Oh, *that* formula.... Well, infinity defies apprehension, and it is about as silly to try to as was my initial answer to the question.
Oh, it was stated by the original poster that every member of the group rolls a dice an infinite number of times - didn't you notice it? Of course, it would be fairly hard to perform in real world. All the story is just a pleasant way of saying that, in the probabilistic sense, every member of the group gets a random real number from <0, 1> with continuous uniform distribution on this interval.
Well, I found that formula inconsistent with the claim that someone from the group actually generated the number 1/3 by rolling the dice forever, due to sheer improbability, and so had to assume something more practical (or ridiculous as the case may be), it's also an illogical supposition, as the number generated is never done, so there is no point at which you can stop and say ok, this is the generated number, or else it's no longer infinite... it just keeps going and could be thrown off on any next throw. The only way the story sort of works is to interpret it with a different, much more boring formula, like the fractional assignment I suggested earlier.
If a person rolls the first digit in one unit of time, then the next in half a unit of time, and the next in a fourth of a unit of time, with each digit taking half as long to generate as the previous, it would take only two units of time to generate. This means that it will take only two units of time to generate an infinite number of digits.
Read the original post once again. It is not stated that somebody generated 1/3; on the contrary, there is a question how big a group we might need to have some chance of generating it. As I already said in my first post here, no finite number of people is enough - moreover, infinitely many people also don't have to be sufficient. Therefore, your intuition is right about "sheer improbability".
The second half of your post is more problematic, as you express your disbelief in one of possible constructions of real numbers, essentially denying their existence... No reason to worry, infinity is quite a complicated business. I'm afraid that I'm not the best person to give you advanced maths tutoring, particularly not in my mother tongue.
>Read the original post once again. It is not stated that somebody generated 1/3
Pasted from original post:
"Each individual in a certain group of people rolls a fair ten sided dice an infinite number of times in order to generate a real number between 0 and 1. Using this method one individual generates the number 1/3"
What about that second sentence?
As for the problems in the second part of my post... I don't see that I'm denying the existence of numbers, only with the assertion that you can come up with an actual value (1/3 for instance) when the digits keep coming randomly, because you can't stop to evaluate the number when calculating an "infinite" number of rolls. An infinity of randomness can not be precisely evaluated. The possibility for deviation from any specific target number can never be eliminated, indeed it must be assumed.
>If a person rolls the first digit in one unit of time, then the next in half a unit of time,
>and the next in a fourth of a unit of time, with each digit taking half as long to generate
>as the previous, it would take only two units of time to generate. This means that it will
>take only two units of time to generate an infinite number of digits.
Wouldn't Zeno's paradox prevent that?
This hurts my head 🤕🤕🤕
Zeno's paradox isn't a true paradox as you can have an infinite sum that adds to a finite number provided that each number is smaller than the previous number.
>Zeno's paradox isn't a true paradox as you can have an infinite sum
>that adds to a finite number provided that each number is smaller
>than the previous number.
But it's about duration... even the fastest supercomputer can't generate digits fast enough to get an infinite and random set within any finite period of time. An infinite and random set cannot and does nowhere exist except theoretically, so there is no actual basis for any precise evaluation of an observable subset (a specific value can never be apprehended until finito) , but we were talking about having to roll dice to generate the digits.
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