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Wait, you lost me in the first part. For simplicity sake, let’s have two sets of numbers. Set A has the numbers 4, 5, and 6, a total of 3 whole numbers. Set B has the numbers 1 and 2, a total of 2 whole numbers. The number 4 from set A can be divided by 2, giving us the unique number 2 from set B. Set A and set B still have different amounts of numbers in them.
My husband is also chiming in, to simplify my original statement. Set C is [0, 1], an infinite range. Set D includes both [0, 1] and the number 2. Subtract set C from set D, you are left with just the number 2. Therefore, the number of elements in set D is exactly one larger than set C, even with both sets being infinite.
Now we're speaking the same language, I'll try to reformulate what I was saying.
Let's say you have the set [0,2], and I have the set [0, 1]. To check which is bigger, we play a "game" where you pick a number from your set, and I respond with a number from mine. Whoever runs out first has the smaller set. What I do, is that every time you say a number, I just divide it by two, and respond with what I get. That way, I can find a number in [0, 1] for every number in [0, 2], so [0, 1] can't be smaller. If we flip the situation, you can take whatever number I say, and multiply it by two to get a number in your set, so [0, 2] can't be smaller. Since none is smaller, they must be the exact same size.
Now I'm on thin ice, but I would love to know if there's an error in the following argument: We play the same "game", but now you have the set [0, 1] + {2}. For every number you say, I can still divide it by two to get a number in my set, so my set still isn't smaller. For every number I say you can:
That way, you can get every number in your set from a number in mine, and opposite, so the set [0, 1] + {2} is the same size as the set [0, 1]. In other words, an uncountable infinity + 1 is the same size as it was before (might have something to do with the uncountable part).
I believe what we have done is create a bijection, that is: find a way to map every unique number in one set to a unique number in the other.
Okay, heads up that my husband and I are both sick right now and have a bit of brain fog, and he’s WFH while I have the day off so he can’t spare as much time to this. We see the logic in your argument and agree with your math. I’m trying to link this all back to the multiverse discussion so I can hopefully wrap my head around it.
Expanding on the idea that many universes were created in the Big Bang, I will pose a lot of questions that I don’t have answers to and will wrap up with a summary of possibilities.
Would the Big Bang create a finite or infinite number of universes? For there to be infinite universes, there would have had to be an infinite amount of mass and energy packed into a singular point before the Big Bang. Intuitively, and from my measly B.S. level of chemistry and physics classes, that feels wrong—but intuition, especially when it comes to infinities, is not worth much.
If there are an infinite number of universes, is this a countable or uncountable infinity (basically ℵ0 or ℵ1, I think)? Do we consider the number of all possible outcomes to be a countable or uncountable infinity?
Uncountable infinities are definitely larger than countable infinities. But are there different sizes of uncountable infinities? Your comment leads me to believe no, because we have no way of assigning a size to an uncountable number, but reading this article leads me to believe that there might be cardinalities beyond ℵ1. Your statements seem to agree with Woodin (and I think most of the math world at this point), while my idea of different sizes of infinities matches with Asperó and Schindler. If the top math minds of the world are this torn on the potential existence of different sizes of uncountable infinities, I can’t expect myself to understand it haha.
Summary of ideas:
My gut says that if we do somehow have multiverses then it must be a finite amount, and the possible number of outcomes is infinite (can’t decide if countable or uncountable)—therefore there can’t be a universe for every possible variation.
For there to be infinite universes that represent every possible permutation of events, I think we would be assuming that these are uncountable infinities, and that there is only one size of uncountable infinities (basically ℵ1 being the highest cardinality, I think).
If we say there are an infinite number of universes and an infinite number of possible outcomes, BUT there does not exist a universe that represents every possible outcome, this would rely 1 of 2 possibilities:
3a) the number of universes is a countable infinite while the number of possible outcomes is an uncountable infinite, or;
3b) that both the number of universes and the number of possible outcomes are both uncountable infinities, that the mathematical theory presented in the article above of different sizes of uncountable infinities (ℵ2 and beyond) is accurate, and therefore that the infinite number of possible outcomes is greater than the infinite number of universes.
I’ve tried writing out my thoughts several times and I keep erasing them, can’t keep track of how convoluted this is. I think I finally got it down though. Please tell me this isn’t complete nonsense lol, I need a nap
Haha, wow! Thanks for a really well thought out reply :) I think you nailed down where we were talking past each other, and I had no idea that the math world was divided on the sizes of uncountable infinities. Like you, I'm going to say that if the mathematicians are divided, I'm probably just going to accept that.
As for the "number of universes": I agree on the possible ways we could have multiple universes, without having one for every possibility. But I want to spin a bit back to what we mean by "multiple universes". I like the idea that if we assume that the universe is infinite, but we know that our observable universe is finite, that implies (without the assumption that "multiple universes" were created in the Big Bang) that there can be en infinite number of "observable universes" that fit within our infinite universe, that are simply moving so fast away from each other that they are completely separated (space between them is expanding faster that the speed of light).
That, in a way, leads back to one of your (our) questions: Does an infinite universe contain a countable or uncountable number of finite, observable, universes? Intuitively I would think the answer is "uncountable", just like there is an uncountable number of finite, non-overlapping intervals on the real numbers (I think?). That leads us back to your (our) other question/condition: Can uncountable infinities have different sizes? And like you said: If the mathematicians are divided on that, I'm not even going to try to answer.
So I don't think we'll get much further until the mathematicians conclude, but it's fun thinking about the possibilities :)
I also like the idea that these “other universes” might be within the realm of our infinite universe, beyond the reach of our finite observable universe. And I agree that we’ve probably reached as far as our logic can take us :) thanks for taking the time and effort to think this through with me, it was very fun!