Anonymous
10/4/2025, 1:09:12 PM
No.16805839
>>16805794
>I think it only proves that there is no surjective function from N to R that you can write down.
>the real numbers are, in some sense, countable since there will only be countably many finite expressions.
You got that the wrong way around. The real numbers *that you can write down* are indeed countable. The set of all real numbers is not. The diagonal argument proves this, as it makes no assumption about being able to write down a real number; it only assumes the existence of a surjection.
(Technically, the version you depict in OP shows the uncountability of the sequences of bits, not the real numbers. Applying that to the real numbers is an extra step. But not a difficult one, and the argument can indeed be adapted that way.)
>There are also places where it disagrees with finite sets.
Oh? Can you give an example?
>You can have a sets which have the same size as proper subsets according to this definition.
Not finite ones.
>Is the set of even numbers really the same size as the set of all naturals?
You will note that those sets are not in fact finite.
>I think it only proves that there is no surjective function from N to R that you can write down.
>the real numbers are, in some sense, countable since there will only be countably many finite expressions.
You got that the wrong way around. The real numbers *that you can write down* are indeed countable. The set of all real numbers is not. The diagonal argument proves this, as it makes no assumption about being able to write down a real number; it only assumes the existence of a surjection.
(Technically, the version you depict in OP shows the uncountability of the sequences of bits, not the real numbers. Applying that to the real numbers is an extra step. But not a difficult one, and the argument can indeed be adapted that way.)
>There are also places where it disagrees with finite sets.
Oh? Can you give an example?
>You can have a sets which have the same size as proper subsets according to this definition.
Not finite ones.
>Is the set of even numbers really the same size as the set of all naturals?
You will note that those sets are not in fact finite.