February 17th, 2018, 10:34 PM  #1 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 374 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: 439 Thanks: 147 
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: 374 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: 7,851 Thanks: 1075 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: 2,089 Thanks: 1085  
February 19th, 2018, 08:25 PM  #6  
Senior Member Joined: Aug 2012 Posts: 1,998 Thanks: 569  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: 7,355 Thanks: 2469 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: 1,431 Thanks: 105 
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: 7,355 Thanks: 2469 Math Focus: Mainly analysis and algebra 
You are talking the same old unmitigated nonsense again.


Tags 
cardinality, limits, reals 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
bases of the reals^2  Jaket1  Linear Algebra  5  October 18th, 2017 07:15 PM 
Noncomputable Reals  AplanisTophet  Number Theory  30  June 21st, 2017 09:25 AM 
Reals  Lalitha183  Abstract Algebra  3  June 2nd, 2017 10:02 PM 
Cardinality of integers equals cardinality of reals  BenFRayfield  Number Theory  0  February 15th, 2014 02:55 PM 
Independence of Reals  mathbalarka  Number Theory  1  May 9th, 2013 05:51 AM 