Thread: Infinite Supersets

Something has been nagging me about the Universe. Uh-oh!

If a true random number generator could produce random numbers for an infinite amount of time, isn't it true that this RNG could produce an infinite list of numbers of the same value?

Or to put it in more common terms; if on an infinite universe there must be an earth where a chimpanzee wrote Macbeth in flawless Esperanto by typing randomly on a keyboard, isn't it true that there will be an infinite number of earths where this happened?

Originally Posted by Yarin

"Probably" (read: Yes)

Nice what you did there, hehe. You should have left out the Yes.

Originally Posted by Yarin

Yes, but only if you assume your hypothetical Multiverse allows it. It could just as well ignore repeats, creating only one universe for each unique number produced by the RNG.

Seems a little too arbitrary of a choice to me. But in any case, I'm not considering a multiverse. Just a single verse entity. So in our universe, if it was infinite, there would be an infinite number of earths that would have lucky chimps.

But I want to go back to the RNG because it think it is easier to conceptualize and finally present the kicker that is what has been bothering me about set theory and that I cannot seem to find an explanation in text books...

So, assuming an infinite RNG will eventually generate an infinite succession of the number 49, that RNG will not generate any other number from then on. However, that same RNG will also eventually generate an infinite succession of the number 2, and of the number 17, and... and infinite number of infinite successions.

So how can the RNG be described in terms of a superset of all those infinities, if the first infinity will stop all others from ever happening? Or is it instead the case that infinite supersets can only be described in terms of the mathematical formal systems and cannot describe any physical system, like an RNG? And the kicker is, if the last question is positive, doesn't that mean that infinity is purely a mathematical concept and that it doesn't exist in our physical universe?

I read your first post again and realized your RNG sequentially produces scalars. For some reason I assumed it could produce nested sequences. I blame George Cantor.

Does your RNG produce rationals, reals, or a closed range of integers?

Are we allowed to describe not just planets, but other RNGs with these numbers?

So how can the RNG be described in terms of a superset of all those infinities, if the first infinity will stop all others from ever happening?

It's a paradox. If there can only be one RNG generating the same number over and over infinitely, then you can logically believe (instead of a stronger word like prove) that we know the RNG produces other sets part of the super set. I'm not really trained in maths, but this seems similar to the supertask: if you wanted to perform an infinite number of tasks in 10 minutes, spend the first 5 doing the first thing, then 2 1/2 minutes doing the next, and so on. It seems like the number of tasks, or the number of sets in a superset, is a countable infinity.

Anyway, that's me babbling. I almost deleted it, but it could be important to keep in case I made a mistake somewhere.

Or is it instead the case that infinite supersets can only be described in terms of the mathematical formal systems and cannot describe any physical system, like an RNG? And the kicker is, if the last question is positive, doesn't that mean that infinity is purely a mathematical concept and that it doesn't exist in our physical universe?

It's not the first time math got something useful out of a paradox. To prove that there is an infinity in the universe, you would have to show me something, an ideal curve or ideal circle, for instance. I don't really count the universe as infinite. The observable universe is shrinking all the time. From a pragmatic point of view, the way we handle infinity in terms of limits, as well as being able to understand different kinds of infinity makes it true as long as we can use it, and I would extend this belief to infinite supersets.

To put it another way - why does everything we know have to be real? I guess my answer is "no," because even thinking about it now, it still doesn't bother me. It is philosophically interesting, though.

Originally Posted by Mario F.

Or to put it in more common terms; if on an infinite universe there must be an earth where a chimpanzee wrote Macbeth in flawless Esperanto by typing randomly on a keyboard, isn't it true that there will be an infinite number of earths where this happened?

Theoretically yes, but thanks to universe's entropy and drake's equation, we are talking odds of finding a needle in all the known galaxies in the observable universe.

Originally Posted by megafiddle

Perhaps the actual probability of an infinite sequence of definite type is the product of the infinitely small and the infinitely large.

You got it! And I can't believe I didn't think of the obvious. Was so enthralled in trying to solve the paradox that I didn't question the mathematical mistake of my assumptions. Classical newb mistake.

That's just it. In laymen terms there is simply an infinitely small probability of generating an infinite large sequence of the same numbers. And this itself an infinite superset. There is a likewise infinitely small probability of generating an infinite sequence of another number, or any number.

My problem with the whole thing is that I could see the exception, but not the bug. I was reaching a conclusion that seemed to negate the possibility of an infinity in the physical world (assuming an infinite RNG is a possible construct). But this was going against some concepts of infinity in the universe that have already been described, such as Singularities, certain measurements in fluid mechanics, geometric points in a plane, etc. It was bothering me because I was moving towards Aristotles' negation of actual infinities and the constructivist mathematicians that couldn't understand Cantor. I'd hate to be associated with either.

Whiteflags and stevesmithx posts were already hinting in the right direction. But was so embroidered in the concepts that I failed to realize I was doing a basic mistake. Thanks for that.