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Comment by hn_throwaway_99 | Hacker News Reader

I'm not a mathematician, nor a physicist, but I don't understand this statement: similar to, for example, the zero probability that a random real number in some interval is a rational number. Is it just a consequence of rational numbers being countable and real numbers not being countable? Seems like an odd way to define probability in this instance.

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It is a consequence of countability, but I like to think about it this way: a real number is, roughly, an infinite sequence of "random" digits. What is the probability that your infinite sequence of random digits will randomly settle into an infinite repeating pattern? That's obviously, zero, right? No way that a random process will produce an infinitely repeating pattern.

But just because it's "probabilistically impossible" to pick a rational number doesn't mean that they don't exist.

Thanks, this was the explanation that made sense to me.

Countability does have to do with it. Measure theory [0] allows us to extend the classical notions of discrete probability to continuous domains.

[0]

https://en.wikipedia.org/wiki/Measure_(mathematics)

https://en.wikipedia.org/wiki/Probability_measure

It is a consequence of the rationals being countable, in that all countable sets necessarily have this property.

The intuition (well, one way of looking at it) is that if you have a line of length 40, and a subset of that line which has total length 2, then a randomly-chosen point on the line has a 5% chance (2 in 40) of being part of that subset. It's not important that the subset be connected; the intervals (3,4) and (28,29) have length 1 each, length 2 in total. We want to think in terms of intervals because they're very easy to measure: the interval (low,high) is (high-low) amount of stuff.

It turns out that there are so few rational numbers that you can define a series of intervals with the following properties:

- every rational number is contained within one or more of the intervals

- the (infinite) sum of the lengths of each interval is arbitrarily small

So for any value, you can capture the entire set of rationals in a set of intervals whose total length is less than that value. This means the "amount of stuff" in the set of rational numbers cannot be bounded away from 0; removing the rationals from a line of probability-stuff leaves no less probability-stuff than was there before.

I think the point being made is that whilst the probability of picking one at random is "zero" the rational numbers exists.

It comes from how we define infinity/the number line. Any number divided by infinity is 0, that's easy. But between any two points there is an infinite number of rational numbers and between any two rational numbers you can derive an infinite number of irrational numbers. I know this isn't kosher, but it is pretty much infinity/(infinity ^ infinity). That leads us back to the probability of picking a rational number as 1/infinity, because for every rational number there are infinity irrational numbers that we also could have chosen.

David Foster Wallace had a really good book called Everything and More where he explored the history of infinity. Might be of interest to you.

That's a really bad argument, since between any two irrational numbers, there's an infinite number of rational numbers.

Ed: For an explicit construction of such numbers --

Suppose we have a<b, two irrational numbers.

Let b-a = d, the difference between them.

Then there exists N, an integer, such that 10^(-N) < d.

Let b1 be b truncated at the (N+1)th digit past the decimal point.

Then b1 is rational (finitely many digits) and b-b1 < d (and b1 < b), so b1 is in (a,b).

You can then add back one digit of b at a time, to get b2, b3, etc which form an infinite sequence of rational numbers converging to b from below. (We get infinite unique numbers from the fact b is irrational. Not all the b_n need be distinct.)

> Any number divided by infinity is 0, that's easy

That's a dirty hack for the sake of argument, but it isn't really true. The convergence limit of division with infinitely increasing divisors is zero.

Corollary division by zero isn't infinity, it's simply not defined.

I wish there was a third alternative to upvote/downvote -- "sounds like a good chap acting in good faith, but knows less than nothing of this subject".

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