$, but nobody has any idea what cardinal that is. You asked a great question, that's a fact. Quote:
Is anything I wrote helpful? I'm not sure what is the intent of your questions. Last edited by Maschke; February 19th, 2018 at 09:14 PM.  
February 20th, 2018, 02:31 AM  #8  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,232 Thanks: 2411 Math Focus: Mainly analysis and algebra  Quote:
The "cardinality of a limit" is simply the cardinality of the set from which you select your epsilon under this definition. But it's not a property that is intrinsic to a limit as far as I'm aware. Last edited by skipjack; February 20th, 2018 at 12:18 PM.  
February 20th, 2018, 09:55 AM  #9 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,296 Thanks: 93 
The real numbers are given by the unique limiting sequence (not the limit of the sequence), as n approaches "infinity," of n place decimals, which can be counted for all n. Cardinality of the reals: countable. EDIT: Limits of the reals are a subset of the reals. .3333...3m, any fixed m, to n places of 3's, has the limit as n approaches infinity, 1/3 Last edited by zylo; February 20th, 2018 at 10:15 AM. 
February 20th, 2018, 10:28 AM  #10 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,232 Thanks: 2411 Math Focus: Mainly analysis and algebra 
You are talking the same old unmitigated nonsense again.


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