How to pronounce “countably” in English

equinumerous with a set of natural numbers is just the same thing as to fit into Hilbert's Hotel. And so what we've shown is thatif you have two countably infinite sets, then their union is also countably infinite. If you put them together and form a new setwith all of the elements of either of them, then that union set is still only countably infinite. It didn't get bigger.

- Have you been able to sort of internalize a good intuition about countable infinity? Because that is a pretty weird thing. You can have a countablyinfinite set of countably infinite sets, and you can shove it all in and it still is a countable infinite set.- Yeah, that's exactly right. I mean, I guess, of course, when you work with these notions, the argument of

"That shining, everything else shines.It's not a countably different entity apart from consciousness, just as all the things that we see in the dream are not countably different,

if you have two countably infinite sets, then their union is also countably infinite. If you put them together and form a new setwith all of the elements of either of them, then that union set is still only countably infinite. It didn't get bigger.And that's a remarkable property for a notion of infinity to have, I suppose. But if you thought that there was only

train, infinite number of cars, with each car having infinite number of seats.- Exactly right. We've proved that if you have countably many countable sets, then the union of those sets, putting all those sets together into one giant set,is still countable. You know, because the train cars are each countable, plus the current hotel. It's sort of like another train car, if you want to think about it that way. The current

There's a clear starting point, zero, and all their subsets also have a definitive starting point.Cantor had successfully well ordered a set that was infinite in both directions, but it was only countably infinite.In his next book, he published his well ordering theorem.

is still countable. You know, because the train cars are each countable, plus the current hotel. It's sort of like another train car, if you want to think about it that way. The currentoccupants of the hotel could, you know, have the same number as any of the train cars. So putting countably many countable sets together to make one big union set isstill countable. It's quite remarkable, I think. I mean when I first learned this many, many years ago, I was completely shocked by it and

more elements in terms of its size because it's still just a countable infinity and it fits into Hilbert's Hotel.- Have you been able to sort of internalize a good intuition about countable infinity? Because that is a pretty weird thing. You can have a countablyinfinite set of countably infinite sets, and you can shove it all in and it still is a countable infinite set.

themselves. So it's a kind of different picture. Before, we used this 3 to the C, 5 times 5 to the S, which is a kind of, youknow, overly arithmetic way to think about it. But there's a kind of direct way to understand that it's still a countable infinity when you have countably many countable sets, because you can juststart putting them on this list. And as long as you give each of the infinite collections a chance to add one more person to the list, then

occupants of the hotel could, you know, have the same number as any of the train cars. So putting countably many countable sets together to make one big union set isstill countable. It's quite remarkable, I think. I mean when I first learned this many, many years ago, I was completely shocked by it andtransfixed by it. It was quite amazing to me that this notion of countable infinity could be closed under this process of

- Exactly right. We've proved that if you have countably many countable sets, then the union of those sets, putting all those sets together into one giant set,is still countable. You know, because the train cars are each countable, plus the current hotel. It's sort of like another train car, if you want to think about it that way. The currentoccupants of the hotel could, you know, have the same number as any of the train cars. So putting countably many countable sets together to make one big union set is

set that we built is... has many more elements than the old set in the sense that there are additional elements, but it doesn't have manymore elements in terms of its size because it's still just a countable infinity and it fits into Hilbert's Hotel.- Have you been able to sort of internalize a good intuition about countable infinity? Because that is a pretty weird thing. You can have a countably