“Cimön is infinitely far away. For the dreamer this poses no problem, but for the fully discorporate…”
“Cimön is alef-null miles from Earth. Alef-null is the first infinite number. It’s like One, Two, Three,…Alef-Null. The three dots stand for forever.”
- - -
克蒙（Cimön）距地球无穷远
On December 13, 1873, the 28-year-old Georg Cantor brought the Continuum Problem to light by proving that there are more points in space than there are natural numbers. The problem is how much more?
Any continuous piece of space is called a continuum. A line segment, the surface of a balloon, the space inside your head, the endless universe…all these are continua. Cantor discovered that viewed as sets of points all continua have the same degree of infinity, which he called c. The degree of infinity of the set of all natural numbers is called alef-null, and the next larger degree of infinity is called alef-one. In 1873, Cantor gave the first proof that c is greater than alef-null. Even if you lived forever and a day, you would not be able to assign a natural number to each and every point in space. The Continuum Problem is to decide how muchgreater c is than alef-null. Cantor thought c should be alef-one, the next infinity. But no one knows if he was right.
- - -
连续统假设
In a way I had dreamed my way past alef-null. I wondered what it would take to reach alef-one, to go on and on through all the levels of infinity, out towards the unattainable Absolute Infinite…
- - -
存在不同层级的无限
“Your body is with…friends. Mount On is on Cimön. It’s infinite, Absolutely Infinite, but you’ll find a way to the end.”
My thoughts turned to the Absolute Infinity. That was bigger than alef-null, bigger than alef-one…bigger than any conceivable level. I was supposed to go to Cimön and climb a mountain Absolutely Infinite in height.
“We’re going to climb a mountain higher than all the infinite numbers. I’ve already figured out how to get started.”
- - -
克蒙上的On山绝对无限高
Kathy was doubtful. “In four hours we’ll be infinitely far away from Earth? Milestone alef-null?”
I nodded.
- - -
主角团四小时跨越无穷距离
With a final effort we turned the universe into a single blinding point of light. I stopped pushing and the point unfurled into a flat vertical landscape. An infinite half-plane. The lower edge was sea and the upper half was an endless mountain.
People go white and disappear, but maybe they just rematerialize somewhere else in Cimön. After all, it’s infinite…
- - -
克蒙无限大
Kathy and I had flown past alef-null. Out here infinity was as real as a pie in the face.
- - -
凯西和雷曼短时间跨越无穷距离
“La,” I said, “La, La, La,…” I did that thing with my mind and let my voice speed up into a high-pitched gabble. A few seconds later I had finished saying alef-null “La"s
Next I tried to count through all the natural numbers, but I got hung up trying to pronounce 217,876,234,110,899,720,123,650,123,124,687,857. I decided to use a simpler system and started over. “One. One plus one. One plus one plus one….” In a minute I was done. I had counted up to alef-null.
I looked critically at the distance between me and the hotel. I sharpened my vision and began counting the boulders dotting the meadow. Sure enough there were alef-null of them to pass. No wonder I hadn’t felt like I was getting any closer. If I walked past ten more or a thousand more boulders there’d still be alef-null of them left.
But my tongue had been able to do alef-null things when I’d counted out loud. ......The endless energy I needed to keep my body moving faster and faster flowed into me from the landscape around me. ......Two minutes later I was standing in the grounds of the hotel, more than a little out of breath
- - -
雷曼一分钟数到ℵ0，两分钟移动无穷远
Although the hotel was only two hundred feet high, it had infinitely many floors.
“You have infinitely many rooms.”
“Yes,” the clerk said, his teeth flashing deep in his beard. “But we have infinitely many guests as well. One in each room. How could we fit you in?”......
“Make the person in Room 1 move into Room 2. Make the guy in Room 2 move into Room 3. And so on. Each guest moves out of his room and into the room with the next higher number. Room 1 is left empty. You can put me there.”
There was a huge lurch, the lights came back on and we were zipping back down past the alef-null floors of the Hilbert Hotel.
But now that I was on it I could see that everything shrank as it approached the middle…so that there were actually alef-null rings of tables around the terrace’s center.
- - -
希尔伯特旅馆
He handed me a slim leather-bound volume and turned to speak into a microphone.
I riffled through the register, noticing a famous name here and there among the alien scrawls. I found a blank page and signed my name. Wondering how many pages were left, I began trying to flip through to the end of the book.
It soon became clear that there were infinitely many pages. I went into a speed-up and flipped past alef-null of them. There were still more. I peeled off alef-null more, and alef-null more again. There were still plenty of pages left. ......
“You’ll never reach the end at that rate. There’s alef-one pages.”
In White Light, I describe a book with as many pages as there are real numbers.
——《Infinity and the Mind》
- - -
雷曼拿起ℵ1页的书，且一次可翻过ℵ0页
As far as I could see the steep meadows stretched up, interrupted regularly by bands of rock. Counting the stripes of rock I could make out many infinite stretches. Infinitely many infinite stretches, and infinitely many infinite stretches of infinite stretches.
- - -
On山的岩石有无穷多的无穷延伸，无穷地无穷延伸的无穷延伸
“This is only the beginning of the second number class. Beyond lie all the alefs. And beyond that is the Absolute, the Absolute Infinite where,where…”
- - -
绝对无穷超越所有阿列夫数
“That’s because we come from a universe with infinitely many inhabited star systems,”
- - -
宇宙有无穷多个恒星系
Boundless energy flowed through us out of Mount On, and we zapped past our first alef-null cliffs. ......We stopped to look back after those first alef-null cliffs.
We kept at it for a long time, folding level after level of speed-ups into each other, passing infinity within infinity of cliffs.
- - -
On山上无穷又无穷的悬崖
“I think it’s what you’d get by raising alef-null to the alef-null power alef-null times in a row.” “You mean epsilon-zero?”
Franx gave an affirmative leg-twitch. “That’s what they called it.”
- - -
ε0
And even if we fold speed up after speed-up together, we’re still just going to get some limit o countably many stages. Alef-one can’t be reached by any countable process. We’re never going to get out of the second number class.
- - -
ℵ1
The Guide’s first energy blast had set a part of the meadow a fire, and the infinite-leaved little plants were giving off a dense, almost liquid smoke.
- - -
向导点燃草地上有着无穷叶子的植物
Ordinarily a writer has to leave things out. If he mentions his pen he doesn’t tell you who sold it to him, what the salesgirl ate for lunch, where her tuna came from, how the ocean was formed.
To include every detail, every associated fact, leads to including the whole universe. It all sticks together like an old dish of hard candy. And to describe the whole universe, an infinity of words are needed. But alef-null was no longer a barrier for me. I could make Proust’s dream come true.
I slid into a speed-up. The thoughts flowed through my fingers and onto the page. Every detail was there, every fleeting association was explained, and the whole infinity that was my life so far was there on the page.
It shot up out of the typewriter. I plucked it out of the air and scanned it with satisfaction. I had it all down in alef-null lines. There was a shrinking field in the typing ball, so that each line was 49/50 as high as the one above it. There was always room for fifty more lines.
- - -
雷曼短时间打出ℵ0行字，以描绘其生活的每一个细节，每一个相关的事件，乃至整个宇宙
Whenever I let the book fall open I would find a single page in the middle, isolated between two topless heaps of pages. The pages were packed in just like points on a line segment. There were c of them.
I did not see how I could have written it all, but each page I looked at seemed familiar. The visions after I’d inhaled that smoke were all here. I had seen every possible variation of my life, and I had proceeded to describe each one of them in endless detail. I had described a whole continuum of parallel worlds…somehow I had pulled the Many into One.
Occasionally I found two pages that differed from one another only in a single name, but usually the differences were much greater. In some lives the narrator could fly, in some he was paralyzed; in some he was a genius, in others he was insane. Somehow they were all me. ......
Carrying the book in my hand I went over and tried the door to my cubicle again.
And my sidekick here has a book with c pages right in his parka.
- - -
雷曼写下、拿起页数相当于连续基数的书，书中描述了他生活的每一种可能的变化，以及平行世界的连续统一体
On impulse I squeezed my book into the shelf in front of me and looked at some of the other volumes. One called Dogscaught my eye, and I took it down. On each page there was a story about a dog. They were all alef-null words long, and sometimes made cumbersome reading. One of them really got to me though. It was sort of like Call of the Wild, and it was all I could do to keep from howling when I finished.
After all the book had c pages, and c is strictly greater than alef-null. Which meant that I could never look at every page unless I could somehow regain that white-light ability to handle the uncountable infinities.
- - -
雷曼拿起一本页数相当于连续基数的故事书，每一页上都有ℵ0个词;白光能力者能处理不可数无穷规模的事情
It really would be interesting to compare your c pages to those alef-one cliffs.
The alef-one cliffs and the c pages were two different uncountable collections,each with its own natural ordering…and there was no obvious way to compare them. ......To work through all alef-one cliffs I’d have to go white again.
- - -
On山上ℵ1的悬崖
Meanwhile Franx kept flying up Mount On. He saw the One, merged in, and then used the Reflection Principle toget to alef-one.
- - -
弗兰克斯使用反射定理达到ℵ1
There’s lots of tunnels to alef-one, and if you’re willing to chance going into the Desert you’ll find tunnels to alef-two, to alef-alef-null, to the inaccessible cardinals… I’ve seen them,
- - -
ℵ_ℵ0、不可达基数
But wouldn’t it be possible to finally reach perfect enlightenment, total union with the Absolute, the One, the ground of all being, God Himself?
- - -
上帝是万物之基
The Absolute is Everything, but it’s Nothing, too.
- - -
绝对者（上帝）既是一切也是虚无
“That’s called the Reflection Principle.”
“Please elaborate. This is new to me.”“
The Reflection Principle is an old theological notion which we use in Set Theory. Any specific description of the full universe of Set Theory also applies to some little set inside the universe. Any description of the Absolute also applies to some limited, relative thing.”
- - -
反射定理
Franx and Ellie finished the duet with a quick arpeggio of alef-null notes, and I clapped lightly.
- - -
弗兰克斯和艾莉完成ℵ0音符长度的二重奏
I cut the deck and set the two halves on the table. “Now Franx, we’ll go into a speed-up. You’ll pick alef-null cards in a row, and when you’re done I’ll have singled out a card which you didn’t pick.”......
After alef-null steps, Franx had picked what looked like every blonde in the deck, and the untouched packet still held a card. A cheeky girl lying demurely on a white bed. For a second I thought it was April.
- - -
弗兰克斯短时间做出ℵ0次抽卡动作
“It’s not a blur,” I protested. “There’s alef-null lines on each page.”They looked blankly at me. “There’s infinitely many lines on each page, and there’s a larger infinity of pages.”
- - -
雷曼所写的书，每页有ℵ0行，页数是更大的无限
The pattern came to me in a right-brain flash. “We’re going towards Zero, Kathy. Nothing. On the other side of Cimön this direction leads towards the Absolute Infinite. Zero and Infinity. They’re the same at the Absolute…”
I still believe the basic premises of White Light: that God is a blinding white light that is possible for a human to directly perceive, and that this cosmic One is located at a nexus where Zero and Infinity are the same.
——《Afterword》
- - -
对于绝对者（上帝）而言，零与无限是同一的
It went something like this:The three-dimensional space of our universe consists of a continuum of c idealized mathematical points. There are two types of substance moving about in this space: mass and aether.
We all have a pretty good idea of what mass is, but aether?
Aether is a very tenuous sort of substance associated with the transfer of energy. We do not necessarily assume that aether fills up all of the space between the bits of mass. All we know is that some regions of space contain mass, some contain aether, and some are empty.
Now any massive object…such as a rock…can be endlessly cut into smaller and smaller pieces. In the limit one ends up with alef-nullinfinitely small bits of mass. These indivisible bits are called mass-monads. In general, then, any massive object is an arrangement of alef-null point-sized mass-monads.
Aether is also infinitely divisible…but even more so! Any aethereal object is to be thought of as an arrangement of alef-one point-sized aether-monads. Since we have c points in space, and since c is at least as great as alef-one, there is certainly room for all these monads.
At any instant, then, the state of affairs in our universe can be specified by stating which of the c possible locations in our space is occupied by a mass-monad, and which locations are occupied by aether-monads. To put it another way, space contains a set M of alef-null points occupied by mass, and a set A of alef-one points occupied by aether. The state of the universe at any instant depends only on the properties of the two sets of points M and A.
Cantor spends most of the 1885 paper describing a special way of splitting M and A into five significant subsets. He closes with these words: “The next step will be to see if the relations between these distinct sets can account for the various modes of existenceand actionexhibited by matter – such asphysical state, chemical differences, light and heat, electricityand magnetism. I prefer not to explicitly state my further speculations along these lines until I have subjected them to a more careful consideration.” Cantor loved italics.
- - -
宇宙的三维部分包含了连续基数（>=ℵ1）个点/单子
I slipped out of my physical body and began running around and around the room. I did alef-null laps, took Kathy’s hand and did alef-null more. ......All the while our physical bodies sat slack-jawed in their chairs. We whipped through alef-null more laps and fell into a laughing heap in a corner of the classroom, too excited to talk.
- - -
脱离肉体后的雷曼等人在房间里跑了ℵ0圈
It seemed reasonable to assume that most of the things I’d seen in Cimön were made of aether. Alef-one aether-monads each.
- - -
克蒙上的大多数事物都由ℵ1个以太单子组成
“I’d like to do some more lab-work. There should be a third level of substance, too. Endlessly many. The number of points in space is Absolutely Infinite. It’s just a matter of…”
- - -
空间中绝对无穷的点
Nick and April frowned at him, but I began talking. “I was in this big factory, with all kinds of weird machines humming along. They weren’t really machines. I mean some were just electronic patterns. But they were all lined up along the walls of this enormous room. There was a real big white-haired guy…”
“God?” Stuart said, smiling.
“Of course. Not the Godhead, just the Father. He was showing me the machines. Some were ideas…like one was Zeno’s Paradox and one was the Continuum Problem. Others were places…there was our Universe and there was Cimön. And there were little machines, too, that were just a person or an atom. There was one of everything.” This was the first time I’d told anyone about what I’d seen during the coma, and they were keeping still to hear my whisper.
“I noticed that each machine had a wire coming out of it. Like an electric cord. So I asked God what they ran on. He says, 'Do you want to see?’
“All the wires seemed to lead into a manhole in the middle of the floor, and we walked over there together. While we were walking, I noticed that God and I each had wires running out of us and down into the manhole too.”
- - -
圣父所在的空间包含宇宙、克蒙、芝诺悖论和连续统假设
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