Yes, you can add new entries to an infinite set, even an infinity of them. However, you have not made a greater infinity from doing this, so saying that we’ve made the infinity “larger” by doing this would be misleading.
After all, imagine that we do the opposite of the Hilbert hotel: we ask the guest in room 1 to leave, and everyone else to move down one room. Have we made the hotel smaller? But it’s still a fully occupied, infinite hotel…
To make an infinity larger you’d have to move from one “aleph” to another, e.g. from the set of integers to the set of reals.
Problem I see with the Hilbert Hotel is that it isn’t physical. When we ask a guest to move, there is an “and then” step, and there is either time or space (or both) between an infinitude of steps. Whilst it is a neat logical construct, it isn’t helpful discussing physical realities. At least not ones with local action.
This underlies why infinity is not easy to grasp in these discussions. If we say the universe is infinite in size, the infinity we are using isn’t any of Cantor’s cardinals. We don’t say the universe is \aleph_0 in diameter. It makes no sense. Nor is it \aleph_1 or \beth_0 in size. Or even \aleph_\aleph
Pretty much. The rules for infinities are different. They are not normal numbers. Many operations we define on numbers just don’t apply.
If we are talking about a number, as in a number of things, Infinity isn’t a well defined number, but what is generally agreed on when you want to get a bit more formal is the idea of cardinality - the number of things in a set. So how may elements does the set of all integers have in it? That isn’t easy to to get a handle on - most people would say, well there are an infinite number of numbers, so the answer must be infinity. Mathematicians are more formal and call that number Aleph zero (or aleph null) aka \aleph_0 which is the first uncountable number. That is, it is essentially the smallest number that you cannot count up to.
This number has remarkable properties. Because it is the cardinality of the set of all integers, we can do really interesting things with it. Since we have all the natural numbers (aka counting numbers), we can use them to label other things. That is we can count them. Which allows us to prove that not only are there \aleph_0 integers, but there are also \aleph_0 even numbers, \aleph_0 odd numbers, and \aleph_0 rational numbers. Which sounds insane.
But consider, we can start counting the members of any of these sets of numbers. We can place each of the members in a one-to-one relationship with an integer. If we can do that, we know that there are the same number of whatever we are counting as there are integers, and thus there are \aleph_0 of them.
But if we want to talk infinity, and not Cantor’s cardinals, and his family of transfinite numbers, we have a different infinity. \infty This guy is an extrema. Not a count. You can usefully have -\infty There is no -\aleph_0
The other guy to watch out for is undefined. He gets confused with \infty but they are not the same thing. 1\over 0 is not \infty
How far away do parallel lines meet? Depending on your geometry that can have a lot of different answers. A point at infinity can be a perfectly reasonable thing to reason with. It is the same point as the point twice as far away.
Nitpick: I wouldn’t call it that, since a set with cardinality aleph-null is still a countable set according to the terminology used by many mathematicians (cite).
But, as you noted earlier, “infinity” is not just a very large number. If, at every nanosecond since the Big Bang, each existing universe had split into a million different universes, then the number of universes would be staggeringly large but not infinite.
What if at every nanosecond since the Big Bang each existing universe had split into an infinite number of universes? This would exceed infinity times infinity plus one which is the maximum number of 'are so’s and 'am not’s possible.
Every once in a while some Physicists who don’t understand a bit of Cantor’s work suggest this.
And they are flat our wrong, just on the Math alone. Throw in the Physics of the history of a 2nd Earth having to be identical as well, quantum events and all, and you’re talking about a literally laughable conjecture.
Whenever I see someone talking about “could there be an infinite number of …” the first question is what do you mean by “infinite” in terms of the Cantor hierarchy? There could be countably infinite universes, or a continuum equivalent number, or even higher.
Does this not depend on whether the universe (which in the context of the Many Worlds Interpretation is a very slippery concept) is finite or infinite? If there are a finite number of particles in the universe then yes there will be a very much larger and constantly increasing, but still finite, number of new universes created by the number of possible outcomes of the particles interactions. If there are an infinite number of particles, even if only a countable infinity, then there must also be a countable infinity of many world universes.
Serious answer: if by “an infinite number” you mean \aleph_0, then I think the number of existing universes would be \aleph_0 raised to some very large but finite power, which would still be \aleph_0.
Cantor’s work is irrelevant to this, because it’s all about finite numbers. The chances of another planet being indistinguishable from Earth are really, really small, but they’re not zero.
And the Many World framework just might require an infinite number of universes spawning from a single decision point. Is it possible for an event to have an irrational probability of occurrence?
I think that conceivable “thing” does not have a valid relationship with infinity. 1 is an infinitely small unity, but first you have to find 1 thing that is unitary, which is really difficult. There is evidence of point-like unitary objects, but since we cannot actually observe those things except by inference, we cannot confirm that they are truly unitary and not somehow composite.
In a sense, we might say that alternate universes do exist simply because we can imagine them. And the extent of the possibilities that we can imagine is vast, but it is not infinite. Even if you add in all the other possible critters that can imagine things in ways we cannot, the horizon of imaginable universes recedes, but it remains bounded by the physical limitations of thought.
Hence, even if we include non-real possible universes, the number remains countable, because infinity is not a number.
That is the definition of countable.
However \aleph_0 isn’t an integer. For it to itself be countable there must be a way of pairing it 1-1 with some integer. One that does not itself already get used counting the integers. This isn’t possible.
This is how \aleph_0 becomes the first of the uncountable numbers. One can then go on to show that 2^{\aleph_0} or \aleph_1 is the second uncountable number. And so on.
Of course the set \{\aleph_0\} has a cardinality of one.
\aleph_0 is a countable cardinality, because it is the cardinality of a countable set, and the set is countable because every element of the set can be put into a 1 to 1 correspondence with an integer. \aleph_0 itself does not correspond to an integer unless it is an element of a set which itself has cardinality \aleph_0 or less.
(and it could be an element of such a set: For instance, there is a set {\aleph_0, \aleph_1, \aleph_2, \aleph_3, …}. In that set, the obvious correspondence is to say that \aleph_0 corresponds to 0.)