Re: Does the number of nines increase?

On 7/5/2024 6:26 AM, Moebius wrote:

Am 05.07.2024 um 06:40 schrieb Chris M. Thomasson:

On 7/4/2024 1:30 PM, Moebius wrote:

Am 04.07.2024 um 22:23 schrieb Chris M. Thomasson:
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I was just thinking that infinite is infinite,

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No, it isn't.

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For some reason I like to think of the density of infinity. The Natural numbers are not dense at all when compared to the reals...

 Yeah, but this idea might be rather missleading!
 Hint: The natural numbers are not dense at all when compared to the rational numbers either, no?
  But both sets, the set of natural numbers and the set of rational numbers are _countably infinite_, while the set of real numbers is _uncountably infinite_
 

there is an infinite number of natural numbers,

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Right, _countably_ infinitely many.
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there are an infinite amount of [real] numbers between say, .0000001 and .00000001

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_Uncountably_ infinitely many.

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Okay. I get a little confused by that sometimes. Trying to count the reals is not possible because of all those infinite infinities that are embedded in them...

 Yeah, a very good metaphor!
 

However The naturals have no infinities between say, 1 and 2. Make any sense to you?

 Yeah, somehow.
 Still, the rational numbers are countable! (Not enough "infinite infinities embedded in them"!)

Humm... Not sure about that. How many embedded infinite infinities are "needed" _before_ it can be deemed uncountable? Say between 0 and 1. There seems to be an infinite number of rationals that can fill in the "gap", so to speak.
0 + (1/8 + 1/8 + 1/4 + 1/2) = 1
0 + (1/16 + 1/16 + 1/8 + 1/4 + 1/2) = 1
0 + (1/16 + 1/16 + 1/8 + 2/3 + 1/3 - 1/4) = 1
These are all rational, right?