Randomly selecting from an infinite number of choices

We're talking about demons and angels, zanket. Supernatural beings such as this are certainly capable of performing a supertask.

 

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Ummm, no. You chose from a finite pool. There was a 100% certainty that the numbers chosen by your method would be less then a megistron, for example.

 

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No, not just as easily, which proves my point. Took you longer to choose this one.

It also proves that you didn't select from an infinite number of choices.

 

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Good to know about a supertask. Wikipedia rocks!

 

To be random you'd have to have equal odds of choosing any of those in the pool. Doesn't look like that was the case in your example.

 

No, it's enough to know that it takes more time for you to randomly select a number, the more digits it can have. That's all I need to know, to know that any number you choose was not chosen from an infinite pool.

You're being argumentative now.

 

No, I shouldn't be able to tell that.

When you're being argumentative, sure. It's not up to me to prove that your megistron number wasn't chosen at random. It's enough for me to know that any algorithm that randomly chooses from an infinite pool will never return a number. For a pseudo-random number generator on a computer, for example, you'd need an infinite-bit computer, or an emulator of that. There'd be an infinite loop or process somewhere.

 

Yeah zanket, but as DH pointed out, you're counting angels on pinheads now. The question is clearly answered, and you're quibbling over the implementation of the algorithm.

 

Sub-topics were introduced. I'm talking about randomly selecting from an infinite number of integers. No angels or supertasks. When an angel or demon can do a supertask, the problem isn't very interesting; probably all sorts of contradictions can be shown then.

 

Zanket is correct. Truly choosing a random number from an infinite list is not possible. There is no way in reality to represent the infinite choices from which to choose. Trippy's example of pointing to a point on a circle (or any other example) does not work because it involves measuring a point to infinite precision - as in INFINITELY "smaller" than the Planck length.

The only place this is possible is in our heads because we can think in abstract terms such as infinity. We cannot truly comprehend infinity, though, and any number that Trippy comes up with may in fact belong to the set of infinite integers however it also belongs to the SUBSET of those numbers that humans are able to conceive of, which necessarily excludes most integers with an infinite number of digits.

 

Trippy, you only need to be able to represent the entire list IF you are requiring that the number be chosen at random from that entire list. In this case RANDOM means that each choice has an equally likely chance of being chosen. You could say "prove that the number 3 was not chosen randomly from an infinite list", and the fact that you cannot possibly think of, write down, or even fathom in your head ONE of the infinite choices of integers which have an infinite number of digits in them means that the number 3 was in fact chosen from a SUBSET of that infinite list. Yes, 3 belongs to the infinite list of integers set but no, it was not chosen randomly from that set. It was chosen "randomly" from a subset consisting of those numbers which we are capable of comprehending.

Also, how does pointing to a location on a number line representing an infinite number of points not require infinite measuring precision? Not possible as far as I know!

 

You don't need to know the precise size of something to know it's not infinite.
But, I gave you an upper bound earlier, and I'll give it to you again - there are fewer than a megistron numbers in the pool of numbers that are possible for you to select. Yes, there are numbers that are larger than a megistron in that pool, but there are many(!) more numbers less than a megistron that are not.

That's a different question. A random selection from a non-uniform infinite distribution is easy. As stated earlier in the thread, just flip a coin, and count the number of flips it takes until heads comes up.

The problem at hand is taking a random selection from a uniform distribution of the naturals.

And how does that specify a location to infinite precision? Just how large are the error bars on a pointing finger?
And even if you do have an infinitely precise pointer, how does that help you select a random natural number? This is not irrelevant - as I pointed out earlier, I suspect choosing randomly from a continuum is a fundamentally different problem to choosing from a discrete distribution. At least, I can't find a way of getting from one to the other... but if anyone else can find a way, I'd be grateful.

 

Or, try this:

Let's say that you have a keyboard with k keys, that you strike r keys per second while composing a post, which takes t seconds.
There are \(k^{rt}\) possible posts that could result.
We know that k, r, and t are finite, therefore \(k^{rt}\) is finite.


Or more specifically:
For a 100 key keyboard and a keyboard-mashing rate of 100 keys per second, then a post 100 years in the making gives you an upper bound of only \(10^{10^{12}}\) or so for the size of the pool of numbers that you could possibly choose using that method.

In my next post, I'll extend this idea to the more abstract realm of random-number-generating algorithms.