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July 4th, 2018, 09:48 AM   #2
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Quote:
 Originally Posted by pwi So what do you think?
We're not paid to think....

July 4th, 2018, 10:31 AM   #3
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Quote:
 Originally Posted by pwi If they can't the continuum would be pointless.
Pretty good pun!

 July 4th, 2018, 11:24 AM #4 Senior Member   Joined: Oct 2009 Posts: 439 Thanks: 147 So what is the average of 1 and 0.999999...? Thanks from studiot
July 4th, 2018, 02:24 PM   #5
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Math Focus: Mainly analysis and algebra
Quote:
 Originally Posted by pwi I say that is true there is no point between them because they are points next to each other.
Maths doesn't work on "what I say" or "what you think". It requires proof.

Quote:
 Originally Posted by pwi If there is such a thing as "continuous", points have to be next to each other.
No. You are using finite reasoning for the infinite. Your intuition, like everybody else's, is badly flawed and doesn't know what it's talking about. This is why maths requires proof.

Quote:
 Originally Posted by pwi That would leave rational and irrational points alternating if they could be next to each other.
But the cardinality of the irrationals is greater than that of the rationals. There is no 1-1 mapping between the two sets. So they don't alternate. It's a phrase that has no meaning anyway as there is no real number that is "next to" any other.

Quote:
 Originally Posted by pwi What are the best arguments you have that 0.999... is NOT next to 1? I say they always leave out the infinitesimal that is defined to not be equal to zero.
The real numbers are the limits of convergent sequences of rational numbers. $0.999\ldots$ represents the series $\sum \frac9{10^n}$ and the limit of that sequence is $1$.

The real numbers do not include infinitesimals.

Last edited by skipjack; September 7th, 2018 at 05:32 AM.

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