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>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 n
by windows_tips 8y ago
>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?
- leephillips 8y agoThat's the definition of uncountably infinite: can not be put in 1-to-1 correspondence with the integers (can not be listed).
- filmor 8y agoThat is pretty much the definition of uncountability :)
- deleted 8y ago[deleted]
- cyphar 8y agoYes, 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 https://www.youtube.com/watch?v=elvOZm0d4H0.
- windows_tips 8y agoSo, you're saying there are numbers that we can't write down because we don't have enough integers to specify them?