Are real numbers really real?

Re:Are real numbers really real? by cyphar on Tuesday June 12, 2018 @02:59PM (#17296718)

Your proposed enumeration of decimals (counter-intuitively) would miss an uncountably infinite number of values. This enumeration is not possible in any fashion, which you can prove fairly easily with Cantor's Diagonal Argument[1].

The punch-line is that there are more real numbers between 0 and 1 than there are whole numbers greater than 0. The proof basically shows that even if you imagine that you had a countably-infinitely-long list of all the real numbers between 0 and 1, you can construct a number that must be missing from the list -- which is a contradiction, and thus no such list can exist (implying that there are more numbers between 0 and 1 than numbers greater than 0).

For instance, your enumeration is missing every irrational (or even recurring decimal) number. That set of numbers is uncountably infinite.

[1]: https://en.wikipedia.org/wiki/Cantor's_diagonal_argument

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Re:Are real numbers really real? by windows_tips on Tuesday June 12, 2018 @04:18PM (#17297579)

>The proof basically shows that even if you imagine that you had a countably-infinitely-long list of all the real numbers between 0 and 1, you can construct a number that must be missing from the list -- which is a contradiction, and thus no such list can exist (implying that there are more numbers between 0 and 1 than numbers greater than 0).

So, there is no list of all the real numbers between 0 and 1, but you are claiming there are uncountably infinite of them?

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Re:Are real numbers really real? by leephillips on Tuesday June 12, 2018 @04:38PM (#17297743)

That's the definition of uncountably infinite: can not be put in 1-to-1 correspondence with the integers (can not be listed).

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Re:Are real numbers really real? by cyphar on Wednesday June 13, 2018 @02:07AM (#17300643)

Yes, because if there were such a list then you could assign a 1-to-1 mapping (by list index) of each number between 0 and 1 to a whole number which would prove they have the same size. If it is impossible to have a complete list of all real numbers between 0 and 1 then there cannot be such a mapping, and thus you'll "run out" of integers before you've counted all of the reals (more correctly you're showing that the size must be larger because if it was smaller or of equal size it could fit in such a list).

Numberphile did a pretty good introductory video to this if you want to learn more: https://www.youtube.com/watch?v=elvOZm0d4H0.

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Re:Are real numbers really real? by filmor on Tuesday June 12, 2018 @04:39PM (#17297752)

That is pretty much the definition of uncountability :)

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