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I did not know that this was an unsolved problem. It is quite paradoxical. You can map the subset of rational numbers in the range [0, 1) to the set of natural
by deckar01 9y ago
I did not know that this was an unsolved problem. It is quite paradoxical. You can map the subset of rational numbers in the range [0, 1) to the set of natural numbers, so there are an infinite number of disjoint subsets of the rational numbers that map to the natural numbers. Intuitively a 2nd dimension of infinity should be bigger, but the count of a set is defined to be one dimensional regardless of the dimensionality of the members.
Edit: real -> rational
- threatofrain 9y agoAnother thing which has bothered me in the past was that a set of 2-tuple integers could be mapped by a set of 1-tuple integers, seemingly without any information loss.
- semi-extrinsic 9y agoSee Cantor's diagonalization argument for exactly how to do this mapping.
- rhart92 9y agoI am not sure if you have heard of them, but space-filling curves are a way to demonstrate this. This is a great video on the Hilbert curve https://www.youtube.com/watch?v=3s7h2MHQtxc https://www.youtube.com/watch?v=3s7h2MHQtxc
- kgwgk 9y agoThe "seemingly" suggests that you remain unconvinced. If you define for example some kind of distance between an arbitrary (x,y) and (0,0), for any (x,y) you can easily count how many points are closer to the origin. There may be ties but it's easy to define a rule to break them. This means you can order all the pairs of integers: (0,0), (1,0), (0,1), (-1,0), (0, -1), (1,1), (1,-1), (-1,-1), (-1,1), (2,0),....
- deleted 9y ago[deleted]
- jakef 9y agoThis picture is for natural numbers but it shows you one way to do the mapping: https://plus.maths.org/issue47/features/macgregor/diagram3.gif https://plus.maths.org/issue47/features/macgregor/diagram3.g...
- protonfish 9y agoHow? Which real number maps to the natural number 42?
- deleted 9y ago[deleted]
- georgeam 9y agoActually if you meant to say that the real numbers in [0,1) map bijectively to the set of natural numbers, that is incorrect. See https://en.wikipedia.org/wiki/Unit_interval https://en.wikipedia.org/wiki/Unit_interval under the section "Cardinality".
- agf 9y agoNo, this is incorrect. You can map the set of rational numbers to the set of natural numbers. The set of reals in that range is shown to be uncountably infinite by Cantor diagonalization. https://www.reddit.com/r/explainlikeimfive/comments/1r5aya/eli5_how_there_are_more_numbers_between_0_and_1/ https://www.reddit.com/r/explainlikeimfive/comments/1r5aya/e...
- deckar01 9y agoAh, I was thinking of the rationals. Thanks for the ELI5 link.
- breck 9y agoIt also seems obvious to me. There is no "infinity", there are just infinite loops, and they are all equivalent.