I considered deleting the post, but this seems more cowardly than just admitting I was wrong. But TIL something!
this post was submitted on 06 Jan 2024
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Similar problem: which set is bigger, the set of all real numbers, or the set of all real numbers between 0 and 1?
Not quite because it's easily shown that the set of all real numbers contains the set of all real numbers between 0-1, but the set of all real numbers from 0-1 does not contain the set of all real numbers. It's like taking a piece of an infinite pie: the slice may be infinite as well, but it's a "smaller" infinite than the whole pie.
This is more like two infinite hoses, but one has a higher pressure. Ones flowing faster than the other, but they're both flowing infinitely.
actually you can for each real number you can exhaustively map a uninque number from the interval (0,1) onto it. (there are many such examples, you can find one way by playing around with the function tanx)
this means these two sets are of the same size by the mathematical definition of cardinality :)
You mean integers and real numbers between 0 and 1.
All real numbers would start at 0, 0.1, 0.001, 0.0001.... (a 1:1 match with the set between 0 and 1) all the way to 1, 1.1, 1.01.... Etc.
no, there aren't enough integers to map onto the interval (0,1).
probably the most famous proof for this is Cantor's diagonalisation argument. though as it usually shows how the cardinality of the naturals is small than this interval, you'll also need to prove that the cardinality of the integers is the same as that of the naturals too (which is usually seen when you go about constructing the set of integers to begin with)
No, ey mean real numbers and real numbers. Any interval of real numbers will have enough numbers to be equivalent to any other (infinite ones included)