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Comment by phkahler | Hacker News Reader

You can map all the real numbers to the interval 0 <= x < 1. To map one of those reals to an integer, simply write it's trailing digits in reverse order on the left side of the decimal point. You may then drop any leading zeros.

0.0 => 0

0.1 => 1

0.2 => 2

...

0.14159 => 95141

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? Can there be an "infinitely large integer?" or are there just infinitely many of them? I think this question gets to the core of the problem.

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> Can there be an "infinitely large integer?"

Check out p-adic numbers (https://math.uchicago.edu/~tghyde/Hyde%20--%20Introduction%2...), which are related to the two's-complement representation of signed integers. For example, in the 10-adic numbers, ...9999 + 1 = 0.

> 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}?

> then there would be a well-defined sorting of the reals between 0 and 1 based on their integer representation

Why is that?

Because the integers are ordered?

I 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.

Any 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.

Again, 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.

I think the point is that the example with irrational numbers guarantees that this mapping/sort can't be done.

> You can map all the real numbers to the interval 0 <= x < 1.

Even easier use a well-established function such as arctan to map the reals bijectively to the interval -π/2 < x < π/2, and then scale and shift the interval to 0 < x < 1.

This allows you to map the real range (0, 1) to (-infinity, infinity). It does not allow you to map the integers to the reals.

> Can there be an "infinitely large integer?"

What could that even mean? Aren't integers all defined as Succ^n(x) for a finite n and a base x?

> simply write it's trailing digits in reverse order

so how do you map say pi/4? Which digit do you start with?

That would be:

...61893587

We start writing the digits right to left an pi/4 is about 0.78539816...

So you found a flaw in Cantor's diagonal argument?

https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument

They are just saying “but what if we replace the set of integers with a larger set”. I think they are also defining this larger set in a base 10 specific way.

I think they basically mean the 10-adic numbers, which, unlike the p-adics for p a prime number, uh, I forget exactly what goes wrong, but I suspect it has like, zero divisors or something like that.

"Can there be an infinitely large integer"

No there can't there's always a larger integer. Anyone who claims otherwise is just making stuff up. It's just mind blowing to me that anyone think otherwise, it's like asking what the final digit of pi is, you'd get laughed out of any college class for insisting there might be one, rightly so.

But if you assign them randomly, then it should work

1->5.85916

2->8.7599

...

Or even randomize the order for the integers

5->7.52256951

77->848.455

...

You can't, this is Cantor's diagonal counterexample.

What does ⅓ map to?

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