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> The argument against this is that there are real numbers with an infinite number of digits, which will not have a specific integer associated with them. Or do
by plus 5y ago
> The argument against this is that there are real numbers with an infinite number of digits, which will not have a specific integer associated with them. Or do they?
If it was possible to construct your mapping, then there would be a well-defined sorting of the reals between 0 and 1 based on their integer representation (e.g. we could sort the set {0.05, 0.1, 0.2} => {50, 1, 2} to [0.1, 0.2, 0.05] => [1, 2, 50]).
How would you sort the list {0.5, 1 / sqrt(2), pi - 3}?
- deleted 5y ago[deleted]
- DangitBobby 5y ago> then there would be a well-defined sorting of the reals between 0 and 1 based on their integer representation Why is that?
- alanbernstein 5y agoBecause the integers are ordered?
- DangitBobby 5y agoI took well-defined to mean "if it exists, we know what it is." So really I guess what I want to know is why it matters that we can't actually calculate the mapping.
- plus 5y agoAny finite number of randomly chosen integers can be sorted in increasing order. The quantities invented for the proposed mapping do not share this property, and thus cannot possibly be integers.
- DangitBobby 5y agoAgain, you don't know how to compute the function which creates the mapping, so you don't know the results, so you don't know how to sort the results. I don't follow the logic which claims your inability to compute something is now an intrinsic property to the resulting set. I do understand that sortability is a property intrinsic to integers.
- alanbernstein 5y agoI think the point is that the example with irrational numbers guarantees that this mapping/sort can't be done.