My Math Forum Limits' and reals' cardinality

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 February 21st, 2018, 08:46 AM #21 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,757 Thanks: 1008 Math Focus: Elementary mathematics and beyond Alright then - I'll reopen the thread but be advised that it will be strictly moderated.
February 21st, 2018, 09:26 AM   #22
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Quote:
 Originally Posted by greg1313 Alright then - I'll reopen the thread but be advised that it will be strictly moderated.
Greg1313, What puzzles me is why you let FLT and Collatz threads go on for months but react so strongly to this one particular poster. If the so-called anti-crank standards were more evenly enforced, they would not seem so unfair and directed at one single individual.

 February 21st, 2018, 10:03 AM #23 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,757 Thanks: 1008 Math Focus: Elementary mathematics and beyond The FLT and Collatz threads are following the rules. The guidelines I've recently posted are being followed so if someone thinks they have an elementary proof of FLT/Collatz there's really not much I can do. zylo remains congenial but is given to ignoring input from other members and making obscure references to other posts. As I said the forums will be strictly moderated when it comes to things of this sort, from here on in. I will give some credibility to your concern - I was somewhat heavy-handed with zylo, but only because I think he could do so much better. I've been impressed by several posts of his. Thank you for your concern, greg1313 Thanks from Maschke and topsquark
 February 22nd, 2018, 06:01 AM #24 Senior Member   Joined: Mar 2015 From: New Jersey Posts: 1,282 Thanks: 93 1) What is a real number? a) Cut of the rational numbers b) Point on a line c) Infinite decimal 2) What is limit of a function? $\displaystyle \lim_{x\rightarrow a}$f(x) =A if, given any $\displaystyle \varepsilon$, $\displaystyle \delta$ exists st |f(x)-A| < $\displaystyle \varepsilon$ when |x-a|< $\displaystyle \delta$ n-place decimals in [0,1) can be arranged in numerical order, which makes them countable for all n (cardinality of real numbers), and therefore countable in any sub-interval (cardinality of an interval}. Bijective map of line to a finite surface : Divide the surface into n-strips. Map [0,1) to CL of first strip, [1,0) to CL of second strip, ......, [n-1,n) to center line of nth strip. Let n $\displaystyle \rightarrow$ approach infinity (induction). Bijective map of line to finite volume: Divide the volume into n^2 square rods. Map [0,1) to CL of first rod, etc. Let n $\displaystyle \rightarrow$ infinity. Last edited by zylo; February 22nd, 2018 at 06:12 AM. Reason: change intervl to interval and add space
 February 22nd, 2018, 08:06 AM #25 Global Moderator   Joined: Dec 2006 Posts: 18,691 Thanks: 1523 Please explain what you mean by "n-place decimal". Is there some other type of decimal that isn't an n-place decimal?
February 22nd, 2018, 12:23 PM   #26
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Quote:
 Originally Posted by zylo n-place decimals in [0,1) can be arranged in numerical order, which makes them countable for all n (cardinality of real numbers), and therefore countable in any sub-interval (cardinality of an interval}.
If, by "n-place decimals" you mean those decimal that have n decimal places (with only '0' after) then yes, for every n they are countable. However, the collection of all such "n-place decimals", for all n, is precisely the set of all rational numbers, not the real numbers. $\pi$, for example, is not in that set because there is no n for which $\pi$ is a "n-place decimal".

February 22nd, 2018, 12:54 PM   #27
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For a given maximum value of n, the set of decimals with up to n decimal places (other than trailing zeros) is finite.

Quote:
 Originally Posted by Country Boy . . . is precisely the set of all rational numbers
No, it's a proper subset of the rationals.

February 22nd, 2018, 04:23 PM   #28
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Quote:
 Originally Posted by zylo 1) What is a real number? a) Cut of the rational numbers b) Point on a line c) Infinite decimal
You missed "limit of a sequence (or sequences) of rational numbers", which I think is the best definition.
Quote:
 Originally Posted by zylo countable for all n (cardinality of real numbers)
Here we go again. Same old nonsense, no attempt to understand why it's nonsense. I'm not going to bother explaining again. Neither do I care what nonsense you are basing on it. Suffice to say that none of it is mathematics.

February 24th, 2018, 08:18 AM   #29
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Quote:
Originally Posted by skipjack
For a given maximum value of n, the set of decimals with up to n decimal places (other than trailing zeros) is finite.

Quote:
 Originally Posted by Country Boy . . . is precisely the set of all rational numbers
No, it's a proper subset of the rationals.
Ouch! Yes, you are right of course. Thanks for the correction. But they certainly are not all real numbers!

Last edited by skipjack; February 24th, 2018 at 10:28 AM.

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