Mathematicians Measure Infinities, Find They’re Equal

Re:Mathematicians Measure Infinities, Find They’re Equal by zelah on Wednesday September 13, 2017 @07:00AM (#15237109)

>I's much easier to consider the infinite strings of digits like "0.765653625367523765..." or "0.5265362556..." or "0.000073468763478..." and also the one with repetitions like "0.0006767000000..." or "0.0072257822222222...". This is essentially a copy of the real number, but in this copy "0.2999999999999..." is different from "0.300000000000000..." This trick makes much easier to prove that the diagonal ´+1 in each one digit is not in the list. Then it's possible to fix the details and use the real numbers instead of the infinite strings of digits.

What am I missing?

"0.765653625367523765..." could be assigned the transfinite natural number beginning "1765653625367523765..."

"0.000073468763478..." could be assigned the transfinite natural number beginning "1000073468763478..."

Transfinite natural numbers must exist otherwise you do not have an infinite set.

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Re:Mathematicians Measure Infinities, Find They’re Equal by zardo on Wednesday September 13, 2017 @12:16PM (#15239720)

>Transfinite natural numbers must exist otherwise you do not have an infinite set

Transfinite

This word does not mean what you think it means. I'm not entirely sure what you think it means, but it's definitely not what it means to everyone else.

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Re:Mathematicians Measure Infinities, Find They’re Equal by relate on Wednesday September 13, 2017 @11:34AM (#15239251)

Suppose I have the set F = { all integers x where x can be written with a finite number of digits }

Are you saying the set F is finite?

Or are you saying that the set F contains transfinite numbers?

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Re:Mathematicians Measure Infinities, Find They’re Equal by pitaj on Wednesday September 13, 2017 @11:36AM (#15239282)

The difference is that `0.7656536...` is smaller than 0.7656537, whereas a similar number with every digit moved left of the decimal point has no bound.

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Re:Mathematicians Measure Infinities, Find They’re Equal by Cyph0n on Wednesday September 13, 2017 @11:32AM (#15239222)

Yes they could, but note that these real numbers are all between 0 and 1. So the "infiniteness" of R is "higher" than that of N.

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